Properties

Label 5070.2.a.bt.1.1
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $1$
Dimension $3$
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: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.69202 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.55496 q^{11} -1.00000 q^{12} -1.69202 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.35690 q^{17} +1.00000 q^{18} -6.51573 q^{19} +1.00000 q^{20} +1.69202 q^{21} -4.55496 q^{22} +8.94869 q^{23} -1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} -1.69202 q^{28} +9.07606 q^{29} -1.00000 q^{30} -10.6528 q^{31} +1.00000 q^{32} +4.55496 q^{33} +2.35690 q^{34} -1.69202 q^{35} +1.00000 q^{36} +6.18598 q^{37} -6.51573 q^{38} +1.00000 q^{40} +3.00969 q^{41} +1.69202 q^{42} -4.93900 q^{43} -4.55496 q^{44} +1.00000 q^{45} +8.94869 q^{46} -4.28621 q^{47} -1.00000 q^{48} -4.13706 q^{49} +1.00000 q^{50} -2.35690 q^{51} -3.40581 q^{53} -1.00000 q^{54} -4.55496 q^{55} -1.69202 q^{56} +6.51573 q^{57} +9.07606 q^{58} +4.32304 q^{59} -1.00000 q^{60} -14.3666 q^{61} -10.6528 q^{62} -1.69202 q^{63} +1.00000 q^{64} +4.55496 q^{66} -3.24698 q^{67} +2.35690 q^{68} -8.94869 q^{69} -1.69202 q^{70} -14.9487 q^{71} +1.00000 q^{72} -6.72886 q^{73} +6.18598 q^{74} -1.00000 q^{75} -6.51573 q^{76} +7.70709 q^{77} +5.67994 q^{79} +1.00000 q^{80} +1.00000 q^{81} +3.00969 q^{82} -7.71917 q^{83} +1.69202 q^{84} +2.35690 q^{85} -4.93900 q^{86} -9.07606 q^{87} -4.55496 q^{88} -9.12498 q^{89} +1.00000 q^{90} +8.94869 q^{92} +10.6528 q^{93} -4.28621 q^{94} -6.51573 q^{95} -1.00000 q^{96} -4.40581 q^{97} -4.13706 q^{98} -4.55496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} - 3q^{6} + 3q^{8} + 3q^{9} + 3q^{10} - 14q^{11} - 3q^{12} - 3q^{15} + 3q^{16} + 3q^{17} + 3q^{18} - 7q^{19} + 3q^{20} - 14q^{22} - 5q^{23} - 3q^{24} + 3q^{25} - 3q^{27} + 12q^{29} - 3q^{30} - 14q^{31} + 3q^{32} + 14q^{33} + 3q^{34} + 3q^{36} + 4q^{37} - 7q^{38} + 3q^{40} - 13q^{41} - 5q^{43} - 14q^{44} + 3q^{45} - 5q^{46} - 21q^{47} - 3q^{48} - 7q^{49} + 3q^{50} - 3q^{51} + 3q^{53} - 3q^{54} - 14q^{55} + 7q^{57} + 12q^{58} - 7q^{59} - 3q^{60} - 17q^{61} - 14q^{62} + 3q^{64} + 14q^{66} - 5q^{67} + 3q^{68} + 5q^{69} - 13q^{71} + 3q^{72} + 13q^{73} + 4q^{74} - 3q^{75} - 7q^{76} - 7q^{77} - 7q^{79} + 3q^{80} + 3q^{81} - 13q^{82} - 12q^{83} + 3q^{85} - 5q^{86} - 12q^{87} - 14q^{88} - 3q^{89} + 3q^{90} - 5q^{92} + 14q^{93} - 21q^{94} - 7q^{95} - 3q^{96} - 7q^{98} - 14q^{99} + O(q^{100}) \)

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
\(7\) −1.69202 −0.639524 −0.319762 0.947498i \(-0.603603\pi\)
−0.319762 + 0.947498i \(0.603603\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.55496 −1.37337 −0.686686 0.726954i \(-0.740935\pi\)
−0.686686 + 0.726954i \(0.740935\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.69202 −0.452212
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.35690 0.571631 0.285816 0.958285i \(-0.407736\pi\)
0.285816 + 0.958285i \(0.407736\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.51573 −1.49481 −0.747405 0.664368i \(-0.768701\pi\)
−0.747405 + 0.664368i \(0.768701\pi\)
\(20\) 1.00000 0.223607
\(21\) 1.69202 0.369229
\(22\) −4.55496 −0.971120
\(23\) 8.94869 1.86593 0.932965 0.359966i \(-0.117212\pi\)
0.932965 + 0.359966i \(0.117212\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −1.69202 −0.319762
\(29\) 9.07606 1.68538 0.842691 0.538397i \(-0.180970\pi\)
0.842691 + 0.538397i \(0.180970\pi\)
\(30\) −1.00000 −0.182574
\(31\) −10.6528 −1.91330 −0.956649 0.291243i \(-0.905931\pi\)
−0.956649 + 0.291243i \(0.905931\pi\)
\(32\) 1.00000 0.176777
\(33\) 4.55496 0.792916
\(34\) 2.35690 0.404204
\(35\) −1.69202 −0.286004
\(36\) 1.00000 0.166667
\(37\) 6.18598 1.01697 0.508484 0.861071i \(-0.330206\pi\)
0.508484 + 0.861071i \(0.330206\pi\)
\(38\) −6.51573 −1.05699
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 3.00969 0.470034 0.235017 0.971991i \(-0.424485\pi\)
0.235017 + 0.971991i \(0.424485\pi\)
\(42\) 1.69202 0.261085
\(43\) −4.93900 −0.753191 −0.376595 0.926378i \(-0.622905\pi\)
−0.376595 + 0.926378i \(0.622905\pi\)
\(44\) −4.55496 −0.686686
\(45\) 1.00000 0.149071
\(46\) 8.94869 1.31941
\(47\) −4.28621 −0.625208 −0.312604 0.949884i \(-0.601201\pi\)
−0.312604 + 0.949884i \(0.601201\pi\)
\(48\) −1.00000 −0.144338
\(49\) −4.13706 −0.591009
\(50\) 1.00000 0.141421
\(51\) −2.35690 −0.330031
\(52\) 0 0
\(53\) −3.40581 −0.467824 −0.233912 0.972258i \(-0.575153\pi\)
−0.233912 + 0.972258i \(0.575153\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.55496 −0.614190
\(56\) −1.69202 −0.226106
\(57\) 6.51573 0.863029
\(58\) 9.07606 1.19175
\(59\) 4.32304 0.562812 0.281406 0.959589i \(-0.409199\pi\)
0.281406 + 0.959589i \(0.409199\pi\)
\(60\) −1.00000 −0.129099
\(61\) −14.3666 −1.83945 −0.919726 0.392560i \(-0.871589\pi\)
−0.919726 + 0.392560i \(0.871589\pi\)
\(62\) −10.6528 −1.35291
\(63\) −1.69202 −0.213175
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 4.55496 0.560677
\(67\) −3.24698 −0.396682 −0.198341 0.980133i \(-0.563555\pi\)
