Properties

Label 43.2.a.b.1.2
Level $43$
Weight $2$
Character 43.1
Self dual yes
Analytic conductor $0.343$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.343356728692\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{2} -1.41421 q^{3} +0.585786 q^{5} -2.00000 q^{6} -0.585786 q^{7} -2.82843 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} -1.41421 q^{3} +0.585786 q^{5} -2.00000 q^{6} -0.585786 q^{7} -2.82843 q^{8} -1.00000 q^{9} +0.828427 q^{10} +1.82843 q^{11} +3.82843 q^{13} -0.828427 q^{14} -0.828427 q^{15} -4.00000 q^{16} +7.82843 q^{17} -1.41421 q^{18} -4.82843 q^{19} +0.828427 q^{21} +2.58579 q^{22} -4.65685 q^{23} +4.00000 q^{24} -4.65685 q^{25} +5.41421 q^{26} +5.65685 q^{27} +4.24264 q^{29} -1.17157 q^{30} -3.00000 q^{31} -2.58579 q^{33} +11.0711 q^{34} -0.343146 q^{35} -8.48528 q^{37} -6.82843 q^{38} -5.41421 q^{39} -1.65685 q^{40} -3.82843 q^{41} +1.17157 q^{42} +1.00000 q^{43} -0.585786 q^{45} -6.58579 q^{46} +6.00000 q^{47} +5.65685 q^{48} -6.65685 q^{49} -6.58579 q^{50} -11.0711 q^{51} +8.17157 q^{53} +8.00000 q^{54} +1.07107 q^{55} +1.65685 q^{56} +6.82843 q^{57} +6.00000 q^{58} +0.828427 q^{59} +8.24264 q^{61} -4.24264 q^{62} +0.585786 q^{63} +8.00000 q^{64} +2.24264 q^{65} -3.65685 q^{66} +9.48528 q^{67} +6.58579 q^{69} -0.485281 q^{70} -8.82843 q^{71} +2.82843 q^{72} -7.75736 q^{73} -12.0000 q^{74} +6.58579 q^{75} -1.07107 q^{77} -7.65685 q^{78} -0.828427 q^{79} -2.34315 q^{80} -5.00000 q^{81} -5.41421 q^{82} +14.6569 q^{83} +4.58579 q^{85} +1.41421 q^{86} -6.00000 q^{87} -5.17157 q^{88} -10.2426 q^{89} -0.828427 q^{90} -2.24264 q^{91} +4.24264 q^{93} +8.48528 q^{94} -2.82843 q^{95} -3.82843 q^{97} -9.41421 q^{98} -1.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 4 q^{6} - 4 q^{7} - 2 q^{9} + O(q^{10}) \) \( 2 q + 4 q^{5} - 4 q^{6} - 4 q^{7} - 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{13} + 4 q^{14} + 4 q^{15} - 8 q^{16} + 10 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{22} + 2 q^{23} + 8 q^{24} + 2 q^{25} + 8 q^{26} - 8 q^{30} - 6 q^{31} - 8 q^{33} + 8 q^{34} - 12 q^{35} - 8 q^{38} - 8 q^{39} + 8 q^{40} - 2 q^{41} + 8 q^{42} + 2 q^{43} - 4 q^{45} - 16 q^{46} + 12 q^{47} - 2 q^{49} - 16 q^{50} - 8 q^{51} + 22 q^{53} + 16 q^{54} - 12 q^{55} - 8 q^{56} + 8 q^{57} + 12 q^{58} - 4 q^{59} + 8 q^{61} + 4 q^{63} + 16 q^{64} - 4 q^{65} + 4 q^{66} + 2 q^{67} + 16 q^{69} + 16 q^{70} - 12 q^{71} - 24 q^{73} - 24 q^{74} + 16 q^{75} + 12 q^{77} - 4 q^{78} + 4 q^{79} - 16 q^{80} - 10 q^{81} - 8 q^{82} + 18 q^{83} + 12 q^{85} - 12 q^{87} - 16 q^{88} - 12 q^{89} + 4 q^{90} + 4 q^{91} - 2 q^{97} - 16 q^{98} + 2 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 0 0
\(5\) 0.585786 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(6\) −2.00000 −0.816497
\(7\) −0.585786 −0.221406 −0.110703 0.993854i \(-0.535310\pi\)
−0.110703 + 0.993854i \(0.535310\pi\)
\(8\) −2.82843 −1.00000
\(9\) −1.00000 −0.333333
\(10\) 0.828427 0.261972
\(11\) 1.82843 0.551292 0.275646 0.961259i \(-0.411108\pi\)
0.275646 + 0.961259i \(0.411108\pi\)
\(12\) 0 0
\(13\) 3.82843 1.06181 0.530907 0.847430i \(-0.321851\pi\)
0.530907 + 0.847430i \(0.321851\pi\)
\(14\) −0.828427 −0.221406
\(15\) −0.828427 −0.213899
\(16\) −4.00000 −1.00000
\(17\) 7.82843 1.89867 0.949336 0.314262i \(-0.101757\pi\)
0.949336 + 0.314262i \(0.101757\pi\)
\(18\) −1.41421 −0.333333
\(19\) −4.82843 −1.10772 −0.553859 0.832611i \(-0.686845\pi\)
−0.553859 + 0.832611i \(0.686845\pi\)
\(20\) 0 0
\(21\) 0.828427 0.180778
\(22\) 2.58579 0.551292
\(23\) −4.65685 −0.971021 −0.485511 0.874231i \(-0.661366\pi\)
−0.485511 + 0.874231i \(0.661366\pi\)
\(24\) 4.00000 0.816497
\(25\) −4.65685 −0.931371
\(26\) 5.41421 1.06181
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) 4.24264 0.787839 0.393919 0.919145i \(-0.371119\pi\)
0.393919 + 0.919145i \(0.371119\pi\)
\(30\) −1.17157 −0.213899
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) −2.58579 −0.450128
\(34\) 11.0711 1.89867
\(35\) −0.343146 −0.0580022
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) −6.82843 −1.10772
\(39\) −5.41421 −0.866968
\(40\) −1.65685 −0.261972
\(41\) −3.82843 −0.597900 −0.298950 0.954269i \(-0.596636\pi\)
−0.298950 + 0.954269i \(0.596636\pi\)
\(42\) 1.17157 0.180778
\(43\) 1.00000 0.152499
\(44\) 0 0
\(45\) −0.585786 −0.0873239
\(46\) −6.58579 −0.971021
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 5.65685 0.816497
\(49\) −6.65685 −0.950979
\(50\) −6.58579 −0.931371
\(51\) −11.0711 −1.55026
\(52\) 0 0
\(53\) 8.17157 1.12245 0.561226 0.827663i \(-0.310330\pi\)
0.561226 + 0.827663i \(0.310330\pi\)
\(54\) 8.00000 1.08866
\(55\) 1.07107 0.144423
\(56\) 1.65685 0.221406
\(57\) 6.82843 0.904447
\(58\) 6.00000 0.787839
\(59\) 0.828427 0.107852 0.0539260 0.998545i \(-0.482826\pi\)
0.0539260 + 0.998545i \(0.482826\pi\)
\(60\) 0 0
\(61\) 8.24264 1.05536 0.527681 0.849443i \(-0.323062\pi\)
0.527681 + 0.849443i \(0.323062\pi\)
