Properties

Label 507.2.a.g.1.2
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +0.561553 q^{5} +2.56155 q^{6} -3.56155 q^{7} +6.56155 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +0.561553 q^{5} +2.56155 q^{6} -3.56155 q^{7} +6.56155 q^{8} +1.00000 q^{9} +1.43845 q^{10} -2.00000 q^{11} +4.56155 q^{12} -9.12311 q^{14} +0.561553 q^{15} +7.68466 q^{16} +2.56155 q^{17} +2.56155 q^{18} -1.12311 q^{19} +2.56155 q^{20} -3.56155 q^{21} -5.12311 q^{22} +2.00000 q^{23} +6.56155 q^{24} -4.68466 q^{25} +1.00000 q^{27} -16.2462 q^{28} -5.68466 q^{29} +1.43845 q^{30} -1.56155 q^{31} +6.56155 q^{32} -2.00000 q^{33} +6.56155 q^{34} -2.00000 q^{35} +4.56155 q^{36} +3.43845 q^{37} -2.87689 q^{38} +3.68466 q^{40} +2.56155 q^{41} -9.12311 q^{42} +0.438447 q^{43} -9.12311 q^{44} +0.561553 q^{45} +5.12311 q^{46} -8.24621 q^{47} +7.68466 q^{48} +5.68466 q^{49} -12.0000 q^{50} +2.56155 q^{51} +11.6847 q^{53} +2.56155 q^{54} -1.12311 q^{55} -23.3693 q^{56} -1.12311 q^{57} -14.5616 q^{58} -11.1231 q^{59} +2.56155 q^{60} +12.1231 q^{61} -4.00000 q^{62} -3.56155 q^{63} +1.43845 q^{64} -5.12311 q^{66} +0.438447 q^{67} +11.6847 q^{68} +2.00000 q^{69} -5.12311 q^{70} +14.0000 q^{71} +6.56155 q^{72} -1.87689 q^{73} +8.80776 q^{74} -4.68466 q^{75} -5.12311 q^{76} +7.12311 q^{77} +9.56155 q^{79} +4.31534 q^{80} +1.00000 q^{81} +6.56155 q^{82} -9.12311 q^{83} -16.2462 q^{84} +1.43845 q^{85} +1.12311 q^{86} -5.68466 q^{87} -13.1231 q^{88} +13.1231 q^{89} +1.43845 q^{90} +9.12311 q^{92} -1.56155 q^{93} -21.1231 q^{94} -0.630683 q^{95} +6.56155 q^{96} -4.43845 q^{97} +14.5616 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 3 q^{7} + 9 q^{8} + 2 q^{9} + O(q^{10}) \) \( 2 q + q^{2} + 2 q^{3} + 5 q^{4} - 3 q^{5} + q^{6} - 3 q^{7} + 9 q^{8} + 2 q^{9} + 7 q^{10} - 4 q^{11} + 5 q^{12} - 10 q^{14} - 3 q^{15} + 3 q^{16} + q^{17} + q^{18} + 6 q^{19} + q^{20} - 3 q^{21} - 2 q^{22} + 4 q^{23} + 9 q^{24} + 3 q^{25} + 2 q^{27} - 16 q^{28} + q^{29} + 7 q^{30} + q^{31} + 9 q^{32} - 4 q^{33} + 9 q^{34} - 4 q^{35} + 5 q^{36} + 11 q^{37} - 14 q^{38} - 5 q^{40} + q^{41} - 10 q^{42} + 5 q^{43} - 10 q^{44} - 3 q^{45} + 2 q^{46} + 3 q^{48} - q^{49} - 24 q^{50} + q^{51} + 11 q^{53} + q^{54} + 6 q^{55} - 22 q^{56} + 6 q^{57} - 25 q^{58} - 14 q^{59} + q^{60} + 16 q^{61} - 8 q^{62} - 3 q^{63} + 7 q^{64} - 2 q^{66} + 5 q^{67} + 11 q^{68} + 4 q^{69} - 2 q^{70} + 28 q^{71} + 9 q^{72} - 12 q^{73} - 3 q^{74} + 3 q^{75} - 2 q^{76} + 6 q^{77} + 15 q^{79} + 21 q^{80} + 2 q^{81} + 9 q^{82} - 10 q^{83} - 16 q^{84} + 7 q^{85} - 6 q^{86} + q^{87} - 18 q^{88} + 18 q^{89} + 7 q^{90} + 10 q^{92} + q^{93} - 34 q^{94} - 26 q^{95} + 9 q^{96} - 13 q^{97} + 25 q^{98} - 4 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 1.81129 0.905646 0.424035i \(-0.139387\pi\)
0.905646 + 0.424035i \(0.139387\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.56155 2.28078
\(5\) 0.561553 0.251134 0.125567 0.992085i \(-0.459925\pi\)
0.125567 + 0.992085i \(0.459925\pi\)
\(6\) 2.56155 1.04575