−0.198341 + 0.980133i \(0.563555\pi\)
\(68\) 2.35690 0.285816
\(69\) −8.94869 −1.07730
\(70\) −1.69202 −0.202235
\(71\) −14.9487 −1.77408 −0.887042 0.461690i \(-0.847243\pi\)
−0.887042 + 0.461690i \(0.847243\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.72886 −0.787553 −0.393777 0.919206i \(-0.628832\pi\)
−0.393777 + 0.919206i \(0.628832\pi\)
\(74\) 6.18598 0.719106
\(75\) −1.00000 −0.115470
\(76\) −6.51573 −0.747405
\(77\) 7.70709 0.878304
\(78\) 0 0
\(79\) 5.67994 0.639043 0.319522 0.947579i \(-0.396478\pi\)
0.319522 + 0.947579i \(0.396478\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) 3.00969 0.332365
\(83\) −7.71917 −0.847289 −0.423644 0.905829i \(-0.639249\pi\)
−0.423644 + 0.905829i \(0.639249\pi\)
\(84\) 1.69202 0.184615
\(85\) 2.35690 0.255641
\(86\) −4.93900 −0.532586
\(87\) −9.07606 −0.973056
\(88\) −4.55496 −0.485560
\(89\) −9.12498 −0.967246 −0.483623 0.875276i \(-0.660679\pi\)
−0.483623 + 0.875276i \(0.660679\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) 8.94869 0.932965
\(93\) 10.6528 1.10464
\(94\) −4.28621 −0.442089
\(95\) −6.51573 −0.668500
\(96\) −1.00000 −0.102062
\(97\) −4.40581 −0.447343 −0.223671 0.974665i \(-0.571804\pi\)
−0.223671 + 0.974665i \(0.571804\pi\)
\(98\) −4.13706 −0.417907
\(99\) −4.55496 −0.457791
\(100\) 1.00000 0.100000
\(101\) 2.31767 0.230617 0.115308 0.993330i \(-0.463214\pi\)
0.115308 + 0.993330i \(0.463214\pi\)
\(102\) −2.35690 −0.233367
\(103\) −9.00000 −0.886796 −0.443398 0.896325i \(-0.646227\pi\)
−0.443398 + 0.896325i \(0.646227\pi\)
\(104\) 0 0
\(105\) 1.69202 0.165124
\(106\) −3.40581 −0.330802
\(107\) −3.06100 −0.295918 −0.147959 0.988994i \(-0.547270\pi\)
−0.147959 + 0.988994i \(0.547270\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 7.45473 0.714034 0.357017 0.934098i \(-0.383794\pi\)
0.357017 + 0.934098i \(0.383794\pi\)
\(110\) −4.55496 −0.434298
\(111\) −6.18598 −0.587147
\(112\) −1.69202 −0.159881
\(113\) −19.9812 −1.87967 −0.939837 0.341623i \(-0.889024\pi\)
−0.939837 + 0.341623i \(0.889024\pi\)
\(114\) 6.51573 0.610254
\(115\) 8.94869 0.834470
\(116\) 9.07606 0.842691
\(117\) 0 0
\(118\) 4.32304 0.397968
\(119\) −3.98792 −0.365572
\(120\) −1.00000 −0.0912871
\(121\) 9.74764 0.886149
\(122\) −14.3666 −1.30069
\(123\) −3.00969 −0.271374
\(124\) −10.6528 −0.956649
\(125\) 1.00000 0.0894427
\(126\) −1.69202 −0.150737
\(127\) −3.14244 −0.278846 −0.139423 0.990233i \(-0.544525\pi\)
−0.139423 + 0.990233i \(0.544525\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.93900 0.434855
\(130\) 0 0
\(131\) −3.77479 −0.329805 −0.164902 0.986310i \(-0.552731\pi\)
−0.164902 + 0.986310i \(0.552731\pi\)
\(132\) 4.55496 0.396458
\(133\) 11.0248 0.955967
\(134\) −3.24698 −0.280496
\(135\) −1.00000 −0.0860663
\(136\) 2.35690 0.202102
\(137\) 14.5894 1.24646 0.623228 0.782040i \(-0.285821\pi\)
0.623228 + 0.782040i \(0.285821\pi\)
\(138\) −8.94869 −0.761763
\(139\) −14.3599 −1.21799 −0.608995 0.793174i \(-0.708427\pi\)
−0.608995 + 0.793174i \(0.708427\pi\)
\(140\) −1.69202 −0.143002
\(141\) 4.28621 0.360964
\(142\) −14.9487 −1.25447
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 9.07606 0.753726
\(146\) −6.72886 −0.556884
\(147\) 4.13706 0.341219
\(148\) 6.18598 0.508484
\(149\) 0.252356 0.0206738 0.0103369 0.999947i \(-0.496710\pi\)
0.0103369 + 0.999947i \(0.496710\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 11.0804 0.901708 0.450854 0.892598i \(-0.351119\pi\)
0.450854 + 0.892598i \(0.351119\pi\)
\(152\) −6.51573 −0.528495
\(153\) 2.35690 0.190544
\(154\) 7.70709 0.621055
\(155\) −10.6528 −0.855653
\(156\) 0 0
\(157\) 5.03923 0.402174 0.201087 0.979573i \(-0.435553\pi\)
0.201087 + 0.979573i \(0.435553\pi\)
\(158\) 5.67994 0.451872
\(159\) 3.40581 0.270099
\(160\) 1.00000 0.0790569
\(161\) −15.1414 −1.19331
\(162\) 1.00000 0.0785674
\(163\) 20.9825 1.64348 0.821740 0.569863i \(-0.193004\pi\)
0.821740 + 0.569863i \(0.193004\pi\)
\(164\) 3.00969 0.235017
\(165\) 4.55496 0.354603
\(166\) −7.71917 −0.599124
\(167\) −22.2325 −1.72040 −0.860201 0.509954i \(-0.829662\pi\)
−0.860201 + 0.509954i \(0.829662\pi\)
\(168\) 1.69202 0.130542
\(169\) 0 0
\(170\) 2.35690 0.180766
\(171\) −6.51573 −0.498270
\(172\) −4.93900 −0.376595
\(173\) −11.6286 −0.884108 −0.442054 0.896988i \(-0.645750\pi\)
−0.442054 + 0.896988i \(0.645750\pi\)
\(174\) −9.07606 −0.688055
\(175\) −1.69202 −0.127905
\(176\) −4.55496 −0.343343
\(177\) −4.32304 −0.324940
\(178\) −9.12498 −0.683946
\(179\) −15.7875 −1.18001 −0.590005 0.807399i \(-0.700874\pi\)
−0.590005 + 0.807399i \(0.700874\pi\)
\(180\) 1.00000 0.0745356
\(181\) −1.91185 −0.142107 −0.0710535 0.997473i \(-0.522636\pi\)
−0.0710535 + 0.997473i \(0.522636\pi\)
\(182\) 0 0
\(183\) 14.3666 1.06201
\(184\) 8.94869 0.659706
\(185\) 6.18598 0.454802
\(186\) 10.6528 0.781101
\(187\) −10.7356 −0.785062
\(188\) −4.28621 −0.312604
\(189\) 1.69202 0.123076
\(190\) −6.51573 −0.472701
\(191\) 24.3260 1.76017 0.880085 0.474817i \(-0.157486\pi\)