\(62\) −4.24264 −0.538816
\(63\) 0.585786 0.0738022
\(64\) 8.00000 1.00000
\(65\) 2.24264 0.278165
\(66\) −3.65685 −0.450128
\(67\) 9.48528 1.15881 0.579406 0.815039i \(-0.303285\pi\)
0.579406 + 0.815039i \(0.303285\pi\)
\(68\) 0 0
\(69\) 6.58579 0.792836
\(70\) −0.485281 −0.0580022
\(71\) −8.82843 −1.04774 −0.523871 0.851798i \(-0.675513\pi\)
−0.523871 + 0.851798i \(0.675513\pi\)
\(72\) 2.82843 0.333333
\(73\) −7.75736 −0.907930 −0.453965 0.891019i \(-0.649991\pi\)
−0.453965 + 0.891019i \(0.649991\pi\)
\(74\) −12.0000 −1.39497
\(75\) 6.58579 0.760461
\(76\) 0 0
\(77\) −1.07107 −0.122060
\(78\) −7.65685 −0.866968
\(79\) −0.828427 −0.0932053 −0.0466027 0.998914i \(-0.514839\pi\)
−0.0466027 + 0.998914i \(0.514839\pi\)
\(80\) −2.34315 −0.261972
\(81\) −5.00000 −0.555556
\(82\) −5.41421 −0.597900
\(83\) 14.6569 1.60880 0.804399 0.594089i \(-0.202487\pi\)
0.804399 + 0.594089i \(0.202487\pi\)
\(84\) 0 0
\(85\) 4.58579 0.497398
\(86\) 1.41421 0.152499
\(87\) −6.00000 −0.643268
\(88\) −5.17157 −0.551292
\(89\) −10.2426 −1.08572 −0.542859 0.839824i \(-0.682658\pi\)
−0.542859 + 0.839824i \(0.682658\pi\)
\(90\) −0.828427 −0.0873239
\(91\) −2.24264 −0.235093
\(92\) 0 0
\(93\) 4.24264 0.439941
\(94\) 8.48528 0.875190
\(95\) −2.82843 −0.290191
\(96\) 0 0
\(97\) −3.82843 −0.388718 −0.194359 0.980930i \(-0.562263\pi\)
−0.194359 + 0.980930i \(0.562263\pi\)
\(98\) −9.41421 −0.950979
\(99\) −1.82843 −0.183764
\(100\) 0 0
\(101\) 0.171573 0.0170721 0.00853607 0.999964i \(-0.497283\pi\)
0.00853607 + 0.999964i \(0.497283\pi\)
\(102\) −15.6569 −1.55026
\(103\) 17.4853 1.72288 0.861438 0.507863i \(-0.169564\pi\)
0.861438 + 0.507863i \(0.169564\pi\)
\(104\) −10.8284 −1.06181
\(105\) 0.485281 0.0473586
\(106\) 11.5563 1.12245
\(107\) −11.6569 −1.12691 −0.563455 0.826147i \(-0.690528\pi\)
−0.563455 + 0.826147i \(0.690528\pi\)
\(108\) 0 0
\(109\) 13.9706 1.33814 0.669069 0.743201i \(-0.266693\pi\)
0.669069 + 0.743201i \(0.266693\pi\)
\(110\) 1.51472 0.144423
\(111\) 12.0000 1.13899
\(112\) 2.34315 0.221406
\(113\) −1.17157 −0.110212 −0.0551062 0.998481i \(-0.517550\pi\)
−0.0551062 + 0.998481i \(0.517550\pi\)
\(114\) 9.65685 0.904447
\(115\) −2.72792 −0.254380
\(116\) 0 0
\(117\) −3.82843 −0.353938
\(118\) 1.17157 0.107852
\(119\) −4.58579 −0.420378
\(120\) 2.34315 0.213899
\(121\) −7.65685 −0.696078
\(122\) 11.6569 1.05536
\(123\) 5.41421 0.488183
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0.828427 0.0738022
\(127\) −1.82843 −0.162247 −0.0811233 0.996704i \(-0.525851\pi\)
−0.0811233 + 0.996704i \(0.525851\pi\)
\(128\) 11.3137 1.00000
\(129\) −1.41421 −0.124515
\(130\) 3.17157 0.278165
\(131\) −1.65685 −0.144760 −0.0723800 0.997377i \(-0.523059\pi\)
−0.0723800 + 0.997377i \(0.523059\pi\)
\(132\) 0 0
\(133\) 2.82843 0.245256
\(134\) 13.4142 1.15881
\(135\) 3.31371 0.285199
\(136\) −22.1421 −1.89867
\(137\) 2.48528 0.212332 0.106166 0.994348i \(-0.466143\pi\)
0.106166 + 0.994348i \(0.466143\pi\)
\(138\) 9.31371 0.792836
\(139\) −11.4853 −0.974169 −0.487084 0.873355i \(-0.661940\pi\)
−0.487084 + 0.873355i \(0.661940\pi\)
\(140\) 0 0
\(141\) −8.48528 −0.714590
\(142\) −12.4853 −1.04774
\(143\) 7.00000 0.585369
\(144\) 4.00000 0.333333
\(145\) 2.48528 0.206391
\(146\) −10.9706 −0.907930
\(147\) 9.41421 0.776471
\(148\) 0 0
\(149\) −8.48528 −0.695141 −0.347571 0.937654i \(-0.612993\pi\)
−0.347571 + 0.937654i \(0.612993\pi\)
\(150\) 9.31371 0.760461
\(151\) 9.75736 0.794043 0.397021 0.917809i \(-0.370044\pi\)
0.397021 + 0.917809i \(0.370044\pi\)
\(152\) 13.6569 1.10772
\(153\) −7.82843 −0.632891
\(154\) −1.51472 −0.122060
\(155\) −1.75736 −0.141154
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −1.17157 −0.0932053
\(159\) −11.5563 −0.916478
\(160\) 0 0
\(161\) 2.72792 0.214990
\(162\) −7.07107 −0.555556
\(163\) −20.2426 −1.58553 −0.792763 0.609530i \(-0.791358\pi\)
−0.792763 + 0.609530i \(0.791358\pi\)
\(164\) 0 0
\(165\) −1.51472 −0.117921
\(166\) 20.7279 1.60880
\(167\) 8.31371 0.643334 0.321667 0.946853i \(-0.395757\pi\)
0.321667 + 0.946853i \(0.395757\pi\)
\(168\) −2.34315 −0.180778
\(169\) 1.65685 0.127450
\(170\) 6.48528 0.497398
\(171\) 4.82843 0.369239
\(172\) 0 0
\(173\) 12.3431 0.938432 0.469216 0.883083i \(-0.344537\pi\)
0.469216 + 0.883083i \(0.344537\pi\)
\(174\) −8.48528 −0.643268
\(175\) 2.72792 0.206212
\(176\) −7.31371 −0.551292
\(177\) −1.17157 −0.0880608
\(178\) −14.4853 −1.08572
\(179\) −7.41421 −0.554164 −0.277082 0.960846i \(-0.589367\pi\)
−0.277082 + 0.960846i \(0.589367\pi\)
\(180\) 0 0
\(181\) −15.3137 −1.13826 −0.569129 0.822248i \(-0.692720\pi\)
−0.569129 + 0.822248i \(0.692720\pi\)
\(182\) −3.17157 −0.235093
\(183\) −11.6569 −0.861699
\(184\) 13.1716 0.971021
\(185\) −4.97056 −0.365443
\(186\) 6.00000 0.439941
\(187\) 14.3137 1.04672
\(188\) 0 0
\(189\) −3.31371 −0.241037
\(190\) −4.00000 −0.290191
\(191\) 22.1421 1.60215 0.801074 0.598565i \(-0.204262\pi\)
0.801074 + 0.598565i \(0.204262\pi\)
