Properties

Label 5070.2.b.s.1351.6
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.6
Root \(0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.s.1351.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +1.00000i q^{5} -1.00000i q^{6} +1.69202i q^{7} -1.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.55496i q^{11} +1.00000 q^{12} -1.69202 q^{14} -1.00000i q^{15} +1.00000 q^{16} -2.35690 q^{17} +1.00000i q^{18} -6.51573i q^{19} -1.00000i q^{20} -1.69202i q^{21} -4.55496 q^{22} -8.94869 q^{23} +1.00000i q^{24} -1.00000 q^{25} -1.00000 q^{27} -1.69202i q^{28} +9.07606 q^{29} +1.00000 q^{30} -10.6528i q^{31} +1.00000i q^{32} -4.55496i q^{33} -2.35690i q^{34} -1.69202 q^{35} -1.00000 q^{36} -6.18598i q^{37} +6.51573 q^{38} +1.00000 q^{40} +3.00969i q^{41} +1.69202 q^{42} +4.93900 q^{43} -4.55496i q^{44} +1.00000i q^{45} -8.94869i q^{46} +4.28621i q^{47} -1.00000 q^{48} +4.13706 q^{49} -1.00000i q^{50} +2.35690 q^{51} -3.40581 q^{53} -1.00000i q^{54} -4.55496 q^{55} +1.69202 q^{56} +6.51573i q^{57} +9.07606i q^{58} -4.32304i q^{59} +1.00000i q^{60} -14.3666 q^{61} +10.6528 q^{62} +1.69202i q^{63} -1.00000 q^{64} +4.55496 q^{66} -3.24698i q^{67} +2.35690 q^{68} +8.94869 q^{69} -1.69202i q^{70} -14.9487i q^{71} -1.00000i q^{72} +6.72886i q^{73} +6.18598 q^{74} +1.00000 q^{75} +6.51573i q^{76} -7.70709 q^{77} +5.67994 q^{79} +1.00000i q^{80} +1.00000 q^{81} -3.00969 q^{82} -7.71917i q^{83} +1.69202i q^{84} -2.35690i q^{85} +4.93900i q^{86} -9.07606 q^{87} +4.55496 q^{88} +9.12498i q^{89} -1.00000 q^{90} +8.94869 q^{92} +10.6528i q^{93} -4.28621 q^{94} +6.51573 q^{95} -1.00000i q^{96} -4.40581i q^{97} +4.13706i q^{98} +4.55496i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} + O(q^{10}) \) \( 6q - 6q^{3} - 6q^{4} + 6q^{9} - 6q^{10} + 6q^{12} + 6q^{16} - 6q^{17} - 28q^{22} + 10q^{23} - 6q^{25} - 6q^{27} + 24q^{29} + 6q^{30} - 6q^{36} + 14q^{38} + 6q^{40} + 10q^{43} - 6q^{48} + 14q^{49} + 6q^{51} + 6q^{53} - 28q^{55} - 34q^{61} + 28q^{62} - 6q^{64} + 28q^{66} + 6q^{68} - 10q^{69} + 8q^{74} + 6q^{75} + 14q^{77} - 14q^{79} + 6q^{81} + 26q^{82} - 24q^{87} + 28q^{88} - 6q^{90} - 10q^{92} - 42q^{94} + 14q^{95} + O(q^{100}) \)

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
\(7\) 1.69202i 0.639524i 0.947498 + 0.319762i \(0.103603\pi\)
−0.947498 + 0.319762i \(0.896397\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 4.55496i 1.37337i 0.726954 + 0.686686i \(0.240935\pi\)
−0.726954 + 0.686686i \(0.759065\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.69202 −0.452212
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) −2.35690 −0.571631 −0.285816 0.958285i \(-0.592264\pi\)
−0.285816 + 0.958285i \(0.592264\pi\)
\(18\) 1.00000i 0.235702i
\(19\) − 6.51573i − 1.49481i −0.664368 0.747405i \(-0.731299\pi\)
0.664368 0.747405i \(-0.268701\pi\)
\(20\) − 1.00000i − 0.223607i
\(21\) − 1.69202i − 0.369229i
\(22\) −4.55496 −0.971120
\(23\) −8.94869 −1.86593 −0.932965 0.359966i \(-0.882788\pi\)
−0.932965 + 0.359966i \(0.882788\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) − 1.69202i − 0.319762i
\(29\) 9.07606 1.68538 0.842691 0.538397i \(-0.180970\pi\)
0.842691 + 0.538397i \(0.180970\pi\)
\(30\) 1.00000 0.182574
\(31\) − 10.6528i − 1.91330i −0.291243 0.956649i \(-0.594069\pi\)
0.291243 0.956649i \(-0.405931\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 4.55496i − 0.792916i
\(34\) − 2.35690i − 0.404204i
\(35\) −1.69202 −0.286004
\(36\) −1.00000 −0.166667
\(37\) − 6.18598i − 1.01697i −0.861071 0.508484i \(-0.830206\pi\)
0.861071 0.508484i \(-0.169794\pi\)
\(38\) 6.51573 1.05699
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.00969i 0.470034i 0.971991 + 0.235017i \(0.0755146\pi\)
−0.971991 + 0.235017i \(0.924485\pi\)
\(42\) 1.69202 0.261085
\(43\) 4.93900 0.753191 0.376595 0.926378i \(-0.377095\pi\)
0.376595 + 0.926378i \(0.377095\pi\)
\(44\) − 4.55496i − 0.686686i
\(45\) 1.00000i 0.149071i
\(46\) − 8.94869i − 1.31941i
\(47\) 4.28621i 0.625208i 0.949884 + 0.312604i \(0.101201\pi\)
−0.949884 + 0.312604i \(0.898799\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.13706 0.591009
\(50\) − 1.00000i − 0.141421i
\(51\) 2.35690 0.330031
\(52\) 0 0
\(53\) −3.40581 −0.467824 −0.233912 0.972258i \(-0.575153\pi\)
−0.233912 + 0.972258i \(0.575153\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −4.55496 −0.614190
\(56\) 1.69202 0.226106
\(57\) 6.51573i 0.863029i
\(58\) 9.07606i 1.19175i
\(59\) − 4.32304i − 0.562812i −0.959589 0.281406i \(-0.909199\pi\)
0.959589 0.281406i \(-0.0908008\pi\)
\(60\) 1.00000i 0.129099i
\(61\) −14.3666 −1.83945 −0.919726 0.392560i \(-0.871589\pi\)
−0.919726 + 0.392560i \(0.871589\pi\)
\(62\) 10.6528 1.35291
\(63\) 1.69202i 0.213175i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 4.55496 0.560677
\(67\) − 3.24698i − 0.396682i −0.980133 0.198341i \(-0.936445\pi\)
0.980133 0.198341i \(-0.0635553\pi\)
\(68\) 2.35690 0.285816
\(69\) 8.94869 1.07730
\(70\) − 1.69202i − 0.202235i
\(71\) − 14.9487i − 1.77408i −0.461690 0.887042i \(-0.652757\pi\)
0.461690 0.887042i \(-0.347243\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 6.72886i 0.787553i 0.919206 + 0.393777i \(0.128832\pi\)