0.880085 + 0.474817i \(0.157486\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −0.521106 −0.0375101 −0.0187550 0.999824i \(-0.505970\pi\)
−0.0187550 + 0.999824i \(0.505970\pi\)
\(194\) −4.40581 −0.316319
\(195\) 0 0
\(196\) −4.13706 −0.295505
\(197\) −4.87502 −0.347331 −0.173665 0.984805i \(-0.555561\pi\)
−0.173665 + 0.984805i \(0.555561\pi\)
\(198\) −4.55496 −0.323707
\(199\) −18.4983 −1.31131 −0.655654 0.755062i \(-0.727607\pi\)
−0.655654 + 0.755062i \(0.727607\pi\)
\(200\) 1.00000 0.0707107
\(201\) 3.24698 0.229024
\(202\) 2.31767 0.163070
\(203\) −15.3569 −1.07784
\(204\) −2.35690 −0.165016
\(205\) 3.00969 0.210206
\(206\) −9.00000 −0.627060
\(207\) 8.94869 0.621977
\(208\) 0 0
\(209\) 29.6789 2.05293
\(210\) 1.69202 0.116761
\(211\) 18.2500 1.25638 0.628190 0.778060i \(-0.283796\pi\)
0.628190 + 0.778060i \(0.283796\pi\)
\(212\) −3.40581 −0.233912
\(213\) 14.9487 1.02427
\(214\) −3.06100 −0.209246
\(215\) −4.93900 −0.336837
\(216\) −1.00000 −0.0680414
\(217\) 18.0248 1.22360
\(218\) 7.45473 0.504898
\(219\) 6.72886 0.454694
\(220\) −4.55496 −0.307095
\(221\) 0 0
\(222\) −6.18598 −0.415176
\(223\) 8.54048 0.571913 0.285957 0.958243i \(-0.407689\pi\)
0.285957 + 0.958243i \(0.407689\pi\)
\(224\) −1.69202 −0.113053
\(225\) 1.00000 0.0666667
\(226\) −19.9812 −1.32913
\(227\) −9.61596 −0.638233 −0.319117 0.947715i \(-0.603386\pi\)
−0.319117 + 0.947715i \(0.603386\pi\)
\(228\) 6.51573 0.431515
\(229\) −13.4058 −0.885881 −0.442941 0.896551i \(-0.646065\pi\)
−0.442941 + 0.896551i \(0.646065\pi\)
\(230\) 8.94869 0.590059
\(231\) −7.70709 −0.507089
\(232\) 9.07606 0.595873
\(233\) −0.153457 −0.0100533 −0.00502664 0.999987i \(-0.501600\pi\)
−0.00502664 + 0.999987i \(0.501600\pi\)
\(234\) 0 0
\(235\) −4.28621 −0.279601
\(236\) 4.32304 0.281406
\(237\) −5.67994 −0.368952
\(238\) −3.98792 −0.258498
\(239\) 14.5851 0.943431 0.471715 0.881751i \(-0.343635\pi\)
0.471715 + 0.881751i \(0.343635\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 3.40044 0.219041 0.109521 0.993985i \(-0.465068\pi\)
0.109521 + 0.993985i \(0.465068\pi\)
\(242\) 9.74764 0.626602
\(243\) −1.00000 −0.0641500
\(244\) −14.3666 −0.919726
\(245\) −4.13706 −0.264307
\(246\) −3.00969 −0.191891
\(247\) 0 0
\(248\) −10.6528 −0.676453
\(249\) 7.71917 0.489182
\(250\) 1.00000 0.0632456
\(251\) 1.80864 0.114161 0.0570803 0.998370i \(-0.481821\pi\)
0.0570803 + 0.998370i \(0.481821\pi\)
\(252\) −1.69202 −0.106587
\(253\) −40.7609 −2.56262
\(254\) −3.14244 −0.197174
\(255\) −2.35690 −0.147595
\(256\) 1.00000 0.0625000
\(257\) −22.2664 −1.38894 −0.694469 0.719523i \(-0.744360\pi\)
−0.694469 + 0.719523i \(0.744360\pi\)
\(258\) 4.93900 0.307489
\(259\) −10.4668 −0.650376
\(260\) 0 0
\(261\) 9.07606 0.561794
\(262\) −3.77479 −0.233207
\(263\) 8.08815 0.498736 0.249368 0.968409i \(-0.419777\pi\)
0.249368 + 0.968409i \(0.419777\pi\)
\(264\) 4.55496 0.280338
\(265\) −3.40581 −0.209217
\(266\) 11.0248 0.675971
\(267\) 9.12498 0.558440
\(268\) −3.24698 −0.198341
\(269\) 16.6843 1.01726 0.508628 0.860986i \(-0.330153\pi\)
0.508628 + 0.860986i \(0.330153\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 27.4077 1.66490 0.832451 0.554099i \(-0.186937\pi\)
0.832451 + 0.554099i \(0.186937\pi\)
\(272\) 2.35690 0.142908
\(273\) 0 0
\(274\) 14.5894 0.881378
\(275\) −4.55496 −0.274674
\(276\) −8.94869 −0.538648
\(277\) −16.0411 −0.963819 −0.481910 0.876221i \(-0.660057\pi\)
−0.481910 + 0.876221i \(0.660057\pi\)
\(278\) −14.3599 −0.861248
\(279\) −10.6528 −0.637766
\(280\) −1.69202 −0.101118
\(281\) −10.0935 −0.602129 −0.301065 0.953604i \(-0.597342\pi\)
−0.301065 + 0.953604i \(0.597342\pi\)
\(282\) 4.28621 0.255240
\(283\) 12.1021 0.719398 0.359699 0.933068i \(-0.382879\pi\)
0.359699 + 0.933068i \(0.382879\pi\)
\(284\) −14.9487 −0.887042
\(285\) 6.51573 0.385959
\(286\) 0 0
\(287\) −5.09246 −0.300598
\(288\) 1.00000 0.0589256
\(289\) −11.4450 −0.673238
\(290\) 9.07606 0.532965
\(291\) 4.40581 0.258273
\(292\) −6.72886 −0.393777
\(293\) 4.01075 0.234311 0.117155 0.993114i \(-0.462622\pi\)
0.117155 + 0.993114i \(0.462622\pi\)
\(294\) 4.13706 0.241278
\(295\) 4.32304 0.251697
\(296\) 6.18598 0.359553
\(297\) 4.55496 0.264305
\(298\) 0.252356 0.0146186
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 8.35690 0.481683
\(302\) 11.0804 0.637604
\(303\) −2.31767 −0.133147
\(304\) −6.51573 −0.373703
\(305\) −14.3666 −0.822628
\(306\) 2.35690 0.134735
\(307\) −15.2784 −0.871987 −0.435993 0.899950i \(-0.643603\pi\)
−0.435993 + 0.899950i \(0.643603\pi\)
\(308\) 7.70709 0.439152
\(309\) 9.00000 0.511992
\(310\) −10.6528 −0.605038
\(311\) 25.4330 1.44217 0.721085 0.692846i \(-0.243644\pi\)
0.721085 + 0.692846i \(0.243644\pi\)
\(312\) 0 0
\(313\) −19.1226 −1.08087 −0.540436 0.841385i \(-0.681741\pi\)
−0.540436 + 0.841385i \(0.681741\pi\)
\(314\) 5.03923 0.284380
\(315\) −1.69202 −0.0953346
\(316\) 5.67994 0.319522
\(317\) −20.3230 −1.14146 −0.570728 0.821139i \(-0.693339\pi\)
−0.570728 + 0.821139i \(0.693339\pi\)
\(318\) 3.40581 0.190989
\(319\) −41.3411 −2.31466
\(320\) 1.00000 0.0559017
\(321\) 3.06100 0.170848