\(192\) −11.3137 −0.816497
\(193\) −17.9706 −1.29355 −0.646775 0.762681i \(-0.723883\pi\)
−0.646775 + 0.762681i \(0.723883\pi\)
\(194\) −5.41421 −0.388718
\(195\) −3.17157 −0.227121
\(196\) 0 0
\(197\) 14.1421 1.00759 0.503793 0.863825i \(-0.331938\pi\)
0.503793 + 0.863825i \(0.331938\pi\)
\(198\) −2.58579 −0.183764
\(199\) −3.65685 −0.259228 −0.129614 0.991565i \(-0.541374\pi\)
−0.129614 + 0.991565i \(0.541374\pi\)
\(200\) 13.1716 0.931371
\(201\) −13.4142 −0.946166
\(202\) 0.242641 0.0170721
\(203\) −2.48528 −0.174433
\(204\) 0 0
\(205\) −2.24264 −0.156633
\(206\) 24.7279 1.72288
\(207\) 4.65685 0.323674
\(208\) −15.3137 −1.06181
\(209\) −8.82843 −0.610675
\(210\) 0.686292 0.0473586
\(211\) 28.1421 1.93738 0.968692 0.248265i \(-0.0798602\pi\)
0.968692 + 0.248265i \(0.0798602\pi\)
\(212\) 0 0
\(213\) 12.4853 0.855477
\(214\) −16.4853 −1.12691
\(215\) 0.585786 0.0399503
\(216\) −16.0000 −1.08866
\(217\) 1.75736 0.119297
\(218\) 19.7574 1.33814
\(219\) 10.9706 0.741322
\(220\) 0 0
\(221\) 29.9706 2.01604
\(222\) 16.9706 1.13899
\(223\) 23.8995 1.60043 0.800214 0.599714i \(-0.204719\pi\)
0.800214 + 0.599714i \(0.204719\pi\)
\(224\) 0 0
\(225\) 4.65685 0.310457
\(226\) −1.65685 −0.110212
\(227\) −17.6569 −1.17193 −0.585963 0.810338i \(-0.699284\pi\)
−0.585963 + 0.810338i \(0.699284\pi\)
\(228\) 0 0
\(229\) 7.97056 0.526710 0.263355 0.964699i \(-0.415171\pi\)
0.263355 + 0.964699i \(0.415171\pi\)
\(230\) −3.85786 −0.254380
\(231\) 1.51472 0.0996612
\(232\) −12.0000 −0.787839
\(233\) −0.828427 −0.0542721 −0.0271360 0.999632i \(-0.508639\pi\)
−0.0271360 + 0.999632i \(0.508639\pi\)
\(234\) −5.41421 −0.353938
\(235\) 3.51472 0.229275
\(236\) 0 0
\(237\) 1.17157 0.0761018
\(238\) −6.48528 −0.420378
\(239\) −2.48528 −0.160759 −0.0803797 0.996764i \(-0.525613\pi\)
−0.0803797 + 0.996764i \(0.525613\pi\)
\(240\) 3.31371 0.213899
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) −10.8284 −0.696078
\(243\) −9.89949 −0.635053
\(244\) 0 0
\(245\) −3.89949 −0.249130
\(246\) 7.65685 0.488183
\(247\) −18.4853 −1.17619
\(248\) 8.48528 0.538816
\(249\) −20.7279 −1.31358
\(250\) −8.00000 −0.505964
\(251\) −5.14214 −0.324569 −0.162284 0.986744i \(-0.551886\pi\)
−0.162284 + 0.986744i \(0.551886\pi\)
\(252\) 0 0
\(253\) −8.51472 −0.535316
\(254\) −2.58579 −0.162247
\(255\) −6.48528 −0.406124
\(256\) 0 0
\(257\) −10.2426 −0.638918 −0.319459 0.947600i \(-0.603501\pi\)
−0.319459 + 0.947600i \(0.603501\pi\)
\(258\) −2.00000 −0.124515
\(259\) 4.97056 0.308856
\(260\) 0 0
\(261\) −4.24264 −0.262613
\(262\) −2.34315 −0.144760
\(263\) −28.1421 −1.73532 −0.867659 0.497159i \(-0.834376\pi\)
−0.867659 + 0.497159i \(0.834376\pi\)
\(264\) 7.31371 0.450128
\(265\) 4.78680 0.294051
\(266\) 4.00000 0.245256
\(267\) 14.4853 0.886485
\(268\) 0 0
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 4.68629 0.285199
\(271\) 7.97056 0.484177 0.242089 0.970254i \(-0.422168\pi\)
0.242089 + 0.970254i \(0.422168\pi\)
\(272\) −31.3137 −1.89867
\(273\) 3.17157 0.191952
\(274\) 3.51472 0.212332
\(275\) −8.51472 −0.513457
\(276\) 0 0
\(277\) −11.8995 −0.714971 −0.357486 0.933919i \(-0.616366\pi\)
−0.357486 + 0.933919i \(0.616366\pi\)
\(278\) −16.2426 −0.974169
\(279\) 3.00000 0.179605
\(280\) 0.970563 0.0580022
\(281\) 2.65685 0.158495 0.0792473 0.996855i \(-0.474748\pi\)
0.0792473 + 0.996855i \(0.474748\pi\)
\(282\) −12.0000 −0.714590
\(283\) −4.31371 −0.256423 −0.128212 0.991747i \(-0.540924\pi\)
−0.128212 + 0.991747i \(0.540924\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 9.89949 0.585369
\(287\) 2.24264 0.132379
\(288\) 0 0
\(289\) 44.2843 2.60496
\(290\) 3.51472 0.206391
\(291\) 5.41421 0.317387
\(292\) 0 0
\(293\) 21.6569 1.26521 0.632603 0.774476i \(-0.281986\pi\)
0.632603 + 0.774476i \(0.281986\pi\)
\(294\) 13.3137 0.776471
\(295\) 0.485281 0.0282542
\(296\) 24.0000 1.39497
\(297\) 10.3431 0.600170
\(298\) −12.0000 −0.695141
\(299\) −17.8284 −1.03104
\(300\) 0 0
\(301\) −0.585786 −0.0337642
\(302\) 13.7990 0.794043
\(303\) −0.242641 −0.0139393
\(304\) 19.3137 1.10772
\(305\) 4.82843 0.276475
\(306\) −11.0711 −0.632891
\(307\) −12.7990 −0.730477 −0.365238 0.930914i \(-0.619013\pi\)
−0.365238 + 0.930914i \(0.619013\pi\)
\(308\) 0 0
\(309\) −24.7279 −1.40672
\(310\) −2.48528 −0.141154
\(311\) 19.9706 1.13243 0.566213 0.824259i \(-0.308408\pi\)
0.566213 + 0.824259i \(0.308408\pi\)
\(312\) 15.3137 0.866968
\(313\) 17.2132 0.972948 0.486474 0.873695i \(-0.338283\pi\)
0.486474 + 0.873695i \(0.338283\pi\)
\(314\) −14.1421 −0.798087
\(315\) 0.343146 0.0193341
\(316\) 0 0
\(317\) 20.1716 1.13295 0.566474 0.824079i \(-0.308307\pi\)
0.566474 + 0.824079i \(0.308307\pi\)
\(318\) −16.3431 −0.916478
\(319\) 7.75736 0.434329
\(320\) 4.68629 0.261972
\(321\) 16.4853 0.920119
\(322\) 3.85786 0.214990
\(323\) −37.7990 −2.10319
\(324\) 0 0
\(325\) −17.8284 −0.988943
\(326\) −28.6274 −1.58553
\(327\) −19.7574 −1.09258
\(328\) 10.8284 0.597900
\(329\) −3.51472 −0.193773