−0.919206 + 0.393777i \(0.871168\pi\)
\(74\) 6.18598 0.719106
\(75\) 1.00000 0.115470
\(76\) 6.51573i 0.747405i
\(77\) −7.70709 −0.878304
\(78\) 0 0
\(79\) 5.67994 0.639043 0.319522 0.947579i \(-0.396478\pi\)
0.319522 + 0.947579i \(0.396478\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) −3.00969 −0.332365
\(83\) − 7.71917i − 0.847289i −0.905829 0.423644i \(-0.860751\pi\)
0.905829 0.423644i \(-0.139249\pi\)
\(84\) 1.69202i 0.184615i
\(85\) − 2.35690i − 0.255641i
\(86\) 4.93900i 0.532586i
\(87\) −9.07606 −0.973056
\(88\) 4.55496 0.485560
\(89\) 9.12498i 0.967246i 0.875276 + 0.483623i \(0.160679\pi\)
−0.875276 + 0.483623i \(0.839321\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 8.94869 0.932965
\(93\) 10.6528i 1.10464i
\(94\) −4.28621 −0.442089
\(95\) 6.51573 0.668500
\(96\) − 1.00000i − 0.102062i
\(97\) − 4.40581i − 0.447343i −0.974665 0.223671i \(-0.928196\pi\)
0.974665 0.223671i \(-0.0718042\pi\)
\(98\) 4.13706i 0.417907i
\(99\) 4.55496i 0.457791i
\(100\) 1.00000 0.100000
\(101\) −2.31767 −0.230617 −0.115308 0.993330i \(-0.536786\pi\)
−0.115308 + 0.993330i \(0.536786\pi\)
\(102\) 2.35690i 0.233367i
\(103\) 9.00000 0.886796 0.443398 0.896325i \(-0.353773\pi\)
0.443398 + 0.896325i \(0.353773\pi\)
\(104\) 0 0
\(105\) 1.69202 0.165124
\(106\) − 3.40581i − 0.330802i
\(107\) −3.06100 −0.295918 −0.147959 0.988994i \(-0.547270\pi\)
−0.147959 + 0.988994i \(0.547270\pi\)
\(108\) 1.00000 0.0962250
\(109\) 7.45473i 0.714034i 0.934098 + 0.357017i \(0.116206\pi\)
−0.934098 + 0.357017i \(0.883794\pi\)
\(110\) − 4.55496i − 0.434298i
\(111\) 6.18598i 0.587147i
\(112\) 1.69202i 0.159881i
\(113\) −19.9812 −1.87967 −0.939837 0.341623i \(-0.889024\pi\)
−0.939837 + 0.341623i \(0.889024\pi\)
\(114\) −6.51573 −0.610254
\(115\) − 8.94869i − 0.834470i
\(116\) −9.07606 −0.842691
\(117\) 0 0
\(118\) 4.32304 0.397968
\(119\) − 3.98792i − 0.365572i
\(120\) −1.00000 −0.0912871
\(121\) −9.74764 −0.886149
\(122\) − 14.3666i − 1.30069i
\(123\) − 3.00969i − 0.271374i
\(124\) 10.6528i 0.956649i
\(125\) − 1.00000i − 0.0894427i
\(126\) −1.69202 −0.150737
\(127\) 3.14244 0.278846 0.139423 0.990233i \(-0.455475\pi\)
0.139423 + 0.990233i \(0.455475\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −4.93900 −0.434855
\(130\) 0 0
\(131\) −3.77479 −0.329805 −0.164902 0.986310i \(-0.552731\pi\)
−0.164902 + 0.986310i \(0.552731\pi\)
\(132\) 4.55496i 0.396458i
\(133\) 11.0248 0.955967
\(134\) 3.24698 0.280496
\(135\) − 1.00000i − 0.0860663i
\(136\) 2.35690i 0.202102i
\(137\) − 14.5894i − 1.24646i −0.782040 0.623228i \(-0.785821\pi\)
0.782040 0.623228i \(-0.214179\pi\)
\(138\) 8.94869i 0.761763i
\(139\) −14.3599 −1.21799 −0.608995 0.793174i \(-0.708427\pi\)
−0.608995 + 0.793174i \(0.708427\pi\)
\(140\) 1.69202 0.143002
\(141\) − 4.28621i − 0.360964i
\(142\) 14.9487 1.25447
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 9.07606i 0.753726i
\(146\) −6.72886 −0.556884
\(147\) −4.13706 −0.341219
\(148\) 6.18598i 0.508484i
\(149\) 0.252356i 0.0206738i 0.999947 + 0.0103369i \(0.00329040\pi\)
−0.999947 + 0.0103369i \(0.996710\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 11.0804i − 0.901708i −0.892598 0.450854i \(-0.851119\pi\)
0.892598 0.450854i \(-0.148881\pi\)
\(152\) −6.51573 −0.528495
\(153\) −2.35690 −0.190544
\(154\) − 7.70709i − 0.621055i
\(155\) 10.6528 0.855653
\(156\) 0 0
\(157\) 5.03923 0.402174 0.201087 0.979573i \(-0.435553\pi\)
0.201087 + 0.979573i \(0.435553\pi\)
\(158\) 5.67994i 0.451872i
\(159\) 3.40581 0.270099
\(160\) −1.00000 −0.0790569
\(161\) − 15.1414i − 1.19331i
\(162\) 1.00000i 0.0785674i
\(163\) − 20.9825i − 1.64348i −0.569863 0.821740i \(-0.693004\pi\)
0.569863 0.821740i \(-0.306996\pi\)
\(164\) − 3.00969i − 0.235017i
\(165\) 4.55496 0.354603
\(166\) 7.71917 0.599124
\(167\) 22.2325i 1.72040i 0.509954 + 0.860201i \(0.329662\pi\)
−0.509954 + 0.860201i \(0.670338\pi\)
\(168\) −1.69202 −0.130542
\(169\) 0 0
\(170\) 2.35690 0.180766
\(171\) − 6.51573i − 0.498270i
\(172\) −4.93900 −0.376595
\(173\) 11.6286 0.884108 0.442054 0.896988i \(-0.354250\pi\)
0.442054 + 0.896988i \(0.354250\pi\)
\(174\) − 9.07606i − 0.688055i
\(175\) − 1.69202i − 0.127905i
\(176\) 4.55496i 0.343343i
\(177\) 4.32304i 0.324940i
\(178\) −9.12498 −0.683946
\(179\) 15.7875 1.18001 0.590005 0.807399i \(-0.299126\pi\)
0.590005 + 0.807399i \(0.299126\pi\)
\(180\) − 1.00000i − 0.0745356i
\(181\) 1.91185 0.142107 0.0710535 0.997473i \(-0.477364\pi\)
0.0710535 + 0.997473i \(0.477364\pi\)
\(182\) 0 0
\(183\) 14.3666 1.06201
\(184\) 8.94869i 0.659706i
\(185\) 6.18598 0.454802
\(186\) −10.6528 −0.781101
\(187\) − 10.7356i − 0.785062i
\(188\) − 4.28621i − 0.312604i
\(189\) − 1.69202i − 0.123076i
\(190\) 6.51573i 0.472701i
\(191\) 24.3260 1.76017 0.880085 0.474817i \(-0.157486\pi\)
0.880085 + 0.474817i \(0.157486\pi\)
\(192\) 1.00000 0.0721688
\(193\) 0.521106i 0.0375101i 0.999824 + 0.0187550i \(0.00597026\pi\)
−0.999824 + 0.0187550i \(0.994030\pi\)
\(194\) 4.40581 0.316319
\(195\) 0 0
\(196\) −4.13706 −0.295505
\(197\) − 4.87502i − 0.347331i −0.984805 0.173665i \(-0.944439\pi\)
0.984805 0.173665i \(-0.0555611\pi\)
\(198\) −4.55496 −0.323707