\(322\) −15.1414 −0.843796
\(323\) −15.3569 −0.854481
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.9825 1.16212
\(327\) −7.45473 −0.412248
\(328\) 3.00969 0.166182
\(329\) 7.25236 0.399835
\(330\) 4.55496 0.250742
\(331\) −22.3588 −1.22895 −0.614476 0.788936i \(-0.710632\pi\)
−0.614476 + 0.788936i \(0.710632\pi\)
\(332\) −7.71917 −0.423644
\(333\) 6.18598 0.338990
\(334\) −22.2325 −1.21651
\(335\) −3.24698 −0.177401
\(336\) 1.69202 0.0923073
\(337\) −3.67025 −0.199931 −0.0999657 0.994991i \(-0.531873\pi\)
−0.0999657 + 0.994991i \(0.531873\pi\)
\(338\) 0 0
\(339\) 19.9812 1.08523
\(340\) 2.35690 0.127821
\(341\) 48.5230 2.62767
\(342\) −6.51573 −0.352330
\(343\) 18.8442 1.01749
\(344\) −4.93900 −0.266293
\(345\) −8.94869 −0.481781
\(346\) −11.6286 −0.625159
\(347\) −21.7832 −1.16938 −0.584690 0.811257i \(-0.698784\pi\)
−0.584690 + 0.811257i \(0.698784\pi\)
\(348\) −9.07606 −0.486528
\(349\) −1.74333 −0.0933184 −0.0466592 0.998911i \(-0.514857\pi\)
−0.0466592 + 0.998911i \(0.514857\pi\)
\(350\) −1.69202 −0.0904424
\(351\) 0 0
\(352\) −4.55496 −0.242780
\(353\) 6.09544 0.324428 0.162214 0.986756i \(-0.448137\pi\)
0.162214 + 0.986756i \(0.448137\pi\)
\(354\) −4.32304 −0.229767
\(355\) −14.9487 −0.793394
\(356\) −9.12498 −0.483623
\(357\) 3.98792 0.211063
\(358\) −15.7875 −0.834393
\(359\) 25.9627 1.37026 0.685129 0.728422i \(-0.259746\pi\)
0.685129 + 0.728422i \(0.259746\pi\)
\(360\) 1.00000 0.0527046
\(361\) 23.4547 1.23446
\(362\) −1.91185 −0.100485
\(363\) −9.74764 −0.511619
\(364\) 0 0
\(365\) −6.72886 −0.352204
\(366\) 14.3666 0.750953
\(367\) 27.8799 1.45532 0.727660 0.685938i \(-0.240608\pi\)
0.727660 + 0.685938i \(0.240608\pi\)
\(368\) 8.94869 0.466483
\(369\) 3.00969 0.156678
\(370\) 6.18598 0.321594
\(371\) 5.76271 0.299185
\(372\) 10.6528 0.552322
\(373\) −25.1400 −1.30170 −0.650851 0.759205i \(-0.725588\pi\)
−0.650851 + 0.759205i \(0.725588\pi\)
\(374\) −10.7356 −0.555123
\(375\) −1.00000 −0.0516398
\(376\) −4.28621 −0.221044
\(377\) 0 0
\(378\) 1.69202 0.0870282
\(379\) −15.7463 −0.808834 −0.404417 0.914575i \(-0.632526\pi\)
−0.404417 + 0.914575i \(0.632526\pi\)
\(380\) −6.51573 −0.334250
\(381\) 3.14244 0.160992
\(382\) 24.3260 1.24463
\(383\) 11.8183 0.603889 0.301944 0.953326i \(-0.402364\pi\)
0.301944 + 0.953326i \(0.402364\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 7.70709 0.392790
\(386\) −0.521106 −0.0265236
\(387\) −4.93900 −0.251064
\(388\) −4.40581 −0.223671
\(389\) −6.14675 −0.311653 −0.155826 0.987784i \(-0.549804\pi\)
−0.155826 + 0.987784i \(0.549804\pi\)
\(390\) 0 0
\(391\) 21.0911 1.06662
\(392\) −4.13706 −0.208953
\(393\) 3.77479 0.190413
\(394\) −4.87502 −0.245600
\(395\) 5.67994 0.285789
\(396\) −4.55496 −0.228895
\(397\) −5.92825 −0.297530 −0.148765 0.988873i \(-0.547530\pi\)
−0.148765 + 0.988873i \(0.547530\pi\)
\(398\) −18.4983 −0.927235
\(399\) −11.0248 −0.551928
\(400\) 1.00000 0.0500000
\(401\) 9.49934 0.474374 0.237187 0.971464i \(-0.423775\pi\)
0.237187 + 0.971464i \(0.423775\pi\)
\(402\) 3.24698 0.161945
\(403\) 0 0
\(404\) 2.31767 0.115308
\(405\) 1.00000 0.0496904
\(406\) −15.3569 −0.762150
\(407\) −28.1769 −1.39668
\(408\) −2.35690 −0.116684
\(409\) −9.00000 −0.445021 −0.222511 0.974930i \(-0.571425\pi\)
−0.222511 + 0.974930i \(0.571425\pi\)
\(410\) 3.00969 0.148638
\(411\) −14.5894 −0.719642
\(412\) −9.00000 −0.443398
\(413\) −7.31468 −0.359932
\(414\) 8.94869 0.439804
\(415\) −7.71917 −0.378919
\(416\) 0 0
\(417\) 14.3599 0.703206
\(418\) 29.6789 1.45164
\(419\) 30.6450 1.49711 0.748554 0.663074i \(-0.230749\pi\)
0.748554 + 0.663074i \(0.230749\pi\)
\(420\) 1.69202 0.0825622
\(421\) 4.03252 0.196533 0.0982666 0.995160i \(-0.468670\pi\)
0.0982666 + 0.995160i \(0.468670\pi\)
\(422\) 18.2500 0.888394
\(423\) −4.28621 −0.208403
\(424\) −3.40581 −0.165401
\(425\) 2.35690 0.114326
\(426\) 14.9487 0.724266
\(427\) 24.3086 1.17637
\(428\) −3.06100 −0.147959
\(429\) 0 0
\(430\) −4.93900 −0.238180
\(431\) −27.6179 −1.33031 −0.665153 0.746707i \(-0.731634\pi\)
−0.665153 + 0.746707i \(0.731634\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 10.6756 0.513038 0.256519 0.966539i \(-0.417424\pi\)
0.256519 + 0.966539i \(0.417424\pi\)
\(434\) 18.0248 0.865216
\(435\) −9.07606 −0.435164
\(436\) 7.45473 0.357017
\(437\) −58.3072 −2.78921
\(438\) 6.72886 0.321517
\(439\) 25.9299 1.23757 0.618783 0.785562i \(-0.287626\pi\)
0.618783 + 0.785562i \(0.287626\pi\)
\(440\) −4.55496 −0.217149
\(441\) −4.13706 −0.197003
\(442\) 0 0
\(443\) −35.3749 −1.68071 −0.840357 0.542033i \(-0.817655\pi\)
−0.840357 + 0.542033i \(0.817655\pi\)
\(444\) −6.18598 −0.293574
\(445\) −9.12498 −0.432566
\(446\) 8.54048 0.404404
\(447\) −0.252356 −0.0119360
\(448\) −1.69202 −0.0799405
\(449\) −6.24698 −0.294813 −0.147407 0.989076i \(-0.547093\pi\)
−0.147407 + 0.989076i \(0.547093\pi\)
\(450\) 1.00000 0.0471405
\(451\) −13.7090 −0.645532
\(452\) −19.9812 −0.939837
\(453\) −11.0804 −0.520601
\(454\) −9.61596 −0.451299
\(455\) 0 0
\(456\) 6.51573 0.305127