\(330\) −2.14214 −0.117921
\(331\) 18.5858 1.02157 0.510784 0.859709i \(-0.329355\pi\)
0.510784 + 0.859709i \(0.329355\pi\)
\(332\) 0 0
\(333\) 8.48528 0.464991
\(334\) 11.7574 0.643334
\(335\) 5.55635 0.303576
\(336\) −3.31371 −0.180778
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 2.34315 0.127450
\(339\) 1.65685 0.0899880
\(340\) 0 0
\(341\) −5.48528 −0.297045
\(342\) 6.82843 0.369239
\(343\) 8.00000 0.431959
\(344\) −2.82843 −0.152499
\(345\) 3.85786 0.207700
\(346\) 17.4558 0.938432
\(347\) −7.41421 −0.398016 −0.199008 0.979998i \(-0.563772\pi\)
−0.199008 + 0.979998i \(0.563772\pi\)
\(348\) 0 0
\(349\) −11.7574 −0.629357 −0.314679 0.949198i \(-0.601897\pi\)
−0.314679 + 0.949198i \(0.601897\pi\)
\(350\) 3.85786 0.206212
\(351\) 21.6569 1.15596
\(352\) 0 0
\(353\) 0.171573 0.00913190 0.00456595 0.999990i \(-0.498547\pi\)
0.00456595 + 0.999990i \(0.498547\pi\)
\(354\) −1.65685 −0.0880608
\(355\) −5.17157 −0.274479
\(356\) 0 0
\(357\) 6.48528 0.343237
\(358\) −10.4853 −0.554164
\(359\) −32.6569 −1.72356 −0.861781 0.507280i \(-0.830651\pi\)
−0.861781 + 0.507280i \(0.830651\pi\)
\(360\) 1.65685 0.0873239
\(361\) 4.31371 0.227037
\(362\) −21.6569 −1.13826
\(363\) 10.8284 0.568345
\(364\) 0 0
\(365\) −4.54416 −0.237852
\(366\) −16.4853 −0.861699
\(367\) −29.7990 −1.55549 −0.777747 0.628577i \(-0.783638\pi\)
−0.777747 + 0.628577i \(0.783638\pi\)
\(368\) 18.6274 0.971021
\(369\) 3.82843 0.199300
\(370\) −7.02944 −0.365443
\(371\) −4.78680 −0.248518
\(372\) 0 0
\(373\) −8.48528 −0.439351 −0.219676 0.975573i \(-0.570500\pi\)
−0.219676 + 0.975573i \(0.570500\pi\)
\(374\) 20.2426 1.04672
\(375\) 8.00000 0.413118
\(376\) −16.9706 −0.875190
\(377\) 16.2426 0.836539
\(378\) −4.68629 −0.241037
\(379\) 7.68629 0.394818 0.197409 0.980321i \(-0.436747\pi\)
0.197409 + 0.980321i \(0.436747\pi\)
\(380\) 0 0
\(381\) 2.58579 0.132474
\(382\) 31.3137 1.60215
\(383\) −3.51472 −0.179594 −0.0897969 0.995960i \(-0.528622\pi\)
−0.0897969 + 0.995960i \(0.528622\pi\)
\(384\) −16.0000 −0.816497
\(385\) −0.627417 −0.0319761
\(386\) −25.4142 −1.29355
\(387\) −1.00000 −0.0508329
\(388\) 0 0
\(389\) 28.6274 1.45147 0.725734 0.687976i \(-0.241500\pi\)
0.725734 + 0.687976i \(0.241500\pi\)
\(390\) −4.48528 −0.227121
\(391\) −36.4558 −1.84365
\(392\) 18.8284 0.950979
\(393\) 2.34315 0.118196
\(394\) 20.0000 1.00759
\(395\) −0.485281 −0.0244172
\(396\) 0 0
\(397\) −21.4558 −1.07684 −0.538419 0.842677i \(-0.680978\pi\)
−0.538419 + 0.842677i \(0.680978\pi\)
\(398\) −5.17157 −0.259228
\(399\) −4.00000 −0.200250
\(400\) 18.6274 0.931371
\(401\) −29.4853 −1.47242 −0.736212 0.676751i \(-0.763388\pi\)
−0.736212 + 0.676751i \(0.763388\pi\)
\(402\) −18.9706 −0.946166
\(403\) −11.4853 −0.572123
\(404\) 0 0
\(405\) −2.92893 −0.145540
\(406\) −3.51472 −0.174433
\(407\) −15.5147 −0.769036
\(408\) 31.3137 1.55026
\(409\) 36.8701 1.82311 0.911554 0.411181i \(-0.134884\pi\)
0.911554 + 0.411181i \(0.134884\pi\)
\(410\) −3.17157 −0.156633
\(411\) −3.51472 −0.173368
\(412\) 0 0
\(413\) −0.485281 −0.0238791
\(414\) 6.58579 0.323674
\(415\) 8.58579 0.421460
\(416\) 0 0
\(417\) 16.2426 0.795406
\(418\) −12.4853 −0.610675
\(419\) −4.10051 −0.200323 −0.100161 0.994971i \(-0.531936\pi\)
−0.100161 + 0.994971i \(0.531936\pi\)
\(420\) 0 0
\(421\) −4.34315 −0.211672 −0.105836 0.994384i \(-0.533752\pi\)
−0.105836 + 0.994384i \(0.533752\pi\)
\(422\) 39.7990 1.93738
\(423\) −6.00000 −0.291730
\(424\) −23.1127 −1.12245
\(425\) −36.4558 −1.76837
\(426\) 17.6569 0.855477
\(427\) −4.82843 −0.233664
\(428\) 0 0
\(429\) −9.89949 −0.477952
\(430\) 0.828427 0.0399503
\(431\) −33.2843 −1.60325 −0.801623 0.597829i \(-0.796030\pi\)
−0.801623 + 0.597829i \(0.796030\pi\)
\(432\) −22.6274 −1.08866
\(433\) 30.2426 1.45337 0.726684 0.686972i \(-0.241060\pi\)
0.726684 + 0.686972i \(0.241060\pi\)
\(434\) 2.48528 0.119297
\(435\) −3.51472 −0.168518
\(436\) 0 0
\(437\) 22.4853 1.07562
\(438\) 15.5147 0.741322
\(439\) −5.48528 −0.261798 −0.130899 0.991396i \(-0.541786\pi\)
−0.130899 + 0.991396i \(0.541786\pi\)
\(440\) −3.02944 −0.144423
\(441\) 6.65685 0.316993
\(442\) 42.3848 2.01604
\(443\) 36.1421 1.71716 0.858582 0.512676i \(-0.171346\pi\)
0.858582 + 0.512676i \(0.171346\pi\)
\(444\) 0 0
\(445\) −6.00000 −0.284427
\(446\) 33.7990 1.60043
\(447\) 12.0000 0.567581
\(448\) −4.68629 −0.221406
\(449\) −9.21320 −0.434798 −0.217399 0.976083i \(-0.569757\pi\)
−0.217399 + 0.976083i \(0.569757\pi\)
\(450\) 6.58579 0.310457
\(451\) −7.00000 −0.329617
\(452\) 0 0
\(453\) −13.7990 −0.648333
\(454\) −24.9706 −1.17193
\(455\) −1.31371 −0.0615876
\(456\) −19.3137 −0.904447
\(457\) −24.7279 −1.15672 −0.578362 0.815780i \(-0.696308\pi\)
−0.578362 + 0.815780i \(0.696308\pi\)
\(458\) 11.2721 0.526710
\(459\) 44.2843 2.06701
\(460\) 0 0
\(461\) −2.62742 −0.122371 −0.0611855 0.998126i \(-0.519488\pi\)
−0.0611855 + 0.998126i \(0.519488\pi\)
\(462\) 2.14214 0.0996612