\(7\) −3.56155 −1.34614 −0.673070 0.739579i \(-0.735025\pi\)
−0.673070 + 0.739579i \(0.735025\pi\)
\(8\) 6.56155 2.31986
\(9\) 1.00000 0.333333
\(10\) 1.43845 0.454877
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 4.56155 1.31681
\(13\) 0 0
\(14\) −9.12311 −2.43825
\(15\) 0.561553 0.144992
\(16\) 7.68466 1.92116
\(17\) 2.56155 0.621268 0.310634 0.950530i \(-0.399459\pi\)
0.310634 + 0.950530i \(0.399459\pi\)
\(18\) 2.56155 0.603764
\(19\) −1.12311 −0.257658 −0.128829 0.991667i \(-0.541122\pi\)
−0.128829 + 0.991667i \(0.541122\pi\)
\(20\) 2.56155 0.572781
\(21\) −3.56155 −0.777195
\(22\) −5.12311 −1.09225
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 6.56155 1.33937
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −16.2462 −3.07025
\(29\) −5.68466 −1.05561 −0.527807 0.849364i \(-0.676986\pi\)
−0.527807 + 0.849364i \(0.676986\pi\)
\(30\) 1.43845 0.262623
\(31\) −1.56155 −0.280463 −0.140232 0.990119i \(-0.544785\pi\)
−0.140232 + 0.990119i \(0.544785\pi\)
\(32\) 6.56155 1.15993
\(33\) −2.00000 −0.348155
\(34\) 6.56155 1.12530
\(35\) −2.00000 −0.338062
\(36\) 4.56155 0.760259
\(37\) 3.43845 0.565277 0.282639 0.959226i \(-0.408790\pi\)
0.282639 + 0.959226i \(0.408790\pi\)
\(38\) −2.87689 −0.466694
\(39\) 0 0
\(40\) 3.68466 0.582596
\(41\) 2.56155 0.400047 0.200024 0.979791i \(-0.435898\pi\)
0.200024 + 0.979791i \(0.435898\pi\)
\(42\) −9.12311 −1.40773
\(43\) 0.438447 0.0668626 0.0334313 0.999441i \(-0.489357\pi\)
0.0334313 + 0.999441i \(0.489357\pi\)
\(44\) −9.12311 −1.37536
\(45\) 0.561553 0.0837114
\(46\) 5.12311 0.755361
\(47\) −8.24621 −1.20283 −0.601417 0.798935i \(-0.705397\pi\)
−0.601417 + 0.798935i \(0.705397\pi\)
\(48\) 7.68466 1.10918
\(49\) 5.68466 0.812094
\(50\) −12.0000 −1.69706
\(51\) 2.56155 0.358689
\(52\) 0 0
\(53\) 11.6847 1.60501 0.802506 0.596645i \(-0.203500\pi\)
0.802506 + 0.596645i \(0.203500\pi\)
\(54\) 2.56155 0.348583
\(55\) −1.12311 −0.151440
\(56\) −23.3693 −3.12286
\(57\) −1.12311 −0.148759
\(58\) −14.5616 −1.91203
\(59\) −11.1231 −1.44811 −0.724053 0.689745i \(-0.757723\pi\)
−0.724053 + 0.689745i \(0.757723\pi\)
\(60\) 2.56155 0.330695
\(61\) 12.1231 1.55220 0.776102 0.630607i \(-0.217194\pi\)
0.776102 + 0.630607i \(0.217194\pi\)
\(62\) −4.00000 −0.508001
\(63\) −3.56155 −0.448713
\(64\) 1.43845 0.179806
\(65\) 0 0
\(66\) −5.12311 −0.630611
\(67\) 0.438447 0.0535648 0.0267824 0.999641i \(-0.491474\pi\)
0.0267824 + 0.999641i \(0.491474\pi\)
\(68\) 11.6847 1.41697
\(69\) 2.00000 0.240772
\(70\) −5.12311 −0.612328
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 6.56155 0.773286
\(73\) −1.87689 −0.219674 −0.109837 0.993950i \(-0.535033\pi\)
−0.109837 + 0.993950i \(0.535033\pi\)
\(74\) 8.80776 1.02388
\(75\) −4.68466 −0.540938
\(76\) −5.12311 −0.587661
\(77\) 7.12311 0.811753
\(78\) 0 0
\(79\) 9.56155 1.07576 0.537879 0.843022i \(-0.319226\pi\)
0.537879 + 0.843022i \(0.319226\pi\)
\(80\) 4.31534 0.482470
\(81\) 1.00000 0.111111
\(82\) 6.56155 0.724602