\(457\) −29.6383 −1.38642 −0.693211 0.720735i \(-0.743805\pi\)
−0.693211 + 0.720735i \(0.743805\pi\)
\(458\) −13.4058 −0.626413
\(459\) −2.35690 −0.110010
\(460\) 8.94869 0.417235
\(461\) 34.2911 1.59710 0.798548 0.601931i \(-0.205602\pi\)
0.798548 + 0.601931i \(0.205602\pi\)
\(462\) −7.70709 −0.358566
\(463\) 24.1769 1.12360 0.561798 0.827275i \(-0.310110\pi\)
0.561798 + 0.827275i \(0.310110\pi\)
\(464\) 9.07606 0.421346
\(465\) 10.6528 0.494011
\(466\) −0.153457 −0.00710875
\(467\) −0.521106 −0.0241139 −0.0120570 0.999927i \(-0.503838\pi\)
−0.0120570 + 0.999927i \(0.503838\pi\)
\(468\) 0 0
\(469\) 5.49396 0.253687
\(470\) −4.28621 −0.197708
\(471\) −5.03923 −0.232195
\(472\) 4.32304 0.198984
\(473\) 22.4969 1.03441
\(474\) −5.67994 −0.260888
\(475\) −6.51573 −0.298962
\(476\) −3.98792 −0.182786
\(477\) −3.40581 −0.155941
\(478\) 14.5851 0.667106
\(479\) 10.4168 0.475957 0.237979 0.971270i \(-0.423515\pi\)
0.237979 + 0.971270i \(0.423515\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 3.40044 0.154886
\(483\) 15.1414 0.688956
\(484\) 9.74764 0.443075
\(485\) −4.40581 −0.200058
\(486\) −1.00000 −0.0453609
\(487\) 30.2349 1.37007 0.685037 0.728508i \(-0.259786\pi\)
0.685037 + 0.728508i \(0.259786\pi\)
\(488\) −14.3666 −0.650345
\(489\) −20.9825 −0.948863
\(490\) −4.13706 −0.186893
\(491\) −31.5883 −1.42556 −0.712781 0.701387i \(-0.752565\pi\)
−0.712781 + 0.701387i \(0.752565\pi\)
\(492\) −3.00969 −0.135687
\(493\) 21.3913 0.963417
\(494\) 0 0
\(495\) −4.55496 −0.204730
\(496\) −10.6528 −0.478325
\(497\) 25.2935 1.13457
\(498\) 7.71917 0.345904
\(499\) −27.2862 −1.22150 −0.610749 0.791824i \(-0.709132\pi\)
−0.610749 + 0.791824i \(0.709132\pi\)
\(500\) 1.00000 0.0447214
\(501\) 22.2325 0.993275
\(502\) 1.80864 0.0807237
\(503\) 29.6752 1.32315 0.661575 0.749879i \(-0.269888\pi\)
0.661575 + 0.749879i \(0.269888\pi\)
\(504\) −1.69202 −0.0753686
\(505\) 2.31767 0.103135
\(506\) −40.7609 −1.81204
\(507\) 0 0
\(508\) −3.14244 −0.139423
\(509\) −44.1661 −1.95763 −0.978815 0.204748i \(-0.934362\pi\)
−0.978815 + 0.204748i \(0.934362\pi\)
\(510\) −2.35690 −0.104365
\(511\) 11.3854 0.503659
\(512\) 1.00000 0.0441942
\(513\) 6.51573 0.287676
\(514\) −22.2664 −0.982127
\(515\) −9.00000 −0.396587
\(516\) 4.93900 0.217427
\(517\) 19.5235 0.858643
\(518\) −10.4668 −0.459885
\(519\) 11.6286 0.510440
\(520\) 0 0
\(521\) 20.1153 0.881267 0.440633 0.897687i \(-0.354754\pi\)
0.440633 + 0.897687i \(0.354754\pi\)
\(522\) 9.07606 0.397249
\(523\) 6.46117 0.282527 0.141264 0.989972i \(-0.454883\pi\)
0.141264 + 0.989972i \(0.454883\pi\)
\(524\) −3.77479 −0.164902
\(525\) 1.69202 0.0738459
\(526\) 8.08815 0.352660
\(527\) −25.1075 −1.09370
\(528\) 4.55496 0.198229
\(529\) 57.0790 2.48170
\(530\) −3.40581 −0.147939
\(531\) 4.32304 0.187604
\(532\) 11.0248 0.477984
\(533\) 0 0
\(534\) 9.12498 0.394877
\(535\) −3.06100 −0.132339
\(536\) −3.24698 −0.140248
\(537\) 15.7875 0.681279
\(538\) 16.6843 0.719309
\(539\) 18.8442 0.811675
\(540\) −1.00000 −0.0430331
\(541\) 34.0097 1.46219 0.731095 0.682275i \(-0.239009\pi\)
0.731095 + 0.682275i \(0.239009\pi\)
\(542\) 27.4077 1.17726
\(543\) 1.91185 0.0820455
\(544\) 2.35690 0.101051
\(545\) 7.45473 0.319326
\(546\) 0 0
\(547\) 14.6907 0.628129 0.314064 0.949402i \(-0.398309\pi\)
0.314064 + 0.949402i \(0.398309\pi\)
\(548\) 14.5894 0.623228
\(549\) −14.3666 −0.613151
\(550\) −4.55496 −0.194224
\(551\) −59.1372 −2.51933
\(552\) −8.94869 −0.380882
\(553\) −9.61058 −0.408683
\(554\) −16.0411 −0.681523
\(555\) −6.18598 −0.262580
\(556\) −14.3599 −0.608995
\(557\) 7.00969 0.297010 0.148505 0.988912i \(-0.452554\pi\)
0.148505 + 0.988912i \(0.452554\pi\)
\(558\) −10.6528 −0.450969
\(559\) 0 0
\(560\) −1.69202 −0.0715010
\(561\) 10.7356 0.453256
\(562\) −10.0935 −0.425770
\(563\) −30.2131 −1.27333 −0.636666 0.771140i \(-0.719687\pi\)
−0.636666 + 0.771140i \(0.719687\pi\)
\(564\) 4.28621 0.180482
\(565\) −19.9812 −0.840616
\(566\) 12.1021 0.508691
\(567\) −1.69202 −0.0710582
\(568\) −14.9487 −0.627233
\(569\) 14.8436 0.622274 0.311137 0.950365i \(-0.399290\pi\)
0.311137 + 0.950365i \(0.399290\pi\)
\(570\) 6.51573 0.272914
\(571\) 0.965557 0.0404073 0.0202037 0.999796i \(-0.493569\pi\)
0.0202037 + 0.999796i \(0.493569\pi\)
\(572\) 0 0
\(573\) −24.3260 −1.01623
\(574\) −5.09246 −0.212555
\(575\) 8.94869 0.373186
\(576\) 1.00000 0.0416667
\(577\) 27.8431 1.15912 0.579561 0.814929i \(-0.303224\pi\)
0.579561 + 0.814929i \(0.303224\pi\)
\(578\) −11.4450 −0.476051
\(579\) 0.521106 0.0216564
\(580\) 9.07606 0.376863
\(581\) 13.0610 0.541862
\(582\) 4.40581 0.182627
\(583\) 15.5133 0.642497
\(584\) −6.72886 −0.278442
\(585\) 0 0
\(586\) 4.01075 0.165683
\(587\) −1.47889 −0.0610405 −0.0305202 0.999534i \(-0.509716\pi\)
−0.0305202 + 0.999534i \(0.509716\pi\)
\(588\) 4.13706 0.170610
\(589\) 69.4107 2.86002
\(590\) 4.32304 0.177977
\(591\) 4.87502 0.200531
\(592\) 6.18598 0.254242
\(593\) −16.8552 −0.692159 −0.346079 0.938205i \(-0.612487\pi\)
−0.346079 + 0.938205i \(0.612487\pi\)
\(594\) 4.55496 0.186892
\(595\) −3.98792 −0.163489