\(463\) −5.27208 −0.245014 −0.122507 0.992468i \(-0.539093\pi\)
−0.122507 + 0.992468i \(0.539093\pi\)
\(464\) −16.9706 −0.787839
\(465\) 2.48528 0.115252
\(466\) −1.17157 −0.0542721
\(467\) 12.3431 0.571173 0.285586 0.958353i \(-0.407812\pi\)
0.285586 + 0.958353i \(0.407812\pi\)
\(468\) 0 0
\(469\) −5.55635 −0.256568
\(470\) 4.97056 0.229275
\(471\) 14.1421 0.651635
\(472\) −2.34315 −0.107852
\(473\) 1.82843 0.0840712
\(474\) 1.65685 0.0761018
\(475\) 22.4853 1.03170
\(476\) 0 0
\(477\) −8.17157 −0.374151
\(478\) −3.51472 −0.160759
\(479\) 4.65685 0.212777 0.106389 0.994325i \(-0.466071\pi\)
0.106389 + 0.994325i \(0.466071\pi\)
\(480\) 0 0
\(481\) −32.4853 −1.48120
\(482\) 5.65685 0.257663
\(483\) −3.85786 −0.175539
\(484\) 0 0
\(485\) −2.24264 −0.101833
\(486\) −14.0000 −0.635053
\(487\) 33.6569 1.52514 0.762569 0.646907i \(-0.223938\pi\)
0.762569 + 0.646907i \(0.223938\pi\)
\(488\) −23.3137 −1.05536
\(489\) 28.6274 1.29458
\(490\) −5.51472 −0.249130
\(491\) 10.5858 0.477730 0.238865 0.971053i \(-0.423225\pi\)
0.238865 + 0.971053i \(0.423225\pi\)
\(492\) 0 0
\(493\) 33.2132 1.49585
\(494\) −26.1421 −1.17619
\(495\) −1.07107 −0.0481409
\(496\) 12.0000 0.538816
\(497\) 5.17157 0.231977
\(498\) −29.3137 −1.31358
\(499\) −34.2426 −1.53291 −0.766456 0.642297i \(-0.777981\pi\)
−0.766456 + 0.642297i \(0.777981\pi\)
\(500\) 0 0
\(501\) −11.7574 −0.525280
\(502\) −7.27208 −0.324569
\(503\) 42.7696 1.90700 0.953500 0.301393i \(-0.0974516\pi\)
0.953500 + 0.301393i \(0.0974516\pi\)
\(504\) −1.65685 −0.0738022
\(505\) 0.100505 0.00447242
\(506\) −12.0416 −0.535316
\(507\) −2.34315 −0.104063
\(508\) 0 0
\(509\) 17.4853 0.775021 0.387511 0.921865i \(-0.373335\pi\)
0.387511 + 0.921865i \(0.373335\pi\)
\(510\) −9.17157 −0.406124
\(511\) 4.54416 0.201022
\(512\) −22.6274 −1.00000
\(513\) −27.3137 −1.20593
\(514\) −14.4853 −0.638918
\(515\) 10.2426 0.451345
\(516\) 0 0
\(517\) 10.9706 0.482485
\(518\) 7.02944 0.308856
\(519\) −17.4558 −0.766227
\(520\) −6.34315 −0.278165
\(521\) 31.0711 1.36125 0.680624 0.732633i \(-0.261709\pi\)
0.680624 + 0.732633i \(0.261709\pi\)
\(522\) −6.00000 −0.262613
\(523\) 3.21320 0.140504 0.0702518 0.997529i \(-0.477620\pi\)
0.0702518 + 0.997529i \(0.477620\pi\)
\(524\) 0 0
\(525\) −3.85786 −0.168371
\(526\) −39.7990 −1.73532
\(527\) −23.4853 −1.02303
\(528\) 10.3431 0.450128
\(529\) −1.31371 −0.0571178
\(530\) 6.76955 0.294051
\(531\) −0.828427 −0.0359507
\(532\) 0 0
\(533\) −14.6569 −0.634859
\(534\) 20.4853 0.886485
\(535\) −6.82843 −0.295219
\(536\) −26.8284 −1.15881
\(537\) 10.4853 0.452473
\(538\) −4.24264 −0.182913
\(539\) −12.1716 −0.524267
\(540\) 0 0
\(541\) −30.7990 −1.32415 −0.662076 0.749437i \(-0.730324\pi\)
−0.662076 + 0.749437i \(0.730324\pi\)
\(542\) 11.2721 0.484177
\(543\) 21.6569 0.929385
\(544\) 0 0
\(545\) 8.18377 0.350554
\(546\) 4.48528 0.191952
\(547\) 9.00000 0.384812 0.192406 0.981315i \(-0.438371\pi\)
0.192406 + 0.981315i \(0.438371\pi\)
\(548\) 0 0
\(549\) −8.24264 −0.351787
\(550\) −12.0416 −0.513457
\(551\) −20.4853 −0.872702
\(552\) −18.6274 −0.792836
\(553\) 0.485281 0.0206363
\(554\) −16.8284 −0.714971
\(555\) 7.02944 0.298383
\(556\) 0 0
\(557\) −9.00000 −0.381342 −0.190671 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) 4.24264 0.179605
\(559\) 3.82843 0.161925
\(560\) 1.37258 0.0580022
\(561\) −20.2426 −0.854645
\(562\) 3.75736 0.158495
\(563\) −5.62742 −0.237167 −0.118584 0.992944i \(-0.537835\pi\)
−0.118584 + 0.992944i \(0.537835\pi\)
\(564\) 0 0
\(565\) −0.686292 −0.0288725
\(566\) −6.10051 −0.256423
\(567\) 2.92893 0.123004
\(568\) 24.9706 1.04774
\(569\) 24.6569 1.03367 0.516835 0.856085i \(-0.327110\pi\)
0.516835 + 0.856085i \(0.327110\pi\)
\(570\) 5.65685 0.236940
\(571\) 11.0711 0.463310 0.231655 0.972798i \(-0.425586\pi\)
0.231655 + 0.972798i \(0.425586\pi\)
\(572\) 0 0
\(573\) −31.3137 −1.30815
\(574\) 3.17157 0.132379
\(575\) 21.6863 0.904381
\(576\) −8.00000 −0.333333
\(577\) 18.9289 0.788022 0.394011 0.919106i \(-0.371087\pi\)
0.394011 + 0.919106i \(0.371087\pi\)
\(578\) 62.6274 2.60496
\(579\) 25.4142 1.05618
\(580\) 0 0
\(581\) −8.58579 −0.356198
\(582\) 7.65685 0.317387
\(583\) 14.9411 0.618798
\(584\) 21.9411 0.907930
\(585\) −2.24264 −0.0927218
\(586\) 30.6274 1.26521
\(587\) −5.79899 −0.239350 −0.119675 0.992813i \(-0.538185\pi\)
−0.119675 + 0.992813i \(0.538185\pi\)
\(588\) 0 0
\(589\) 14.4853 0.596856
\(590\) 0.686292 0.0282542
\(591\) −20.0000 −0.822690
\(592\) 33.9411 1.39497
\(593\) 5.07107 0.208244 0.104122 0.994565i \(-0.466797\pi\)
0.104122 + 0.994565i \(0.466797\pi\)
\(594\) 14.6274 0.600170
\(595\) −2.68629 −0.110127
\(596\) 0 0
\(597\) 5.17157 0.211658
\(598\) −25.2132 −1.03104
\(599\) 25.3431 1.03549 0.517746 0.855534i \(-0.326771\pi\)
0.517746 + 0.855534i \(0.326771\pi\)
\(600\) −18.6274 −0.760461
\(601\) 8.97056 0.365917 0.182958 0.983121i \(-0.441433\pi\)
0.182958 + 0.983121i \(0.441433\pi\)
\(602\) −0.828427 −0.0337642
\(603\) −9.48528 −0.386271