\(199\) 18.4983 1.31131 0.655654 0.755062i \(-0.272393\pi\)
0.655654 + 0.755062i \(0.272393\pi\)
\(200\) 1.00000i 0.0707107i
\(201\) 3.24698i 0.229024i
\(202\) − 2.31767i − 0.163070i
\(203\) 15.3569i 1.07784i
\(204\) −2.35690 −0.165016
\(205\) −3.00969 −0.210206
\(206\) 9.00000i 0.627060i
\(207\) −8.94869 −0.621977
\(208\) 0 0
\(209\) 29.6789 2.05293
\(210\) 1.69202i 0.116761i
\(211\) 18.2500 1.25638 0.628190 0.778060i \(-0.283796\pi\)
0.628190 + 0.778060i \(0.283796\pi\)
\(212\) 3.40581 0.233912
\(213\) 14.9487i 1.02427i
\(214\) − 3.06100i − 0.209246i
\(215\) 4.93900i 0.336837i
\(216\) 1.00000i 0.0680414i
\(217\) 18.0248 1.22360
\(218\) −7.45473 −0.504898
\(219\) − 6.72886i − 0.454694i
\(220\) 4.55496 0.307095
\(221\) 0 0
\(222\) −6.18598 −0.415176
\(223\) 8.54048i 0.571913i 0.958243 + 0.285957i \(0.0923113\pi\)
−0.958243 + 0.285957i \(0.907689\pi\)
\(224\) −1.69202 −0.113053
\(225\) −1.00000 −0.0666667
\(226\) − 19.9812i − 1.32913i
\(227\) − 9.61596i − 0.638233i −0.947715 0.319117i \(-0.896614\pi\)
0.947715 0.319117i \(-0.103386\pi\)
\(228\) − 6.51573i − 0.431515i
\(229\) 13.4058i 0.885881i 0.896551 + 0.442941i \(0.146065\pi\)
−0.896551 + 0.442941i \(0.853935\pi\)
\(230\) 8.94869 0.590059
\(231\) 7.70709 0.507089
\(232\) − 9.07606i − 0.595873i
\(233\) 0.153457 0.0100533 0.00502664 0.999987i \(-0.498400\pi\)
0.00502664 + 0.999987i \(0.498400\pi\)
\(234\) 0 0
\(235\) −4.28621 −0.279601
\(236\) 4.32304i 0.281406i
\(237\) −5.67994 −0.368952
\(238\) 3.98792 0.258498
\(239\) 14.5851i 0.943431i 0.881751 + 0.471715i \(0.156365\pi\)
−0.881751 + 0.471715i \(0.843635\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) − 3.40044i − 0.219041i −0.993985 0.109521i \(-0.965068\pi\)
0.993985 0.109521i \(-0.0349316\pi\)
\(242\) − 9.74764i − 0.626602i
\(243\) −1.00000 −0.0641500
\(244\) 14.3666 0.919726
\(245\) 4.13706i 0.264307i
\(246\) 3.00969 0.191891
\(247\) 0 0
\(248\) −10.6528 −0.676453
\(249\) 7.71917i 0.489182i
\(250\) 1.00000 0.0632456
\(251\) −1.80864 −0.114161 −0.0570803 0.998370i \(-0.518179\pi\)
−0.0570803 + 0.998370i \(0.518179\pi\)
\(252\) − 1.69202i − 0.106587i
\(253\) − 40.7609i − 2.56262i
\(254\) 3.14244i 0.197174i
\(255\) 2.35690i 0.147595i
\(256\) 1.00000 0.0625000
\(257\) 22.2664 1.38894 0.694469 0.719523i \(-0.255640\pi\)
0.694469 + 0.719523i \(0.255640\pi\)
\(258\) − 4.93900i − 0.307489i
\(259\) 10.4668 0.650376
\(260\) 0 0
\(261\) 9.07606 0.561794
\(262\) − 3.77479i − 0.233207i
\(263\) 8.08815 0.498736 0.249368 0.968409i \(-0.419777\pi\)
0.249368 + 0.968409i \(0.419777\pi\)
\(264\) −4.55496 −0.280338
\(265\) − 3.40581i − 0.209217i
\(266\) 11.0248i 0.675971i
\(267\) − 9.12498i − 0.558440i
\(268\) 3.24698i 0.198341i
\(269\) 16.6843 1.01726 0.508628 0.860986i \(-0.330153\pi\)
0.508628 + 0.860986i \(0.330153\pi\)
\(270\) 1.00000 0.0608581
\(271\) − 27.4077i − 1.66490i −0.554099 0.832451i \(-0.686937\pi\)
0.554099 0.832451i \(-0.313063\pi\)
\(272\) −2.35690 −0.142908
\(273\) 0 0
\(274\) 14.5894 0.881378
\(275\) − 4.55496i − 0.274674i
\(276\) −8.94869 −0.538648
\(277\) 16.0411 0.963819 0.481910 0.876221i \(-0.339943\pi\)
0.481910 + 0.876221i \(0.339943\pi\)
\(278\) − 14.3599i − 0.861248i
\(279\) − 10.6528i − 0.637766i
\(280\) 1.69202i 0.101118i
\(281\) 10.0935i 0.602129i 0.953604 + 0.301065i \(0.0973420\pi\)
−0.953604 + 0.301065i \(0.902658\pi\)
\(282\) 4.28621 0.255240
\(283\) −12.1021 −0.719398 −0.359699 0.933068i \(-0.617121\pi\)
−0.359699 + 0.933068i \(0.617121\pi\)
\(284\) 14.9487i 0.887042i
\(285\) −6.51573 −0.385959
\(286\) 0 0
\(287\) −5.09246 −0.300598
\(288\) 1.00000i 0.0589256i
\(289\) −11.4450 −0.673238
\(290\) −9.07606 −0.532965
\(291\) 4.40581i 0.258273i
\(292\) − 6.72886i − 0.393777i
\(293\) − 4.01075i − 0.234311i −0.993114 0.117155i \(-0.962622\pi\)
0.993114 0.117155i \(-0.0373775\pi\)
\(294\) − 4.13706i − 0.241278i
\(295\) 4.32304 0.251697
\(296\) −6.18598 −0.359553
\(297\) − 4.55496i − 0.264305i
\(298\) −0.252356 −0.0146186
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 8.35690i 0.481683i
\(302\) 11.0804 0.637604
\(303\) 2.31767 0.133147
\(304\) − 6.51573i − 0.373703i
\(305\) − 14.3666i − 0.822628i
\(306\) − 2.35690i − 0.134735i
\(307\) 15.2784i 0.871987i 0.899950 + 0.435993i \(0.143603\pi\)
−0.899950 + 0.435993i \(0.856397\pi\)
\(308\) 7.70709 0.439152
\(309\) −9.00000 −0.511992
\(310\) 10.6528i 0.605038i
\(311\) −25.4330 −1.44217 −0.721085 0.692846i \(-0.756356\pi\)
−0.721085 + 0.692846i \(0.756356\pi\)
\(312\) 0 0
\(313\) −19.1226 −1.08087 −0.540436 0.841385i \(-0.681741\pi\)
−0.540436 + 0.841385i \(0.681741\pi\)
\(314\) 5.03923i 0.284380i
\(315\) −1.69202 −0.0953346
\(316\) −5.67994 −0.319522
\(317\) − 20.3230i − 1.14146i −0.821139 0.570728i \(-0.806661\pi\)
0.821139 0.570728i \(-0.193339\pi\)
\(318\) 3.40581i 0.190989i
\(319\) 41.3411i 2.31466i
\(320\) − 1.00000i − 0.0559017i
\(321\) 3.06100 0.170848
\(322\) 15.1414 0.843796
\(323\) 15.3569i 0.854481i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 20.9825 1.16212
\(327\) − 7.45473i − 0.412248i
\(328\) 3.00969 0.166182
\(329\) −7.25236 −0.399835
\(330\) 4.55496i 0.250742i
\(331\) − 22.3588i − 1.22895i −0.788936 0.614476i \(-0.789368\pi\)