\(596\) 0.252356 0.0103369
\(597\) 18.4983 0.757084
\(598\) 0 0
\(599\) −24.4862 −1.00048 −0.500239 0.865887i \(-0.666755\pi\)
−0.500239 + 0.865887i \(0.666755\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −9.97344 −0.406825 −0.203413 0.979093i \(-0.565203\pi\)
−0.203413 + 0.979093i \(0.565203\pi\)
\(602\) 8.35690 0.340602
\(603\) −3.24698 −0.132227
\(604\) 11.0804 0.450854
\(605\) 9.74764 0.396298
\(606\) −2.31767 −0.0941488
\(607\) −35.4476 −1.43877 −0.719386 0.694611i \(-0.755577\pi\)
−0.719386 + 0.694611i \(0.755577\pi\)
\(608\) −6.51573 −0.264248
\(609\) 15.3569 0.622293
\(610\) −14.3666 −0.581686
\(611\) 0 0
\(612\) 2.35690 0.0952719
\(613\) 16.1782 0.653432 0.326716 0.945123i \(-0.394058\pi\)
0.326716 + 0.945123i \(0.394058\pi\)
\(614\) −15.2784 −0.616588
\(615\) −3.00969 −0.121362
\(616\) 7.70709 0.310527
\(617\) −17.6920 −0.712254 −0.356127 0.934438i \(-0.615903\pi\)
−0.356127 + 0.934438i \(0.615903\pi\)
\(618\) 9.00000 0.362033
\(619\) 1.23968 0.0498271 0.0249136 0.999690i \(-0.492069\pi\)
0.0249136 + 0.999690i \(0.492069\pi\)
\(620\) −10.6528 −0.427826
\(621\) −8.94869 −0.359099
\(622\) 25.4330 1.01977
\(623\) 15.4397 0.618577
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −19.1226 −0.764292
\(627\) −29.6789 −1.18526
\(628\) 5.03923 0.201087
\(629\) 14.5797 0.581331
\(630\) −1.69202 −0.0674117
\(631\) 33.7845 1.34494 0.672469 0.740125i \(-0.265234\pi\)
0.672469 + 0.740125i \(0.265234\pi\)
\(632\) 5.67994 0.225936
\(633\) −18.2500 −0.725371
\(634\) −20.3230 −0.807131
\(635\) −3.14244 −0.124704
\(636\) 3.40581 0.135049
\(637\) 0 0
\(638\) −41.3411 −1.63671
\(639\) −14.9487 −0.591361
\(640\) 1.00000 0.0395285
\(641\) 12.0785 0.477070 0.238535 0.971134i \(-0.423333\pi\)
0.238535 + 0.971134i \(0.423333\pi\)
\(642\) 3.06100 0.120808
\(643\) −41.1377 −1.62231 −0.811155 0.584831i \(-0.801161\pi\)
−0.811155 + 0.584831i \(0.801161\pi\)
\(644\) −15.1414 −0.596654
\(645\) 4.93900 0.194473
\(646\) −15.3569 −0.604209
\(647\) 23.2054 0.912297 0.456148 0.889904i \(-0.349229\pi\)
0.456148 + 0.889904i \(0.349229\pi\)
\(648\) 1.00000 0.0392837
\(649\) −19.6913 −0.772951
\(650\) 0 0
\(651\) −18.0248 −0.706446
\(652\) 20.9825 0.821740
\(653\) 7.77538 0.304274 0.152137 0.988359i \(-0.451384\pi\)
0.152137 + 0.988359i \(0.451384\pi\)
\(654\) −7.45473 −0.291503
\(655\) −3.77479 −0.147493
\(656\) 3.00969 0.117509
\(657\) −6.72886 −0.262518
\(658\) 7.25236 0.282726
\(659\) −0.349600 −0.0136185 −0.00680924 0.999977i \(-0.502167\pi\)
−0.00680924 + 0.999977i \(0.502167\pi\)
\(660\) 4.55496 0.177302
\(661\) 19.7855 0.769568 0.384784 0.923007i \(-0.374276\pi\)
0.384784 + 0.923007i \(0.374276\pi\)
\(662\) −22.3588 −0.869000
\(663\) 0 0
\(664\) −7.71917 −0.299562
\(665\) 11.0248 0.427522
\(666\) 6.18598 0.239702
\(667\) 81.2189 3.14481
\(668\) −22.2325 −0.860201
\(669\) −8.54048 −0.330194
\(670\) −3.24698 −0.125442
\(671\) 65.4392 2.52625
\(672\) 1.69202 0.0652711
\(673\) −16.6146 −0.640447 −0.320223 0.947342i \(-0.603758\pi\)
−0.320223 + 0.947342i \(0.603758\pi\)
\(674\) −3.67025 −0.141373
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −5.85517 −0.225032 −0.112516 0.993650i \(-0.535891\pi\)
−0.112516 + 0.993650i \(0.535891\pi\)
\(678\) 19.9812 0.767374
\(679\) 7.45473 0.286086
\(680\) 2.35690 0.0903828
\(681\) 9.61596 0.368484
\(682\) 48.5230 1.85804
\(683\) −2.19029 −0.0838092 −0.0419046 0.999122i \(-0.513343\pi\)
−0.0419046 + 0.999122i \(0.513343\pi\)
\(684\) −6.51573 −0.249135
\(685\) 14.5894 0.557432
\(686\) 18.8442 0.719473
\(687\) 13.4058 0.511464
\(688\) −4.93900 −0.188298
\(689\) 0 0
\(690\) −8.94869 −0.340671
\(691\) 11.8576 0.451083 0.225541 0.974234i \(-0.427585\pi\)
0.225541 + 0.974234i \(0.427585\pi\)
\(692\) −11.6286 −0.442054
\(693\) 7.70709 0.292768
\(694\) −21.7832 −0.826877
\(695\) −14.3599 −0.544701
\(696\) −9.07606 −0.344027
\(697\) 7.09352 0.268686
\(698\) −1.74333 −0.0659861
\(699\) 0.153457 0.00580427
\(700\) −1.69202 −0.0639524
\(701\) 22.7627 0.859736 0.429868 0.902892i \(-0.358560\pi\)
0.429868 + 0.902892i \(0.358560\pi\)
\(702\) 0 0
\(703\) −40.3062 −1.52018
\(704\) −4.55496 −0.171671
\(705\) 4.28621 0.161428
\(706\) 6.09544 0.229405
\(707\) −3.92154 −0.147485
\(708\) −4.32304 −0.162470
\(709\) 18.5854 0.697988 0.348994 0.937125i \(-0.386523\pi\)
0.348994 + 0.937125i \(0.386523\pi\)
\(710\) −14.9487 −0.561014
\(711\) 5.67994 0.213014
\(712\) −9.12498 −0.341973
\(713\) −95.3285 −3.57008
\(714\) 3.98792 0.149244
\(715\) 0 0
\(716\) −15.7875 −0.590005
\(717\) −14.5851 −0.544690
\(718\) 25.9627 0.968919
\(719\) −36.1575 −1.34845 −0.674224 0.738527i \(-0.735522\pi\)
−0.674224 + 0.738527i \(0.735522\pi\)
\(720\) 1.00000 0.0372678
\(721\) 15.2282 0.567128
\(722\) 23.4547 0.872895
\(723\) −3.40044 −0.126464
\(724\) −1.91185 −0.0710535
\(725\) 9.07606 0.337077
\(726\) −9.74764 −0.361769
\(727\) 19.0164 0.705279 0.352639 0.935759i \(-0.385284\pi\)
0.352639 + 0.935759i \(0.385284\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −6.72886 −0.249046
\(731\) −11.6407 −0.430547