\(604\) 0 0
\(605\) −4.48528 −0.182353
\(606\) −0.343146 −0.0139393
\(607\) 0.970563 0.0393939 0.0196970 0.999806i \(-0.493730\pi\)
0.0196970 + 0.999806i \(0.493730\pi\)
\(608\) 0 0
\(609\) 3.51472 0.142424
\(610\) 6.82843 0.276475
\(611\) 22.9706 0.929289
\(612\) 0 0
\(613\) 1.17157 0.0473194 0.0236597 0.999720i \(-0.492468\pi\)
0.0236597 + 0.999720i \(0.492468\pi\)
\(614\) −18.1005 −0.730477
\(615\) 3.17157 0.127890
\(616\) 3.02944 0.122060
\(617\) −13.9706 −0.562434 −0.281217 0.959644i \(-0.590738\pi\)
−0.281217 + 0.959644i \(0.590738\pi\)
\(618\) −34.9706 −1.40672
\(619\) −4.97056 −0.199784 −0.0998919 0.994998i \(-0.531850\pi\)
−0.0998919 + 0.994998i \(0.531850\pi\)
\(620\) 0 0
\(621\) −26.3431 −1.05711
\(622\) 28.2426 1.13243
\(623\) 6.00000 0.240385
\(624\) 21.6569 0.866968
\(625\) 19.9706 0.798823
\(626\) 24.3431 0.972948
\(627\) 12.4853 0.498614
\(628\) 0 0
\(629\) −66.4264 −2.64859
\(630\) 0.485281 0.0193341
\(631\) −26.7696 −1.06568 −0.532840 0.846216i \(-0.678875\pi\)
−0.532840 + 0.846216i \(0.678875\pi\)
\(632\) 2.34315 0.0932053
\(633\) −39.7990 −1.58187
\(634\) 28.5269 1.13295
\(635\) −1.07107 −0.0425040
\(636\) 0 0
\(637\) −25.4853 −1.00976
\(638\) 10.9706 0.434329
\(639\) 8.82843 0.349247
\(640\) 6.62742 0.261972
\(641\) 1.55635 0.0614721 0.0307360 0.999528i \(-0.490215\pi\)
0.0307360 + 0.999528i \(0.490215\pi\)
\(642\) 23.3137 0.920119
\(643\) −9.51472 −0.375224 −0.187612 0.982243i \(-0.560075\pi\)
−0.187612 + 0.982243i \(0.560075\pi\)
\(644\) 0 0
\(645\) −0.828427 −0.0326193
\(646\) −53.4558 −2.10319
\(647\) −6.82843 −0.268453 −0.134227 0.990951i \(-0.542855\pi\)
−0.134227 + 0.990951i \(0.542855\pi\)
\(648\) 14.1421 0.555556
\(649\) 1.51472 0.0594579
\(650\) −25.2132 −0.988943
\(651\) −2.48528 −0.0974059
\(652\) 0 0
\(653\) 20.8284 0.815079 0.407540 0.913188i \(-0.366387\pi\)
0.407540 + 0.913188i \(0.366387\pi\)
\(654\) −27.9411 −1.09258
\(655\) −0.970563 −0.0379230
\(656\) 15.3137 0.597900
\(657\) 7.75736 0.302643
\(658\) −4.97056 −0.193773
\(659\) −6.31371 −0.245947 −0.122974 0.992410i \(-0.539243\pi\)
−0.122974 + 0.992410i \(0.539243\pi\)
\(660\) 0 0
\(661\) −17.0000 −0.661223 −0.330612 0.943767i \(-0.607255\pi\)
−0.330612 + 0.943767i \(0.607255\pi\)
\(662\) 26.2843 1.02157
\(663\) −42.3848 −1.64609
\(664\) −41.4558 −1.60880
\(665\) 1.65685 0.0642501
\(666\) 12.0000 0.464991
\(667\) −19.7574 −0.765008
\(668\) 0 0
\(669\) −33.7990 −1.30674
\(670\) 7.85786 0.303576
\(671\) 15.0711 0.581812
\(672\) 0 0
\(673\) 8.72792 0.336437 0.168218 0.985750i \(-0.446199\pi\)
0.168218 + 0.985750i \(0.446199\pi\)
\(674\) −7.07107 −0.272367
\(675\) −26.3431 −1.01395
\(676\) 0 0
\(677\) 16.8284 0.646769 0.323384 0.946268i \(-0.395179\pi\)
0.323384 + 0.946268i \(0.395179\pi\)
\(678\) 2.34315 0.0899880
\(679\) 2.24264 0.0860647
\(680\) −12.9706 −0.497398
\(681\) 24.9706 0.956874
\(682\) −7.75736 −0.297045
\(683\) 4.45584 0.170498 0.0852491 0.996360i \(-0.472831\pi\)
0.0852491 + 0.996360i \(0.472831\pi\)
\(684\) 0 0
\(685\) 1.45584 0.0556249
\(686\) 11.3137 0.431959
\(687\) −11.2721 −0.430057
\(688\) −4.00000 −0.152499
\(689\) 31.2843 1.19184
\(690\) 5.45584 0.207700
\(691\) −1.27208 −0.0483921 −0.0241961 0.999707i \(-0.507703\pi\)
−0.0241961 + 0.999707i \(0.507703\pi\)
\(692\) 0 0
\(693\) 1.07107 0.0406865
\(694\) −10.4853 −0.398016
\(695\) −6.72792 −0.255205
\(696\) 16.9706 0.643268
\(697\) −29.9706 −1.13522
\(698\) −16.6274 −0.629357
\(699\) 1.17157 0.0443130
\(700\) 0 0
\(701\) −18.3431 −0.692811 −0.346406 0.938085i \(-0.612598\pi\)
−0.346406 + 0.938085i \(0.612598\pi\)
\(702\) 30.6274 1.15596
\(703\) 40.9706 1.54523
\(704\) 14.6274 0.551292
\(705\) −4.97056 −0.187202
\(706\) 0.242641 0.00913190
\(707\) −0.100505 −0.00377988
\(708\) 0 0
\(709\) 24.1127 0.905571 0.452786 0.891619i \(-0.350430\pi\)
0.452786 + 0.891619i \(0.350430\pi\)
\(710\) −7.31371 −0.274479
\(711\) 0.828427 0.0310684
\(712\) 28.9706 1.08572
\(713\) 13.9706 0.523202
\(714\) 9.17157 0.343237
\(715\) 4.10051 0.153350
\(716\) 0 0
\(717\) 3.51472 0.131260
\(718\) −46.1838 −1.72356
\(719\) 22.6274 0.843860 0.421930 0.906628i \(-0.361353\pi\)
0.421930 + 0.906628i \(0.361353\pi\)
\(720\) 2.34315 0.0873239
\(721\) −10.2426 −0.381456
\(722\) 6.10051 0.227037
\(723\) −5.65685 −0.210381
\(724\) 0 0
\(725\) −19.7574 −0.733770
\(726\) 15.3137 0.568345
\(727\) 8.97056 0.332700 0.166350 0.986067i \(-0.446802\pi\)
0.166350 + 0.986067i \(0.446802\pi\)
\(728\) 6.34315 0.235093
\(729\) 29.0000 1.07407
\(730\) −6.42641 −0.237852
\(731\) 7.82843 0.289545
\(732\) 0 0
\(733\) −16.9706 −0.626822 −0.313411 0.949618i \(-0.601472\pi\)
−0.313411 + 0.949618i \(0.601472\pi\)
\(734\) −42.1421 −1.55549
\(735\) 5.51472 0.203413
\(736\) 0 0
\(737\) 17.3431 0.638843
\(738\) 5.41421 0.199300
\(739\) 11.4558 0.421410 0.210705 0.977550i \(-0.432424\pi\)
0.210705 + 0.977550i \(0.432424\pi\)
\(740\) 0 0
\(741\) 26.1421 0.960355
\(742\) −6.76955 −0.248518