0.788936 0.614476i \(-0.210632\pi\)
\(332\) 7.71917i 0.423644i
\(333\) − 6.18598i − 0.338990i
\(334\) −22.2325 −1.21651
\(335\) 3.24698 0.177401
\(336\) − 1.69202i − 0.0923073i
\(337\) 3.67025 0.199931 0.0999657 0.994991i \(-0.468127\pi\)
0.0999657 + 0.994991i \(0.468127\pi\)
\(338\) 0 0
\(339\) 19.9812 1.08523
\(340\) 2.35690i 0.127821i
\(341\) 48.5230 2.62767
\(342\) 6.51573 0.352330
\(343\) 18.8442i 1.01749i
\(344\) − 4.93900i − 0.266293i
\(345\) 8.94869i 0.481781i
\(346\) 11.6286i 0.625159i
\(347\) −21.7832 −1.16938 −0.584690 0.811257i \(-0.698784\pi\)
−0.584690 + 0.811257i \(0.698784\pi\)
\(348\) 9.07606 0.486528
\(349\) 1.74333i 0.0933184i 0.998911 + 0.0466592i \(0.0148575\pi\)
−0.998911 + 0.0466592i \(0.985143\pi\)
\(350\) 1.69202 0.0904424
\(351\) 0 0
\(352\) −4.55496 −0.242780
\(353\) 6.09544i 0.324428i 0.986756 + 0.162214i \(0.0518634\pi\)
−0.986756 + 0.162214i \(0.948137\pi\)
\(354\) −4.32304 −0.229767
\(355\) 14.9487 0.793394
\(356\) − 9.12498i − 0.483623i
\(357\) 3.98792i 0.211063i
\(358\) 15.7875i 0.834393i
\(359\) − 25.9627i − 1.37026i −0.728422 0.685129i \(-0.759746\pi\)
0.728422 0.685129i \(-0.240254\pi\)
\(360\) 1.00000 0.0527046
\(361\) −23.4547 −1.23446
\(362\) 1.91185i 0.100485i
\(363\) 9.74764 0.511619
\(364\) 0 0
\(365\) −6.72886 −0.352204
\(366\) 14.3666i 0.750953i
\(367\) 27.8799 1.45532 0.727660 0.685938i \(-0.240608\pi\)
0.727660 + 0.685938i \(0.240608\pi\)
\(368\) −8.94869 −0.466483
\(369\) 3.00969i 0.156678i
\(370\) 6.18598i 0.321594i
\(371\) − 5.76271i − 0.299185i
\(372\) − 10.6528i − 0.552322i
\(373\) −25.1400 −1.30170 −0.650851 0.759205i \(-0.725588\pi\)
−0.650851 + 0.759205i \(0.725588\pi\)
\(374\) 10.7356 0.555123
\(375\) 1.00000i 0.0516398i
\(376\) 4.28621 0.221044
\(377\) 0 0
\(378\) 1.69202 0.0870282
\(379\) − 15.7463i − 0.808834i −0.914575 0.404417i \(-0.867474\pi\)
0.914575 0.404417i \(-0.132526\pi\)
\(380\) −6.51573 −0.334250
\(381\) −3.14244 −0.160992
\(382\) 24.3260i 1.24463i
\(383\) 11.8183i 0.603889i 0.953326 + 0.301944i \(0.0976357\pi\)
−0.953326 + 0.301944i \(0.902364\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 7.70709i − 0.392790i
\(386\) −0.521106 −0.0265236
\(387\) 4.93900 0.251064
\(388\) 4.40581i 0.223671i
\(389\) 6.14675 0.311653 0.155826 0.987784i \(-0.450196\pi\)
0.155826 + 0.987784i \(0.450196\pi\)
\(390\) 0 0
\(391\) 21.0911 1.06662
\(392\) − 4.13706i − 0.208953i
\(393\) 3.77479 0.190413
\(394\) 4.87502 0.245600
\(395\) 5.67994i 0.285789i
\(396\) − 4.55496i − 0.228895i
\(397\) 5.92825i 0.297530i 0.988873 + 0.148765i \(0.0475298\pi\)
−0.988873 + 0.148765i \(0.952470\pi\)
\(398\) 18.4983i 0.927235i
\(399\) −11.0248 −0.551928
\(400\) −1.00000 −0.0500000
\(401\) − 9.49934i − 0.474374i −0.971464 0.237187i \(-0.923775\pi\)
0.971464 0.237187i \(-0.0762254\pi\)
\(402\) −3.24698 −0.161945
\(403\) 0 0
\(404\) 2.31767 0.115308
\(405\) 1.00000i 0.0496904i
\(406\) −15.3569 −0.762150
\(407\) 28.1769 1.39668
\(408\) − 2.35690i − 0.116684i
\(409\) − 9.00000i − 0.445021i −0.974930 0.222511i \(-0.928575\pi\)
0.974930 0.222511i \(-0.0714252\pi\)
\(410\) − 3.00969i − 0.148638i
\(411\) 14.5894i 0.719642i
\(412\) −9.00000 −0.443398
\(413\) 7.31468 0.359932
\(414\) − 8.94869i − 0.439804i
\(415\) 7.71917 0.378919
\(416\) 0 0
\(417\) 14.3599 0.703206
\(418\) 29.6789i 1.45164i
\(419\) 30.6450 1.49711 0.748554 0.663074i \(-0.230749\pi\)
0.748554 + 0.663074i \(0.230749\pi\)
\(420\) −1.69202 −0.0825622
\(421\) 4.03252i 0.196533i 0.995160 + 0.0982666i \(0.0313298\pi\)
−0.995160 + 0.0982666i \(0.968670\pi\)
\(422\) 18.2500i 0.888394i
\(423\) 4.28621i 0.208403i
\(424\) 3.40581i 0.165401i
\(425\) 2.35690 0.114326
\(426\) −14.9487 −0.724266
\(427\) − 24.3086i − 1.17637i
\(428\) 3.06100 0.147959
\(429\) 0 0
\(430\) −4.93900 −0.238180
\(431\) − 27.6179i − 1.33031i −0.746707 0.665153i \(-0.768366\pi\)
0.746707 0.665153i \(-0.231634\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −10.6756 −0.513038 −0.256519 0.966539i \(-0.582576\pi\)
−0.256519 + 0.966539i \(0.582576\pi\)
\(434\) 18.0248i 0.865216i
\(435\) − 9.07606i − 0.435164i
\(436\) − 7.45473i − 0.357017i
\(437\) 58.3072i 2.78921i
\(438\) 6.72886 0.321517
\(439\) −25.9299 −1.23757 −0.618783 0.785562i \(-0.712374\pi\)
−0.618783 + 0.785562i \(0.712374\pi\)
\(440\) 4.55496i 0.217149i
\(441\) 4.13706 0.197003
\(442\) 0 0
\(443\) −35.3749 −1.68071 −0.840357 0.542033i \(-0.817655\pi\)
−0.840357 + 0.542033i \(0.817655\pi\)
\(444\) − 6.18598i − 0.293574i
\(445\) −9.12498 −0.432566
\(446\) −8.54048 −0.404404
\(447\) − 0.252356i − 0.0119360i
\(448\) − 1.69202i − 0.0799405i
\(449\) 6.24698i 0.294813i 0.989076 + 0.147407i \(0.0470926\pi\)
−0.989076 + 0.147407i \(0.952907\pi\)
\(450\) − 1.00000i − 0.0471405i
\(451\) −13.7090 −0.645532
\(452\) 19.9812 0.939837
\(453\) 11.0804i 0.520601i
\(454\) 9.61596 0.451299
\(455\) 0 0
\(456\) 6.51573 0.305127
\(457\) − 29.6383i − 1.38642i −0.720735 0.693211i \(-0.756195\pi\)
0.720735 0.693211i \(-0.243805\pi\)
\(458\) −13.4058 −0.626413
\(459\) 2.35690 0.110010
\(460\) 8.94869i 0.417235i
\(461\) 34.2911i 1.59710i 0.601931 + 0.798548i \(0.294398\pi\)
−0.601931 + 0.798548i \(0.705602\pi\)
\(462\) 7.70709i 0.358566i