\(83\) −9.12311 −1.00139 −0.500695 0.865624i \(-0.666922\pi\)
−0.500695 + 0.865624i \(0.666922\pi\)
\(84\) −16.2462 −1.77261
\(85\) 1.43845 0.156022
\(86\) 1.12311 0.121108
\(87\) −5.68466 −0.609459
\(88\) −13.1231 −1.39893
\(89\) 13.1231 1.39105 0.695523 0.718504i \(-0.255173\pi\)
0.695523 + 0.718504i \(0.255173\pi\)
\(90\) 1.43845 0.151626
\(91\) 0 0
\(92\) 9.12311 0.951150
\(93\) −1.56155 −0.161925
\(94\) −21.1231 −2.17868
\(95\) −0.630683 −0.0647067
\(96\) 6.56155 0.669686
\(97\) −4.43845 −0.450656 −0.225328 0.974283i \(-0.572345\pi\)
−0.225328 + 0.974283i \(0.572345\pi\)
\(98\) 14.5616 1.47094
\(99\) −2.00000 −0.201008
\(100\) −21.3693 −2.13693
\(101\) −3.43845 −0.342138 −0.171069 0.985259i \(-0.554722\pi\)
−0.171069 + 0.985259i \(0.554722\pi\)
\(102\) 6.56155 0.649691
\(103\) −7.56155 −0.745062 −0.372531 0.928020i \(-0.621510\pi\)
−0.372531 + 0.928020i \(0.621510\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 29.9309 2.90714
\(107\) 8.24621 0.797191 0.398596 0.917127i \(-0.369498\pi\)
0.398596 + 0.917127i \(0.369498\pi\)
\(108\) 4.56155 0.438936
\(109\) 17.8078 1.70567 0.852837 0.522177i \(-0.174880\pi\)
0.852837 + 0.522177i \(0.174880\pi\)
\(110\) −2.87689 −0.274301
\(111\) 3.43845 0.326363
\(112\) −27.3693 −2.58616
\(113\) −14.8078 −1.39300 −0.696499 0.717558i \(-0.745260\pi\)
−0.696499 + 0.717558i \(0.745260\pi\)
\(114\) −2.87689 −0.269446
\(115\) 1.12311 0.104730
\(116\) −25.9309 −2.40762
\(117\) 0 0
\(118\) −28.4924 −2.62294
\(119\) −9.12311 −0.836314
\(120\) 3.68466 0.336362
\(121\) −7.00000 −0.636364
\(122\) 31.0540 2.81149
\(123\) 2.56155 0.230967
\(124\) −7.12311 −0.639674
\(125\) −5.43845 −0.486430
\(126\) −9.12311 −0.812751
\(127\) 9.56155 0.848451 0.424225 0.905557i \(-0.360546\pi\)
0.424225 + 0.905557i \(0.360546\pi\)
\(128\) −9.43845 −0.834249
\(129\) 0.438447 0.0386031
\(130\) 0 0
\(131\) −17.3693 −1.51756 −0.758782 0.651345i \(-0.774205\pi\)
−0.758782 + 0.651345i \(0.774205\pi\)
\(132\) −9.12311 −0.794064
\(133\) 4.00000 0.346844
\(134\) 1.12311 0.0970215
\(135\) 0.561553 0.0483308
\(136\) 16.8078 1.44125
\(137\) −1.43845 −0.122895 −0.0614474 0.998110i \(-0.519572\pi\)
−0.0614474 + 0.998110i \(0.519572\pi\)
\(138\) 5.12311 0.436108
\(139\) 10.9309 0.927144 0.463572 0.886059i \(-0.346567\pi\)
0.463572 + 0.886059i \(0.346567\pi\)
\(140\) −9.12311 −0.771043
\(141\) −8.24621 −0.694456
\(142\) 35.8617 3.00945
\(143\) 0 0
\(144\) 7.68466 0.640388
\(145\) −3.19224 −0.265101
\(146\) −4.80776 −0.397893
\(147\) 5.68466 0.468863
\(148\) 15.6847 1.28927
\(149\) −6.56155 −0.537543 −0.268772 0.963204i \(-0.586618\pi\)
−0.268772 + 0.963204i \(0.586618\pi\)
\(150\) −12.0000 −0.979796
\(151\) 15.3693 1.25074 0.625369 0.780329i \(-0.284949\pi\)
0.625369 + 0.780329i \(0.284949\pi\)
\(152\) −7.36932 −0.597731
\(153\) 2.56155 0.207089
\(154\) 18.2462 1.47032
\(155\) −0.876894 −0.0704339
\(156\) 0 0
\(157\) −4.36932 −0.348709 −0.174355 0.984683i \(-0.555784\pi\)
−0.174355 + 0.984683i \(0.555784\pi\)