\(732\) 14.3666 0.531004
\(733\) 8.89248 0.328451 0.164226 0.986423i \(-0.447488\pi\)
0.164226 + 0.986423i \(0.447488\pi\)
\(734\) 27.8799 1.02907
\(735\) 4.13706 0.152598
\(736\) 8.94869 0.329853
\(737\) 14.7899 0.544791
\(738\) 3.00969 0.110788
\(739\) −30.8595 −1.13518 −0.567592 0.823310i \(-0.692125\pi\)
−0.567592 + 0.823310i \(0.692125\pi\)
\(740\) 6.18598 0.227401
\(741\) 0 0
\(742\) 5.76271 0.211556
\(743\) 18.6300 0.683467 0.341733 0.939797i \(-0.388986\pi\)
0.341733 + 0.939797i \(0.388986\pi\)
\(744\) 10.6528 0.390550
\(745\) 0.252356 0.00924562
\(746\) −25.1400 −0.920443
\(747\) −7.71917 −0.282430
\(748\) −10.7356 −0.392531
\(749\) 5.17928 0.189247
\(750\) −1.00000 −0.0365148
\(751\) 12.5147 0.456667 0.228333 0.973583i \(-0.426672\pi\)
0.228333 + 0.973583i \(0.426672\pi\)
\(752\) −4.28621 −0.156302
\(753\) −1.80864 −0.0659106
\(754\) 0 0
\(755\) 11.0804 0.403256
\(756\) 1.69202 0.0615382
\(757\) −4.75494 −0.172821 −0.0864106 0.996260i \(-0.527540\pi\)
−0.0864106 + 0.996260i \(0.527540\pi\)
\(758\) −15.7463 −0.571932
\(759\) 40.7609 1.47953
\(760\) −6.51573 −0.236350
\(761\) −21.7748 −0.789336 −0.394668 0.918824i \(-0.629140\pi\)
−0.394668 + 0.918824i \(0.629140\pi\)
\(762\) 3.14244 0.113839
\(763\) −12.6136 −0.456642
\(764\) 24.3260 0.880085
\(765\) 2.35690 0.0852137
\(766\) 11.8183 0.427014
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 11.4045 0.411256 0.205628 0.978630i \(-0.434076\pi\)
0.205628 + 0.978630i \(0.434076\pi\)
\(770\) 7.70709 0.277744
\(771\) 22.2664 0.801903
\(772\) −0.521106 −0.0187550
\(773\) 49.6118 1.78441 0.892206 0.451630i \(-0.149157\pi\)
0.892206 + 0.451630i \(0.149157\pi\)
\(774\) −4.93900 −0.177529
\(775\) −10.6528 −0.382660
\(776\) −4.40581 −0.158159
\(777\) 10.4668 0.375495
\(778\) −6.14675 −0.220372
\(779\) −19.6103 −0.702613
\(780\) 0 0
\(781\) 68.0907 2.43648
\(782\) 21.0911 0.754217
\(783\) −9.07606 −0.324352
\(784\) −4.13706 −0.147752
\(785\) 5.03923 0.179858
\(786\) 3.77479 0.134642
\(787\) 26.5532 0.946518 0.473259 0.880923i \(-0.343078\pi\)
0.473259 + 0.880923i \(0.343078\pi\)
\(788\) −4.87502 −0.173665
\(789\) −8.08815 −0.287946
\(790\) 5.67994 0.202083
\(791\) 33.8086 1.20210
\(792\) −4.55496 −0.161853
\(793\) 0 0
\(794\) −5.92825 −0.210386
\(795\) 3.40581 0.120792
\(796\) −18.4983 −0.655654
\(797\) 9.44803 0.334666 0.167333 0.985900i \(-0.446484\pi\)
0.167333 + 0.985900i \(0.446484\pi\)
\(798\) −11.0248 −0.390272
\(799\) −10.1021 −0.357388
\(800\) 1.00000 0.0353553
\(801\) −9.12498 −0.322415
\(802\) 9.49934 0.335433
\(803\) 30.6497 1.08160
\(804\) 3.24698 0.114512
\(805\) −15.1414 −0.533663
\(806\) 0 0
\(807\) −16.6843 −0.587313
\(808\) 2.31767 0.0815352
\(809\) −27.3376 −0.961140 −0.480570 0.876956i \(-0.659570\pi\)
−0.480570 + 0.876956i \(0.659570\pi\)
\(810\) 1.00000 0.0351364
\(811\) −2.86592 −0.100636 −0.0503180 0.998733i \(-0.516023\pi\)
−0.0503180 + 0.998733i \(0.516023\pi\)
\(812\) −15.3569 −0.538921
\(813\) −27.4077 −0.961231
\(814\) −28.1769 −0.987599
\(815\) 20.9825 0.734986
\(816\) −2.35690 −0.0825079
\(817\) 32.1812 1.12588
\(818\) −9.00000 −0.314678
\(819\) 0 0
\(820\) 3.00969 0.105103
\(821\) 16.9739 0.592394 0.296197 0.955127i \(-0.404282\pi\)
0.296197 + 0.955127i \(0.404282\pi\)
\(822\) −14.5894 −0.508864
\(823\) −22.2083 −0.774134 −0.387067 0.922052i \(-0.626512\pi\)
−0.387067 + 0.922052i \(0.626512\pi\)
\(824\) −9.00000 −0.313530
\(825\) 4.55496 0.158583
\(826\) −7.31468 −0.254510
\(827\) 36.3618 1.26442 0.632212 0.774796i \(-0.282147\pi\)
0.632212 + 0.774796i \(0.282147\pi\)
\(828\) 8.94869 0.310988
\(829\) −26.5478 −0.922042 −0.461021 0.887389i \(-0.652517\pi\)
−0.461021 + 0.887389i \(0.652517\pi\)
\(830\) −7.71917 −0.267936
\(831\) 16.0411 0.556461
\(832\) 0 0
\(833\) −9.75063 −0.337839
\(834\) 14.3599 0.497242
\(835\) −22.2325 −0.769388
\(836\) 29.6789 1.02647
\(837\) 10.6528 0.368214
\(838\) 30.6450 1.05861
\(839\) 26.7668 0.924091 0.462046 0.886856i \(-0.347116\pi\)
0.462046 + 0.886856i \(0.347116\pi\)
\(840\) 1.69202 0.0583803
\(841\) 53.3749 1.84052
\(842\) 4.03252 0.138970
\(843\) 10.0935 0.347639
\(844\) 18.2500 0.628190
\(845\) 0 0
\(846\) −4.28621 −0.147363
\(847\) −16.4932 −0.566714
\(848\) −3.40581 −0.116956
\(849\) −12.1021 −0.415345
\(850\) 2.35690 0.0808409
\(851\) 55.3564 1.89759
\(852\) 14.9487 0.512134
\(853\) −50.3967 −1.72555 −0.862775 0.505587i \(-0.831276\pi\)
−0.862775 + 0.505587i \(0.831276\pi\)
\(854\) 24.3086 0.831822
\(855\) −6.51573 −0.222833
\(856\) −3.06100 −0.104623
\(857\) 27.5114 0.939772 0.469886 0.882727i \(-0.344295\pi\)
0.469886 + 0.882727i \(0.344295\pi\)
\(858\) 0 0
\(859\) 2.26098 0.0771436 0.0385718 0.999256i \(-0.487719\pi\)
0.0385718 + 0.999256i \(0.487719\pi\)
\(860\) −4.93900 −0.168419
\(861\) 5.09246 0.173550
\(862\) −27.6179 −0.940669
\(863\) 2.18406 0.0743463 0.0371732 0.999309i \(-0.488165\pi\)
0.0371732 + 0.999309i \(0.488165\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −11.6286 −0.395385
\(866\) 10.6756 0.362773
\(867\) 11.4450 0.388694