\(463\) − 24.1769i − 1.12360i −0.827275 0.561798i \(-0.810110\pi\)
0.827275 0.561798i \(-0.189890\pi\)
\(464\) 9.07606 0.421346
\(465\) −10.6528 −0.494011
\(466\) 0.153457i 0.00710875i
\(467\) 0.521106 0.0241139 0.0120570 0.999927i \(-0.496162\pi\)
0.0120570 + 0.999927i \(0.496162\pi\)
\(468\) 0 0
\(469\) 5.49396 0.253687
\(470\) − 4.28621i − 0.197708i
\(471\) −5.03923 −0.232195
\(472\) −4.32304 −0.198984
\(473\) 22.4969i 1.03441i
\(474\) − 5.67994i − 0.260888i
\(475\) 6.51573i 0.298962i
\(476\) 3.98792i 0.182786i
\(477\) −3.40581 −0.155941
\(478\) −14.5851 −0.667106
\(479\) − 10.4168i − 0.475957i −0.971270 0.237979i \(-0.923515\pi\)
0.971270 0.237979i \(-0.0764848\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 3.40044 0.154886
\(483\) 15.1414i 0.688956i
\(484\) 9.74764 0.443075
\(485\) 4.40581 0.200058
\(486\) − 1.00000i − 0.0453609i
\(487\) 30.2349i 1.37007i 0.728508 + 0.685037i \(0.240214\pi\)
−0.728508 + 0.685037i \(0.759786\pi\)
\(488\) 14.3666i 0.650345i
\(489\) 20.9825i 0.948863i
\(490\) −4.13706 −0.186893
\(491\) 31.5883 1.42556 0.712781 0.701387i \(-0.247435\pi\)
0.712781 + 0.701387i \(0.247435\pi\)
\(492\) 3.00969i 0.135687i
\(493\) −21.3913 −0.963417
\(494\) 0 0
\(495\) −4.55496 −0.204730
\(496\) − 10.6528i − 0.478325i
\(497\) 25.2935 1.13457
\(498\) −7.71917 −0.345904
\(499\) − 27.2862i − 1.22150i −0.791824 0.610749i \(-0.790868\pi\)
0.791824 0.610749i \(-0.209132\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) − 22.2325i − 0.993275i
\(502\) − 1.80864i − 0.0807237i
\(503\) 29.6752 1.32315 0.661575 0.749879i \(-0.269888\pi\)
0.661575 + 0.749879i \(0.269888\pi\)
\(504\) 1.69202 0.0753686
\(505\) − 2.31767i − 0.103135i
\(506\) 40.7609 1.81204
\(507\) 0 0
\(508\) −3.14244 −0.139423
\(509\) − 44.1661i − 1.95763i −0.204748 0.978815i \(-0.565638\pi\)
0.204748 0.978815i \(-0.434362\pi\)
\(510\) −2.35690 −0.104365
\(511\) −11.3854 −0.503659
\(512\) 1.00000i 0.0441942i
\(513\) 6.51573i 0.287676i
\(514\) 22.2664i 0.982127i
\(515\) 9.00000i 0.396587i
\(516\) 4.93900 0.217427
\(517\) −19.5235 −0.858643
\(518\) 10.4668i 0.459885i
\(519\) −11.6286 −0.510440
\(520\) 0 0
\(521\) 20.1153 0.881267 0.440633 0.897687i \(-0.354754\pi\)
0.440633 + 0.897687i \(0.354754\pi\)
\(522\) 9.07606i 0.397249i
\(523\) 6.46117 0.282527 0.141264 0.989972i \(-0.454883\pi\)
0.141264 + 0.989972i \(0.454883\pi\)
\(524\) 3.77479 0.164902
\(525\) 1.69202i 0.0738459i
\(526\) 8.08815i 0.352660i
\(527\) 25.1075i 1.09370i
\(528\) − 4.55496i − 0.198229i
\(529\) 57.0790 2.48170
\(530\) 3.40581 0.147939
\(531\) − 4.32304i − 0.187604i
\(532\) −11.0248 −0.477984
\(533\) 0 0
\(534\) 9.12498 0.394877
\(535\) − 3.06100i − 0.132339i
\(536\) −3.24698 −0.140248
\(537\) −15.7875 −0.681279
\(538\) 16.6843i 0.719309i
\(539\) 18.8442i 0.811675i
\(540\) 1.00000i 0.0430331i
\(541\) − 34.0097i − 1.46219i −0.682275 0.731095i \(-0.739009\pi\)
0.682275 0.731095i \(-0.260991\pi\)
\(542\) 27.4077 1.17726
\(543\) −1.91185 −0.0820455
\(544\) − 2.35690i − 0.101051i
\(545\) −7.45473 −0.319326
\(546\) 0 0
\(547\) 14.6907 0.628129 0.314064 0.949402i \(-0.398309\pi\)
0.314064 + 0.949402i \(0.398309\pi\)
\(548\) 14.5894i 0.623228i
\(549\) −14.3666 −0.613151
\(550\) 4.55496 0.194224
\(551\) − 59.1372i − 2.51933i
\(552\) − 8.94869i − 0.380882i
\(553\) 9.61058i 0.408683i
\(554\) 16.0411i 0.681523i
\(555\) −6.18598 −0.262580
\(556\) 14.3599 0.608995
\(557\) − 7.00969i − 0.297010i −0.988912 0.148505i \(-0.952554\pi\)
0.988912 0.148505i \(-0.0474461\pi\)
\(558\) 10.6528 0.450969
\(559\) 0 0
\(560\) −1.69202 −0.0715010
\(561\) 10.7356i 0.453256i
\(562\) −10.0935 −0.425770
\(563\) 30.2131 1.27333 0.636666 0.771140i \(-0.280313\pi\)
0.636666 + 0.771140i \(0.280313\pi\)
\(564\) 4.28621i 0.180482i
\(565\) − 19.9812i − 0.840616i
\(566\) − 12.1021i − 0.508691i
\(567\) 1.69202i 0.0710582i
\(568\) −14.9487 −0.627233
\(569\) −14.8436 −0.622274 −0.311137 0.950365i \(-0.600710\pi\)
−0.311137 + 0.950365i \(0.600710\pi\)
\(570\) − 6.51573i − 0.272914i
\(571\) −0.965557 −0.0404073 −0.0202037 0.999796i \(-0.506431\pi\)
−0.0202037 + 0.999796i \(0.506431\pi\)
\(572\) 0 0
\(573\) −24.3260 −1.01623
\(574\) − 5.09246i − 0.212555i
\(575\) 8.94869 0.373186
\(576\) −1.00000 −0.0416667
\(577\) 27.8431i 1.15912i 0.814929 + 0.579561i \(0.196776\pi\)
−0.814929 + 0.579561i \(0.803224\pi\)
\(578\) − 11.4450i − 0.476051i
\(579\) − 0.521106i − 0.0216564i
\(580\) − 9.07606i − 0.376863i
\(581\) 13.0610 0.541862
\(582\) −4.40581 −0.182627
\(583\) − 15.5133i − 0.642497i
\(584\) 6.72886 0.278442
\(585\) 0 0
\(586\) 4.01075 0.165683
\(587\) − 1.47889i − 0.0610405i −0.999534 0.0305202i \(-0.990284\pi\)
0.999534 0.0305202i \(-0.00971640\pi\)
\(588\) 4.13706 0.170610
\(589\) −69.4107 −2.86002
\(590\) 4.32304i 0.177977i
\(591\) 4.87502i 0.200531i
\(592\) − 6.18598i − 0.254242i
\(593\) 16.8552i 0.692159i 0.938205 + 0.346079i \(0.112487\pi\)
−0.938205 + 0.346079i \(0.887513\pi\)
\(594\) 4.55496 0.186892
\(595\) 3.98792 0.163489
\(596\) − 0.252356i − 0.0103369i
\(597\) −18.4983 −0.757084
\(598\) 0 0
\(599\) −24.4862 −1.00048 −0.500239 0.865887i \(-0.666755\pi\)