\(743\) −47.1127 −1.72840 −0.864199 0.503151i \(-0.832174\pi\)
−0.864199 + 0.503151i \(0.832174\pi\)
\(744\) −12.0000 −0.439941
\(745\) −4.97056 −0.182107
\(746\) −12.0000 −0.439351
\(747\) −14.6569 −0.536266
\(748\) 0 0
\(749\) 6.82843 0.249505
\(750\) 11.3137 0.413118
\(751\) −20.2426 −0.738664 −0.369332 0.929297i \(-0.620414\pi\)
−0.369332 + 0.929297i \(0.620414\pi\)
\(752\) −24.0000 −0.875190
\(753\) 7.27208 0.265009
\(754\) 22.9706 0.836539
\(755\) 5.71573 0.208017
\(756\) 0 0
\(757\) 20.4853 0.744550 0.372275 0.928122i \(-0.378578\pi\)
0.372275 + 0.928122i \(0.378578\pi\)
\(758\) 10.8701 0.394818
\(759\) 12.0416 0.437083
\(760\) 8.00000 0.290191
\(761\) −43.1127 −1.56283 −0.781417 0.624009i \(-0.785503\pi\)
−0.781417 + 0.624009i \(0.785503\pi\)
\(762\) 3.65685 0.132474
\(763\) −8.18377 −0.296272
\(764\) 0 0
\(765\) −4.58579 −0.165799
\(766\) −4.97056 −0.179594
\(767\) 3.17157 0.114519
\(768\) 0 0
\(769\) −28.7696 −1.03746 −0.518728 0.854939i \(-0.673594\pi\)
−0.518728 + 0.854939i \(0.673594\pi\)
\(770\) −0.887302 −0.0319761
\(771\) 14.4853 0.521675
\(772\) 0 0
\(773\) 27.8995 1.00348 0.501738 0.865020i \(-0.332694\pi\)
0.501738 + 0.865020i \(0.332694\pi\)
\(774\) −1.41421 −0.0508329
\(775\) 13.9706 0.501837
\(776\) 10.8284 0.388718
\(777\) −7.02944 −0.252180
\(778\) 40.4853 1.45147
\(779\) 18.4853 0.662304
\(780\) 0 0
\(781\) −16.1421 −0.577611
\(782\) −51.5563 −1.84365
\(783\) 24.0000 0.857690
\(784\) 26.6274 0.950979
\(785\) −5.85786 −0.209076
\(786\) 3.31371 0.118196
\(787\) −29.7990 −1.06222 −0.531110 0.847303i \(-0.678225\pi\)
−0.531110 + 0.847303i \(0.678225\pi\)
\(788\) 0 0
\(789\) 39.7990 1.41688
\(790\) −0.686292 −0.0244172
\(791\) 0.686292 0.0244017
\(792\) 5.17157 0.183764
\(793\) 31.5563 1.12060
\(794\) −30.3431 −1.07684
\(795\) −6.76955 −0.240091
\(796\) 0 0
\(797\) 12.6863 0.449372 0.224686 0.974431i \(-0.427864\pi\)
0.224686 + 0.974431i \(0.427864\pi\)
\(798\) −5.65685 −0.200250
\(799\) 46.9706 1.66170
\(800\) 0 0
\(801\) 10.2426 0.361906
\(802\) −41.6985 −1.47242
\(803\) −14.1838 −0.500534
\(804\) 0 0
\(805\) 1.59798 0.0563214
\(806\) −16.2426 −0.572123
\(807\) 4.24264 0.149348
\(808\) −0.485281 −0.0170721
\(809\) 5.65685 0.198884 0.0994422 0.995043i \(-0.468294\pi\)
0.0994422 + 0.995043i \(0.468294\pi\)
\(810\) −4.14214 −0.145540
\(811\) 48.7279 1.71107 0.855534 0.517746i \(-0.173229\pi\)
0.855534 + 0.517746i \(0.173229\pi\)
\(812\) 0 0
\(813\) −11.2721 −0.395329
\(814\) −21.9411 −0.769036
\(815\) −11.8579 −0.415363
\(816\) 44.2843 1.55026
\(817\) −4.82843 −0.168925
\(818\) 52.1421 1.82311
\(819\) 2.24264 0.0783642
\(820\) 0 0
\(821\) 10.1127 0.352936 0.176468 0.984306i \(-0.443533\pi\)
0.176468 + 0.984306i \(0.443533\pi\)
\(822\) −4.97056 −0.173368
\(823\) −43.3431 −1.51085 −0.755424 0.655237i \(-0.772569\pi\)
−0.755424 + 0.655237i \(0.772569\pi\)
\(824\) −49.4558 −1.72288
\(825\) 12.0416 0.419236
\(826\) −0.686292 −0.0238791
\(827\) 9.65685 0.335802 0.167901 0.985804i \(-0.446301\pi\)
0.167901 + 0.985804i \(0.446301\pi\)
\(828\) 0 0
\(829\) −15.7990 −0.548722 −0.274361 0.961627i \(-0.588466\pi\)
−0.274361 + 0.961627i \(0.588466\pi\)
\(830\) 12.1421 0.421460
\(831\) 16.8284 0.583772
\(832\) 30.6274 1.06181
\(833\) −52.1127 −1.80560
\(834\) 22.9706 0.795406
\(835\) 4.87006 0.168535
\(836\) 0 0
\(837\) −16.9706 −0.586588
\(838\) −5.79899 −0.200323
\(839\) 4.87006 0.168133 0.0840665 0.996460i \(-0.473209\pi\)
0.0840665 + 0.996460i \(0.473209\pi\)
\(840\) −1.37258 −0.0473586
\(841\) −11.0000 −0.379310
\(842\) −6.14214 −0.211672
\(843\) −3.75736 −0.129410
\(844\) 0 0
\(845\) 0.970563 0.0333884
\(846\) −8.48528 −0.291730
\(847\) 4.48528 0.154116
\(848\) −32.6863 −1.12245
\(849\) 6.10051 0.209369
\(850\) −51.5563 −1.76837
\(851\) 39.5147 1.35455
\(852\) 0 0
\(853\) −32.5980 −1.11613 −0.558067 0.829796i \(-0.688457\pi\)
−0.558067 + 0.829796i \(0.688457\pi\)
\(854\) −6.82843 −0.233664
\(855\) 2.82843 0.0967302
\(856\) 32.9706 1.12691
\(857\) −3.65685 −0.124916 −0.0624579 0.998048i \(-0.519894\pi\)
−0.0624579 + 0.998048i \(0.519894\pi\)
\(858\) −14.0000 −0.477952
\(859\) −16.9706 −0.579028 −0.289514 0.957174i \(-0.593494\pi\)
−0.289514 + 0.957174i \(0.593494\pi\)
\(860\) 0 0
\(861\) −3.17157 −0.108087
\(862\) −47.0711 −1.60325
\(863\) 43.2548 1.47241 0.736206 0.676758i \(-0.236616\pi\)
0.736206 + 0.676758i \(0.236616\pi\)
\(864\) 0 0
\(865\) 7.23045 0.245843
\(866\) 42.7696 1.45337
\(867\) −62.6274 −2.12694
\(868\) 0 0
\(869\) −1.51472 −0.0513833
\(870\) −4.97056 −0.168518
\(871\) 36.3137 1.23044
\(872\) −39.5147 −1.33814
\(873\) 3.82843 0.129573
\(874\) 31.7990 1.07562
\(875\) 3.31371 0.112024
\(876\) 0 0
\(877\) −2.79899 −0.0945152 −0.0472576 0.998883i \(-0.515048\pi\)
−0.0472576 + 0.998883i \(0.515048\pi\)
\(878\) −7.75736 −0.261798
\(879\) −30.6274 −1.03304
\(880\) −4.28427 −0.144423
\(881\) −54.2548 −1.82789 −0.913946 0.405836i \(-0.866980\pi\)
−0.913946 + 0.405836i \(0.866980\pi\)