\(868\) 18.0248 0.611800
\(869\) −25.8719 −0.877644
\(870\) −9.07606 −0.307707
\(871\) 0 0
\(872\) 7.45473 0.252449
\(873\) −4.40581 −0.149114
\(874\) −58.3072 −1.97227
\(875\) −1.69202 −0.0572008
\(876\) 6.72886 0.227347
\(877\) 19.9815 0.674727 0.337363 0.941375i \(-0.390465\pi\)
0.337363 + 0.941375i \(0.390465\pi\)
\(878\) 25.9299 0.875092
\(879\) −4.01075 −0.135279
\(880\) −4.55496 −0.153548
\(881\) −38.4801 −1.29643 −0.648213 0.761459i \(-0.724484\pi\)
−0.648213 + 0.761459i \(0.724484\pi\)
\(882\) −4.13706 −0.139302
\(883\) −37.5512 −1.26370 −0.631850 0.775091i \(-0.717704\pi\)
−0.631850 + 0.775091i \(0.717704\pi\)
\(884\) 0 0
\(885\) −4.32304 −0.145318
\(886\) −35.3749 −1.18844
\(887\) 18.5676 0.623440 0.311720 0.950174i \(-0.399095\pi\)
0.311720 + 0.950174i \(0.399095\pi\)
\(888\) −6.18598 −0.207588
\(889\) 5.31708 0.178329
\(890\) −9.12498 −0.305870
\(891\) −4.55496 −0.152597
\(892\) 8.54048 0.285957
\(893\) 27.9278 0.934567
\(894\) −0.252356 −0.00844006
\(895\) −15.7875 −0.527717
\(896\) −1.69202 −0.0565265
\(897\) 0 0
\(898\) −6.24698 −0.208464
\(899\) −96.6854 −3.22464
\(900\) 1.00000 0.0333333
\(901\) −8.02715 −0.267423
\(902\) −13.7090 −0.456460
\(903\) −8.35690 −0.278100
\(904\) −19.9812 −0.664565
\(905\) −1.91185 −0.0635522
\(906\) −11.0804 −0.368121
\(907\) 60.1648 1.99774 0.998870 0.0475319i \(-0.0151356\pi\)
0.998870 + 0.0475319i \(0.0151356\pi\)
\(908\) −9.61596 −0.319117
\(909\) 2.31767 0.0768722
\(910\) 0 0
\(911\) −37.0737 −1.22831 −0.614153 0.789187i \(-0.710502\pi\)
−0.614153 + 0.789187i \(0.710502\pi\)
\(912\) 6.51573 0.215757
\(913\) 35.1605 1.16364
\(914\) −29.6383 −0.980348
\(915\) 14.3666 0.474945
\(916\) −13.4058 −0.442941
\(917\) 6.38703 0.210918
\(918\) −2.35690 −0.0777892
\(919\) −50.5526 −1.66758 −0.833788 0.552085i \(-0.813832\pi\)
−0.833788 + 0.552085i \(0.813832\pi\)
\(920\) 8.94869 0.295030
\(921\) 15.2784 0.503442
\(922\) 34.2911 1.12932
\(923\) 0 0
\(924\) −7.70709 −0.253545
\(925\) 6.18598 0.203394
\(926\) 24.1769 0.794502
\(927\) −9.00000 −0.295599
\(928\) 9.07606 0.297936
\(929\) −12.7269 −0.417557 −0.208779 0.977963i \(-0.566949\pi\)
−0.208779 + 0.977963i \(0.566949\pi\)
\(930\) 10.6528 0.349319
\(931\) 26.9560 0.883447
\(932\) −0.153457 −0.00502664
\(933\) −25.4330 −0.832638
\(934\) −0.521106 −0.0170511
\(935\) −10.7356 −0.351090
\(936\) 0 0
\(937\) −35.3279 −1.15411 −0.577057 0.816704i \(-0.695799\pi\)
−0.577057 + 0.816704i \(0.695799\pi\)
\(938\) 5.49396 0.179384
\(939\) 19.1226 0.624042
\(940\) −4.28621 −0.139801
\(941\) 42.0267 1.37003 0.685015 0.728529i \(-0.259796\pi\)
0.685015 + 0.728529i \(0.259796\pi\)
\(942\) −5.03923 −0.164187
\(943\) 26.9328 0.877052
\(944\) 4.32304 0.140703
\(945\) 1.69202 0.0550415
\(946\) 22.4969 0.731439
\(947\) 4.63640 0.150663 0.0753314 0.997159i \(-0.475999\pi\)
0.0753314 + 0.997159i \(0.475999\pi\)
\(948\) −5.67994 −0.184476
\(949\) 0 0
\(950\) −6.51573 −0.211398
\(951\) 20.3230 0.659020
\(952\) −3.98792 −0.129249
\(953\) −41.8998 −1.35727 −0.678633 0.734477i \(-0.737427\pi\)
−0.678633 + 0.734477i \(0.737427\pi\)
\(954\) −3.40581 −0.110267
\(955\) 24.3260 0.787172
\(956\) 14.5851 0.471715
\(957\) 41.3411 1.33637
\(958\) 10.4168 0.336552
\(959\) −24.6856 −0.797139
\(960\) −1.00000 −0.0322749
\(961\) 82.4820 2.66071
\(962\) 0 0
\(963\) −3.06100 −0.0986393
\(964\) 3.40044 0.109521
\(965\) −0.521106 −0.0167750
\(966\) 15.1414 0.487166
\(967\) −48.3545 −1.55498 −0.777488 0.628898i \(-0.783506\pi\)
−0.777488 + 0.628898i \(0.783506\pi\)
\(968\) 9.74764 0.313301
\(969\) 15.3569 0.493335
\(970\) −4.40581 −0.141462
\(971\) −18.9326 −0.607575 −0.303787 0.952740i \(-0.598251\pi\)
−0.303787 + 0.952740i \(0.598251\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 24.2972 0.778933
\(974\) 30.2349 0.968789
\(975\) 0 0
\(976\) −14.3666 −0.459863
\(977\) −54.7972 −1.75312 −0.876558 0.481296i \(-0.840166\pi\)
−0.876558 + 0.481296i \(0.840166\pi\)
\(978\) −20.9825 −0.670948
\(979\) 41.5639 1.32839
\(980\) −4.13706 −0.132154
\(981\) 7.45473 0.238011
\(982\) −31.5883 −1.00802
\(983\) 24.6431 0.785993 0.392996 0.919540i \(-0.371438\pi\)
0.392996 + 0.919540i \(0.371438\pi\)
\(984\) −3.00969 −0.0959454
\(985\) −4.87502 −0.155331
\(986\) 21.3913 0.681239
\(987\) −7.25236 −0.230845
\(988\) 0 0
\(989\) −44.1976 −1.40540
\(990\) −4.55496 −0.144766
\(991\) 23.1612 0.735741 0.367870 0.929877i \(-0.380087\pi\)
0.367870 + 0.929877i \(0.380087\pi\)
\(992\) −10.6528 −0.338227
\(993\) 22.3588 0.709536
\(994\) 25.2935 0.802261
\(995\) −18.4983 −0.586435
\(996\) 7.71917 0.244591
\(997\) −11.5700 −0.366426 −0.183213 0.983073i \(-0.558650\pi\)
−0.183213 + 0.983073i \(0.558650\pi\)
\(998\) −27.2862 −0.863730
\(999\) −6.18598 −0.195716
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.bt.1.1 yes 3
13.5 odd 4 5070.2.b.s.1351.1 6
13.8 odd 4 5070.2.b.s.1351.6 6
13.12 even 2 5070.2.a.bk.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bk.1.3 3 13.12 even 2
5070.2.a.bt.1.1 yes 3 1.1 even 1 trivial
5070.2.b.s.1351.1 6 13.5 odd 4
5070.2.b.s.1351.6 6 13.8 odd 4