−0.500239 + 0.865887i \(0.666755\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) −9.97344 −0.406825 −0.203413 0.979093i \(-0.565203\pi\)
−0.203413 + 0.979093i \(0.565203\pi\)
\(602\) −8.35690 −0.340602
\(603\) − 3.24698i − 0.132227i
\(604\) 11.0804i 0.450854i
\(605\) − 9.74764i − 0.396298i
\(606\) 2.31767i 0.0941488i
\(607\) −35.4476 −1.43877 −0.719386 0.694611i \(-0.755577\pi\)
−0.719386 + 0.694611i \(0.755577\pi\)
\(608\) 6.51573 0.264248
\(609\) − 15.3569i − 0.622293i
\(610\) 14.3666 0.581686
\(611\) 0 0
\(612\) 2.35690 0.0952719
\(613\) 16.1782i 0.653432i 0.945123 + 0.326716i \(0.105942\pi\)
−0.945123 + 0.326716i \(0.894058\pi\)
\(614\) −15.2784 −0.616588
\(615\) 3.00969 0.121362
\(616\) 7.70709i 0.310527i
\(617\) − 17.6920i − 0.712254i −0.934438 0.356127i \(-0.884097\pi\)
0.934438 0.356127i \(-0.115903\pi\)
\(618\) − 9.00000i − 0.362033i
\(619\) − 1.23968i − 0.0498271i −0.999690 0.0249136i \(-0.992069\pi\)
0.999690 0.0249136i \(-0.00793105\pi\)
\(620\) −10.6528 −0.427826
\(621\) 8.94869 0.359099
\(622\) − 25.4330i − 1.01977i
\(623\) −15.4397 −0.618577
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 19.1226i − 0.764292i
\(627\) −29.6789 −1.18526
\(628\) −5.03923 −0.201087
\(629\) 14.5797i 0.581331i
\(630\) − 1.69202i − 0.0674117i
\(631\) − 33.7845i − 1.34494i −0.740125 0.672469i \(-0.765234\pi\)
0.740125 0.672469i \(-0.234766\pi\)
\(632\) − 5.67994i − 0.225936i
\(633\) −18.2500 −0.725371
\(634\) 20.3230 0.807131
\(635\) 3.14244i 0.124704i
\(636\) −3.40581 −0.135049
\(637\) 0 0
\(638\) −41.3411 −1.63671
\(639\) − 14.9487i − 0.591361i
\(640\) 1.00000 0.0395285
\(641\) −12.0785 −0.477070 −0.238535 0.971134i \(-0.576667\pi\)
−0.238535 + 0.971134i \(0.576667\pi\)
\(642\) 3.06100i 0.120808i
\(643\) − 41.1377i − 1.62231i −0.584831 0.811155i \(-0.698839\pi\)
0.584831 0.811155i \(-0.301161\pi\)
\(644\) 15.1414i 0.596654i
\(645\) − 4.93900i − 0.194473i
\(646\) −15.3569 −0.604209
\(647\) −23.2054 −0.912297 −0.456148 0.889904i \(-0.650771\pi\)
−0.456148 + 0.889904i \(0.650771\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) 19.6913 0.772951
\(650\) 0 0
\(651\) −18.0248 −0.706446
\(652\) 20.9825i 0.821740i
\(653\) 7.77538 0.304274 0.152137 0.988359i \(-0.451384\pi\)
0.152137 + 0.988359i \(0.451384\pi\)
\(654\) 7.45473 0.291503
\(655\) − 3.77479i − 0.147493i
\(656\) 3.00969i 0.117509i
\(657\) 6.72886i 0.262518i
\(658\) − 7.25236i − 0.282726i
\(659\) −0.349600 −0.0136185 −0.00680924 0.999977i \(-0.502167\pi\)
−0.00680924 + 0.999977i \(0.502167\pi\)
\(660\) −4.55496 −0.177302
\(661\) − 19.7855i − 0.769568i −0.923007 0.384784i \(-0.874276\pi\)
0.923007 0.384784i \(-0.125724\pi\)
\(662\) 22.3588 0.869000
\(663\) 0 0
\(664\) −7.71917 −0.299562
\(665\) 11.0248i 0.427522i
\(666\) 6.18598 0.239702
\(667\) −81.2189 −3.14481
\(668\) − 22.2325i − 0.860201i
\(669\) − 8.54048i − 0.330194i
\(670\) 3.24698i 0.125442i
\(671\) − 65.4392i − 2.52625i
\(672\) 1.69202 0.0652711
\(673\) 16.6146 0.640447 0.320223 0.947342i \(-0.396242\pi\)
0.320223 + 0.947342i \(0.396242\pi\)
\(674\) 3.67025i 0.141373i
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −5.85517 −0.225032 −0.112516 0.993650i \(-0.535891\pi\)
−0.112516 + 0.993650i \(0.535891\pi\)
\(678\) 19.9812i 0.767374i
\(679\) 7.45473 0.286086
\(680\) −2.35690 −0.0903828
\(681\) 9.61596i 0.368484i
\(682\) 48.5230i 1.85804i
\(683\) 2.19029i 0.0838092i 0.999122 + 0.0419046i \(0.0133426\pi\)
−0.999122 + 0.0419046i \(0.986657\pi\)
\(684\) 6.51573i 0.249135i
\(685\) 14.5894 0.557432
\(686\) −18.8442 −0.719473
\(687\) − 13.4058i − 0.511464i
\(688\) 4.93900 0.188298
\(689\) 0 0
\(690\) −8.94869 −0.340671
\(691\) 11.8576i 0.451083i 0.974234 + 0.225541i \(0.0724151\pi\)
−0.974234 + 0.225541i \(0.927585\pi\)
\(692\) −11.6286 −0.442054
\(693\) −7.70709 −0.292768
\(694\) − 21.7832i − 0.826877i
\(695\) − 14.3599i − 0.544701i
\(696\) 9.07606i 0.344027i
\(697\) − 7.09352i − 0.268686i
\(698\) −1.74333 −0.0659861
\(699\) −0.153457 −0.00580427
\(700\) 1.69202i 0.0639524i
\(701\) −22.7627 −0.859736 −0.429868 0.902892i \(-0.641440\pi\)
−0.429868 + 0.902892i \(0.641440\pi\)
\(702\) 0 0
\(703\) −40.3062 −1.52018
\(704\) − 4.55496i − 0.171671i
\(705\) 4.28621 0.161428
\(706\) −6.09544 −0.229405
\(707\) − 3.92154i − 0.147485i
\(708\) − 4.32304i − 0.162470i
\(709\) − 18.5854i − 0.697988i −0.937125 0.348994i \(-0.886523\pi\)
0.937125 0.348994i \(-0.113477\pi\)
\(710\) 14.9487i 0.561014i
\(711\) 5.67994 0.213014
\(712\) 9.12498 0.341973
\(713\) 95.3285i 3.57008i
\(714\) −3.98792 −0.149244
\(715\) 0 0
\(716\) −15.7875 −0.590005
\(717\) − 14.5851i − 0.544690i
\(718\) 25.9627 0.968919
\(719\) 36.1575 1.34845 0.674224 0.738527i \(-0.264478\pi\)
0.674224 + 0.738527i \(0.264478\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 15.2282i 0.567128i
\(722\) − 23.4547i − 0.872895i
\(723\) 3.40044i 0.126464i
\(724\) −1.91185 −0.0710535
\(725\) −9.07606 −0.337077
\(726\) 9.74764i 0.361769i
\(727\) −19.0164 −0.705279 −0.352639 0.935759i \(-0.614716\pi\)
−0.352639 + 0.935759i \(0.614716\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 6.72886i − 0.249046i
\(731\) −11.6407 −0.430547
\(732\) −14.3666 −0.531004
\(733\) 8.89248i 0.328451i 0.986423 + 0.164226i \(0.0525125\pi\)