\(882\) 9.41421 0.316993
\(883\) −41.9706 −1.41242 −0.706211 0.708001i \(-0.749597\pi\)
−0.706211 + 0.708001i \(0.749597\pi\)
\(884\) 0 0
\(885\) −0.686292 −0.0230694
\(886\) 51.1127 1.71716
\(887\) 25.0294 0.840406 0.420203 0.907430i \(-0.361959\pi\)
0.420203 + 0.907430i \(0.361959\pi\)
\(888\) −33.9411 −1.13899
\(889\) 1.07107 0.0359225
\(890\) −8.48528 −0.284427
\(891\) −9.14214 −0.306273
\(892\) 0 0
\(893\) −28.9706 −0.969463
\(894\) 16.9706 0.567581
\(895\) −4.34315 −0.145175
\(896\) −6.62742 −0.221406
\(897\) 25.2132 0.841844
\(898\) −13.0294 −0.434798
\(899\) −12.7279 −0.424500
\(900\) 0 0
\(901\) 63.9706 2.13117
\(902\) −9.89949 −0.329617
\(903\) 0.828427 0.0275683
\(904\) 3.31371 0.110212
\(905\) −8.97056 −0.298192
\(906\) −19.5147 −0.648333
\(907\) 13.9706 0.463885 0.231942 0.972730i \(-0.425492\pi\)
0.231942 + 0.972730i \(0.425492\pi\)
\(908\) 0 0
\(909\) −0.171573 −0.00569071
\(910\) −1.85786 −0.0615876
\(911\) 4.24264 0.140565 0.0702825 0.997527i \(-0.477610\pi\)
0.0702825 + 0.997527i \(0.477610\pi\)
\(912\) −27.3137 −0.904447
\(913\) 26.7990 0.886917
\(914\) −34.9706 −1.15672
\(915\) −6.82843 −0.225741
\(916\) 0 0
\(917\) 0.970563 0.0320508
\(918\) 62.6274 2.06701
\(919\) −29.4853 −0.972630 −0.486315 0.873784i \(-0.661659\pi\)
−0.486315 + 0.873784i \(0.661659\pi\)
\(920\) 7.71573 0.254380
\(921\) 18.1005 0.596432
\(922\) −3.71573 −0.122371
\(923\) −33.7990 −1.11251
\(924\) 0 0
\(925\) 39.5147 1.29924
\(926\) −7.45584 −0.245014
\(927\) −17.4853 −0.574292
\(928\) 0 0
\(929\) 44.8284 1.47077 0.735386 0.677648i \(-0.237001\pi\)
0.735386 + 0.677648i \(0.237001\pi\)
\(930\) 3.51472 0.115252
\(931\) 32.1421 1.05342
\(932\) 0 0
\(933\) −28.2426 −0.924623
\(934\) 17.4558 0.571173
\(935\) 8.38478 0.274212
\(936\) 10.8284 0.353938
\(937\) 31.0122 1.01312 0.506562 0.862203i \(-0.330916\pi\)
0.506562 + 0.862203i \(0.330916\pi\)
\(938\) −7.85786 −0.256568
\(939\) −24.3431 −0.794409
\(940\) 0 0
\(941\) 13.6274 0.444241 0.222121 0.975019i \(-0.428702\pi\)
0.222121 + 0.975019i \(0.428702\pi\)
\(942\) 20.0000 0.651635
\(943\) 17.8284 0.580573
\(944\) −3.31371 −0.107852
\(945\) −1.94113 −0.0631448
\(946\) 2.58579 0.0840712
\(947\) −0.171573 −0.00557537 −0.00278768 0.999996i \(-0.500887\pi\)
−0.00278768 + 0.999996i \(0.500887\pi\)
\(948\) 0 0
\(949\) −29.6985 −0.964054
\(950\) 31.7990 1.03170
\(951\) −28.5269 −0.925048
\(952\) 12.9706 0.420378
\(953\) 24.0416 0.778785 0.389392 0.921072i \(-0.372685\pi\)
0.389392 + 0.921072i \(0.372685\pi\)
\(954\) −11.5563 −0.374151
\(955\) 12.9706 0.419718
\(956\) 0 0
\(957\) −10.9706 −0.354628
\(958\) 6.58579 0.212777
\(959\) −1.45584 −0.0470117
\(960\) −6.62742 −0.213899
\(961\) −22.0000 −0.709677
\(962\) −45.9411 −1.48120
\(963\) 11.6569 0.375637
\(964\) 0 0
\(965\) −10.5269 −0.338873
\(966\) −5.45584 −0.175539
\(967\) 26.9411 0.866368 0.433184 0.901305i \(-0.357390\pi\)
0.433184 + 0.901305i \(0.357390\pi\)
\(968\) 21.6569 0.696078
\(969\) 53.4558 1.71725
\(970\) −3.17157 −0.101833
\(971\) 25.1421 0.806850 0.403425 0.915013i \(-0.367820\pi\)
0.403425 + 0.915013i \(0.367820\pi\)
\(972\) 0 0
\(973\) 6.72792 0.215687
\(974\) 47.5980 1.52514
\(975\) 25.2132 0.807469
\(976\) −32.9706 −1.05536
\(977\) −0.686292 −0.0219564 −0.0109782 0.999940i \(-0.503495\pi\)
−0.0109782 + 0.999940i \(0.503495\pi\)
\(978\) 40.4853 1.29458
\(979\) −18.7279 −0.598547
\(980\) 0 0
\(981\) −13.9706 −0.446046
\(982\) 14.9706 0.477730
\(983\) 2.52691 0.0805960 0.0402980 0.999188i \(-0.487169\pi\)
0.0402980 + 0.999188i \(0.487169\pi\)
\(984\) −15.3137 −0.488183
\(985\) 8.28427 0.263959
\(986\) 46.9706 1.49585
\(987\) 4.97056 0.158215
\(988\) 0 0
\(989\) −4.65685 −0.148079
\(990\) −1.51472 −0.0481409
\(991\) −19.5563 −0.621228 −0.310614 0.950536i \(-0.600535\pi\)
−0.310614 + 0.950536i \(0.600535\pi\)
\(992\) 0 0
\(993\) −26.2843 −0.834106
\(994\) 7.31371 0.231977
\(995\) −2.14214 −0.0679103
\(996\) 0 0
\(997\) 17.0711 0.540646 0.270323 0.962770i \(-0.412869\pi\)
0.270323 + 0.962770i \(0.412869\pi\)
\(998\) −48.4264 −1.53291
\(999\) −48.0000 −1.51865
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 43.2.a.b.1.2 2
3.2 odd 2 387.2.a.h.1.1 2
4.3 odd 2 688.2.a.f.1.2 2
5.2 odd 4 1075.2.b.f.474.3 4
5.3 odd 4 1075.2.b.f.474.2 4
5.4 even 2 1075.2.a.i.1.1 2
7.6 odd 2 2107.2.a.b.1.2 2
8.3 odd 2 2752.2.a.m.1.1 2
8.5 even 2 2752.2.a.l.1.2 2
11.10 odd 2 5203.2.a.f.1.1 2
12.11 even 2 6192.2.a.bd.1.2 2
13.12 even 2 7267.2.a.b.1.1 2
15.14 odd 2 9675.2.a.bf.1.2 2
43.42 odd 2 1849.2.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.2.a.b.1.2 2 1.1 even 1 trivial
387.2.a.h.1.1 2 3.2 odd 2
688.2.a.f.1.2 2 4.3 odd 2
1075.2.a.i.1.1 2 5.4 even 2
1075.2.b.f.474.2 4 5.3 odd 4
1075.2.b.f.474.3 4 5.2 odd 4
1849.2.a.f.1.1 2 43.42 odd 2
2107.2.a.b.1.2 2 7.6 odd 2
2752.2.a.l.1.2 2 8.5 even 2
2752.2.a.m.1.1 2 8.3 odd 2
5203.2.a.f.1.1 2 11.10 odd 2
6192.2.a.bd.1.2 2 12.11 even 2
7267.2.a.b.1.1 2 13.12 even 2
9675.2.a.bf.1.2 2 15.14 odd 2