−0.986423 + 0.164226i \(0.947488\pi\)
\(734\) 27.8799i 1.02907i
\(735\) − 4.13706i − 0.152598i
\(736\) − 8.94869i − 0.329853i
\(737\) 14.7899 0.544791
\(738\) −3.00969 −0.110788
\(739\) 30.8595i 1.13518i 0.823310 + 0.567592i \(0.192125\pi\)
−0.823310 + 0.567592i \(0.807875\pi\)
\(740\) −6.18598 −0.227401
\(741\) 0 0
\(742\) 5.76271 0.211556
\(743\) 18.6300i 0.683467i 0.939797 + 0.341733i \(0.111014\pi\)
−0.939797 + 0.341733i \(0.888986\pi\)
\(744\) 10.6528 0.390550
\(745\) −0.252356 −0.00924562
\(746\) − 25.1400i − 0.920443i
\(747\) − 7.71917i − 0.282430i
\(748\) 10.7356i 0.392531i
\(749\) − 5.17928i − 0.189247i
\(750\) −1.00000 −0.0365148
\(751\) −12.5147 −0.456667 −0.228333 0.973583i \(-0.573328\pi\)
−0.228333 + 0.973583i \(0.573328\pi\)
\(752\) 4.28621i 0.156302i
\(753\) 1.80864 0.0659106
\(754\) 0 0
\(755\) 11.0804 0.403256
\(756\) 1.69202i 0.0615382i
\(757\) −4.75494 −0.172821 −0.0864106 0.996260i \(-0.527540\pi\)
−0.0864106 + 0.996260i \(0.527540\pi\)
\(758\) 15.7463 0.571932
\(759\) 40.7609i 1.47953i
\(760\) − 6.51573i − 0.236350i
\(761\) 21.7748i 0.789336i 0.918824 + 0.394668i \(0.129140\pi\)
−0.918824 + 0.394668i \(0.870860\pi\)
\(762\) − 3.14244i − 0.113839i
\(763\) −12.6136 −0.456642
\(764\) −24.3260 −0.880085
\(765\) − 2.35690i − 0.0852137i
\(766\) −11.8183 −0.427014
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 11.4045i 0.411256i 0.978630 + 0.205628i \(0.0659237\pi\)
−0.978630 + 0.205628i \(0.934076\pi\)
\(770\) 7.70709 0.277744
\(771\) −22.2664 −0.801903
\(772\) − 0.521106i − 0.0187550i
\(773\) 49.6118i 1.78441i 0.451630 + 0.892206i \(0.350843\pi\)
−0.451630 + 0.892206i \(0.649157\pi\)
\(774\) 4.93900i 0.177529i
\(775\) 10.6528i 0.382660i
\(776\) −4.40581 −0.158159
\(777\) −10.4668 −0.375495
\(778\) 6.14675i 0.220372i
\(779\) 19.6103 0.702613
\(780\) 0 0
\(781\) 68.0907 2.43648
\(782\) 21.0911i 0.754217i
\(783\) −9.07606 −0.324352
\(784\) 4.13706 0.147752
\(785\) 5.03923i 0.179858i
\(786\) 3.77479i 0.134642i
\(787\) − 26.5532i − 0.946518i −0.880923 0.473259i \(-0.843078\pi\)
0.880923 0.473259i \(-0.156922\pi\)
\(788\) 4.87502i 0.173665i
\(789\) −8.08815 −0.287946
\(790\) −5.67994 −0.202083
\(791\) − 33.8086i − 1.20210i
\(792\) 4.55496 0.161853
\(793\) 0 0
\(794\) −5.92825 −0.210386
\(795\) 3.40581i 0.120792i
\(796\) −18.4983 −0.655654
\(797\) −9.44803 −0.334666 −0.167333 0.985900i \(-0.553516\pi\)
−0.167333 + 0.985900i \(0.553516\pi\)
\(798\) − 11.0248i − 0.390272i
\(799\) − 10.1021i − 0.357388i
\(800\) − 1.00000i − 0.0353553i
\(801\) 9.12498i 0.322415i
\(802\) 9.49934 0.335433
\(803\) −30.6497 −1.08160
\(804\) − 3.24698i − 0.114512i
\(805\) 15.1414 0.533663
\(806\) 0 0
\(807\) −16.6843 −0.587313
\(808\) 2.31767i 0.0815352i
\(809\) −27.3376 −0.961140 −0.480570 0.876956i \(-0.659570\pi\)
−0.480570 + 0.876956i \(0.659570\pi\)
\(810\) −1.00000 −0.0351364
\(811\) − 2.86592i − 0.100636i −0.998733 0.0503180i \(-0.983977\pi\)
0.998733 0.0503180i \(-0.0160235\pi\)
\(812\) − 15.3569i − 0.538921i
\(813\) 27.4077i 0.961231i
\(814\) 28.1769i 0.987599i
\(815\) 20.9825 0.734986
\(816\) 2.35690 0.0825079
\(817\) − 32.1812i − 1.12588i
\(818\) 9.00000 0.314678
\(819\) 0 0
\(820\) 3.00969 0.105103
\(821\) 16.9739i 0.592394i 0.955127 + 0.296197i \(0.0957185\pi\)
−0.955127 + 0.296197i \(0.904282\pi\)
\(822\) −14.5894 −0.508864
\(823\) 22.2083 0.774134 0.387067 0.922052i \(-0.373488\pi\)
0.387067 + 0.922052i \(0.373488\pi\)
\(824\) − 9.00000i − 0.313530i
\(825\) 4.55496i 0.158583i
\(826\) 7.31468i 0.254510i
\(827\) − 36.3618i − 1.26442i −0.774796 0.632212i \(-0.782147\pi\)
0.774796 0.632212i \(-0.217853\pi\)
\(828\) 8.94869 0.310988
\(829\) 26.5478 0.922042 0.461021 0.887389i \(-0.347483\pi\)
0.461021 + 0.887389i \(0.347483\pi\)
\(830\) 7.71917i 0.267936i
\(831\) −16.0411 −0.556461
\(832\) 0 0
\(833\) −9.75063 −0.337839
\(834\) 14.3599i 0.497242i
\(835\) −22.2325 −0.769388
\(836\) −29.6789 −1.02647
\(837\) 10.6528i 0.368214i
\(838\) 30.6450i 1.05861i
\(839\) − 26.7668i − 0.924091i −0.886856 0.462046i \(-0.847116\pi\)
0.886856 0.462046i \(-0.152884\pi\)
\(840\) − 1.69202i − 0.0583803i
\(841\) 53.3749 1.84052
\(842\) −4.03252 −0.138970
\(843\) − 10.0935i − 0.347639i
\(844\) −18.2500 −0.628190
\(845\) 0 0
\(846\) −4.28621 −0.147363
\(847\) − 16.4932i − 0.566714i
\(848\) −3.40581 −0.116956
\(849\) 12.1021 0.415345
\(850\) 2.35690i 0.0808409i
\(851\) 55.3564i 1.89759i
\(852\) − 14.9487i − 0.512134i
\(853\) 50.3967i 1.72555i 0.505587 + 0.862775i \(0.331276\pi\)
−0.505587 + 0.862775i \(0.668724\pi\)
\(854\) 24.3086 0.831822
\(855\) 6.51573 0.222833
\(856\) 3.06100i 0.104623i
\(857\) −27.5114 −0.939772 −0.469886 0.882727i \(-0.655705\pi\)
−0.469886 + 0.882727i \(0.655705\pi\)
\(858\) 0 0
\(859\) 2.26098 0.0771436 0.0385718 0.999256i \(-0.487719\pi\)
0.0385718 + 0.999256i \(0.487719\pi\)
\(860\) − 4.93900i − 0.168419i
\(861\) 5.09246 0.173550
\(862\) 27.6179 0.940669
\(863\) 2.18406i 0.0743463i 0.999309 + 0.0371732i \(0.0118353\pi\)
−0.999309 + 0.0371732i \(0.988165\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 11.6286i 0.395385i
\(866\) − 10.6756i − 0.362773i
\(867\) 11.4450 0.388694
\(868\) −18.0248 −0.611800
\(869\) 25.8719i 0.877644i
\(870\) 9.07606 0.307707
\(871\) 0 0
\(872\) 7.45473 0.252449
\(873\) − 4.40581i − 0.149114i
\(874\) −58.3072 −1.97227
\(875\) 1.69202 0.0572008
\(876\) 6.72886i 0.227347i
\(877\) 19.9815i 0.674727i 0.941375 + 0.337363i \(0.109535\pi\)
−0.941375 + 0.337363i \(0.890465\pi\)
\(878\) − 25.9299i − 0.875092i
\(879\) 4.01075i 0.135279i
\(880\) −4.55496 −0.153548
\(881\) 38.4801 1.29643 0.648213 0.761459i \(-0.275516\pi\)
0.648213 + 0.761459i \(0.275516\pi\)
\(882\) 4.13706i 0.139302i
\(883\) 37.5512 1.26370 0.631850 0.775091i \(-0.282296\pi\)
0.631850 + 0.775091i \(0.282296\pi\)
\(884\) 0 0
\(885\) −4.32304 −0.145318
\(886\) − 35.3749i − 1.18844i
\(887\) 18.5676 0.623440 0.311720 0.950174i \(-0.399095\pi\)
0.311720 + 0.950174i \(0.399095\pi\)
\(888\) 6.18598 0.207588
\(889\) 5.31708i 0.178329i
\(890\) − 9.12498i − 0.305870i
\(891\) 4.55496i 0.152597i
\(892\) − 8.54048i − 0.285957i
\(893\) 27.9278 0.934567
\(894\) 0.252356 0.00844006
\(895\) 15.7875i 0.527717i
\(896\) 1.69202 0.0565265
\(897\) 0 0
\(898\) −6.24698 −0.208464
\(899\) − 96.6854i − 3.22464i
\(900\) 1.00000 0.0333333
\(901\) 8.02715 0.267423
\(902\) − 13.7090i − 0.456460i
\(903\) − 8.35690i − 0.278100i
\(904\) 19.9812i 0.664565i
\(905\) 1.91185i 0.0635522i
\(906\) −11.0804 −0.368121
\(907\) −60.1648 −1.99774 −0.998870 0.0475319i \(-0.984864\pi\)
−0.998870 + 0.0475319i \(0.984864\pi\)
\(908\) 9.61596i 0.319117i
\(909\) −2.31767 −0.0768722
\(910\) 0 0
\(911\) −37.0737 −1.22831 −0.614153 0.789187i \(-0.710502\pi\)
−0.614153 + 0.789187i \(0.710502\pi\)
\(912\) 6.51573i 0.215757i
\(913\) 35.1605 1.16364
\(914\) 29.6383 0.980348
\(915\) 14.3666i 0.474945i
\(916\) − 13.4058i − 0.442941i
\(917\) − 6.38703i − 0.210918i
\(918\) 2.35690i 0.0777892i
\(919\) −50.5526 −1.66758 −0.833788 0.552085i \(-0.813832\pi\)
−0.833788 + 0.552085i \(0.813832\pi\)
\(920\) −8.94869 −0.295030
\(921\) − 15.2784i − 0.503442i
\(922\) −34.2911 −1.12932
\(923\) 0 0
\(924\) −7.70709 −0.253545
\(925\) 6.18598i 0.203394i
\(926\) 24.1769 0.794502
\(927\) 9.00000 0.295599
\(928\) 9.07606i 0.297936i
\(929\) − 12.7269i − 0.417557i −0.977963 0.208779i \(-0.933051\pi\)
0.977963 0.208779i \(-0.0669488\pi\)
\(930\) − 10.6528i − 0.349319i
\(931\) − 26.9560i − 0.883447i
\(932\) −0.153457 −0.00502664
\(933\) 25.4330 0.832638
\(934\) 0.521106i 0.0170511i
\(935\) 10.7356 0.351090
\(936\) 0 0
\(937\) −35.3279 −1.15411 −0.577057 0.816704i \(-0.695799\pi\)
−0.577057 + 0.816704i \(0.695799\pi\)
\(938\) 5.49396i 0.179384i
\(939\) 19.1226 0.624042
\(940\) 4.28621 0.139801
\(941\) 42.0267i 1.37003i 0.728529 + 0.685015i \(0.240204\pi\)
−0.728529 + 0.685015i \(0.759796\pi\)
\(942\) − 5.03923i − 0.164187i
\(943\) − 26.9328i − 0.877052i
\(944\) − 4.32304i − 0.140703i
\(945\) 1.69202 0.0550415
\(946\) −22.4969 −0.731439
\(947\) − 4.63640i − 0.150663i −0.997159 0.0753314i \(-0.975999\pi\)
0.997159 0.0753314i \(-0.0240015\pi\)
\(948\) 5.67994 0.184476
\(949\) 0 0
\(950\) −6.51573 −0.211398
\(951\) 20.3230i 0.659020i
\(952\) −3.98792 −0.129249
\(953\) 41.8998 1.35727 0.678633 0.734477i \(-0.262573\pi\)
0.678633 + 0.734477i \(0.262573\pi\)
\(954\) − 3.40581i − 0.110267i
\(955\) 24.3260i 0.787172i
\(956\) − 14.5851i − 0.471715i
\(957\) − 41.3411i − 1.33637i
\(958\) 10.4168 0.336552
\(959\) 24.6856 0.797139
\(960\) 1.00000i 0.0322749i
\(961\) −82.4820 −2.66071
\(962\) 0 0
\(963\) −3.06100 −0.0986393
\(964\) 3.40044i 0.109521i
\(965\) −0.521106 −0.0167750
\(966\) −15.1414 −0.487166
\(967\) − 48.3545i − 1.55498i −0.628898 0.777488i \(-0.716494\pi\)
0.628898 0.777488i \(-0.283506\pi\)
\(968\) 9.74764i 0.313301i
\(969\) − 15.3569i − 0.493335i
\(970\) 4.40581i 0.141462i
\(971\) −18.9326 −0.607575 −0.303787 0.952740i \(-0.598251\pi\)
−0.303787 + 0.952740i \(0.598251\pi\)
\(972\) 1.00000 0.0320750
\(973\) − 24.2972i − 0.778933i
\(974\) −30.2349 −0.968789
\(975\) 0 0
\(976\) −14.3666 −0.459863
\(977\) − 54.7972i − 1.75312i −0.481296 0.876558i \(-0.659834\pi\)
0.481296 0.876558i \(-0.340166\pi\)
\(978\) −20.9825 −0.670948
\(979\) −41.5639 −1.32839
\(980\) − 4.13706i − 0.132154i
\(981\) 7.45473i 0.238011i
\(982\) 31.5883i 1.00802i
\(983\) − 24.6431i − 0.785993i −0.919540 0.392996i \(-0.871438\pi\)
0.919540 0.392996i \(-0.128562\pi\)
\(984\) −3.00969 −0.0959454
\(985\) 4.87502 0.155331
\(986\) − 21.3913i − 0.681239i
\(987\) 7.25236 0.230845
\(988\) 0 0
\(989\) −44.1976 −1.40540
\(990\) − 4.55496i − 0.144766i
\(991\) 23.1612 0.735741 0.367870 0.929877i \(-0.380087\pi\)
0.367870 + 0.929877i \(0.380087\pi\)
\(992\) 10.6528 0.338227
\(993\) 22.3588i 0.709536i
\(994\) 25.2935i 0.802261i
\(995\) 18.4983i 0.586435i
\(996\) − 7.71917i − 0.244591i
\(997\) −11.5700 −0.366426 −0.183213 0.983073i \(-0.558650\pi\)
−0.183213 + 0.983073i \(0.558650\pi\)
\(998\) 27.2862 0.863730
\(999\) 6.18598i 0.195716i
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.s.1351.6 6
13.5 odd 4 5070.2.a.bt.1.1 yes 3
13.8 odd 4 5070.2.a.bk.1.3 3
13.12 even 2 inner 5070.2.b.s.1351.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bk.1.3 3 13.8 odd 4
5070.2.a.bt.1.1 yes 3 13.5 odd 4
5070.2.b.s.1351.1 6 13.12 even 2 inner
5070.2.b.s.1351.6 6 1.1 even 1 trivial