\(158\) 24.4924 1.94851
\(159\) 11.6847 0.926654
\(160\) 3.68466 0.291298
\(161\) −7.12311 −0.561379
\(162\) 2.56155 0.201255
\(163\) 15.8078 1.23816 0.619080 0.785328i \(-0.287506\pi\)
0.619080 + 0.785328i \(0.287506\pi\)
\(164\) 11.6847 0.912419
\(165\) −1.12311 −0.0874337
\(166\) −23.3693 −1.81381
\(167\) −6.24621 −0.483346 −0.241673 0.970358i \(-0.577696\pi\)
−0.241673 + 0.970358i \(0.577696\pi\)
\(168\) −23.3693 −1.80298
\(169\) 0 0
\(170\) 3.68466 0.282600
\(171\) −1.12311 −0.0858860
\(172\) 2.00000 0.152499
\(173\) 3.75379 0.285395 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(174\) −14.5616 −1.10391
\(175\) 16.6847 1.26124
\(176\) −15.3693 −1.15851
\(177\) −11.1231 −0.836064
\(178\) 33.6155 2.51959
\(179\) −13.1231 −0.980867 −0.490433 0.871479i \(-0.663162\pi\)
−0.490433 + 0.871479i \(0.663162\pi\)
\(180\) 2.56155 0.190927
\(181\) 9.68466 0.719855 0.359927 0.932980i \(-0.382801\pi\)
0.359927 + 0.932980i \(0.382801\pi\)
\(182\) 0 0
\(183\) 12.1231 0.896166
\(184\) 13.1231 0.967448
\(185\) 1.93087 0.141960
\(186\) −4.00000 −0.293294
\(187\) −5.12311 −0.374639
\(188\) −37.6155 −2.74339
\(189\) −3.56155 −0.259065
\(190\) −1.61553 −0.117203
\(191\) −0.876894 −0.0634499 −0.0317249 0.999497i \(-0.510100\pi\)
−0.0317249 + 0.999497i \(0.510100\pi\)
\(192\) 1.43845 0.103811
\(193\) 19.4924 1.40310 0.701548 0.712623i \(-0.252493\pi\)
0.701548 + 0.712623i \(0.252493\pi\)
\(194\) −11.3693 −0.816269
\(195\) 0 0
\(196\) 25.9309 1.85220
\(197\) −11.3693 −0.810030 −0.405015 0.914310i \(-0.632734\pi\)
−0.405015 + 0.914310i \(0.632734\pi\)
\(198\) −5.12311 −0.364083
\(199\) −23.1771 −1.64298 −0.821490 0.570223i \(-0.806857\pi\)
−0.821490 + 0.570223i \(0.806857\pi\)
\(200\) −30.7386 −2.17355
\(201\) 0.438447 0.0309257
\(202\) −8.80776 −0.619712
\(203\) 20.2462 1.42101
\(204\) 11.6847 0.818090
\(205\) 1.43845 0.100466
\(206\) −19.3693 −1.34952
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) 2.24621 0.155374
\(210\) −5.12311 −0.353528
\(211\) 7.31534 0.503609 0.251804 0.967778i \(-0.418976\pi\)
0.251804 + 0.967778i \(0.418976\pi\)
\(212\) 53.3002 3.66067
\(213\) 14.0000 0.959264
\(214\) 21.1231 1.44395
\(215\) 0.246211 0.0167915
\(216\) 6.56155 0.446457
\(217\) 5.56155 0.377543
\(218\) 45.6155 3.08947
\(219\) −1.87689 −0.126829
\(220\) −5.12311 −0.345400
\(221\) 0 0
\(222\) 8.80776 0.591138
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) −23.3693 −1.56143
\(225\) −4.68466 −0.312311
\(226\) −37.9309 −2.52312
\(227\) −1.12311 −0.0745431 −0.0372716 0.999305i \(-0.511867\pi\)
−0.0372716 + 0.999305i \(0.511867\pi\)
\(228\) −5.12311 −0.339286
\(229\) 0.246211 0.0162701 0.00813505 0.999967i \(-0.497411\pi\)
0.00813505 + 0.999967i \(0.497411\pi\)
\(230\) 2.87689 0.189697
\(231\) 7.12311 0.468666
\(232\) −37.3002 −2.44888
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) −4.63068 −0.302072
\(236\) −50.7386 −3.30280
\(237\) 9.56155 0.621090
\(238\) −23.3693 −1.51481
\(239\) −0.630683 −0.0407955 −0.0203977 0.999792i \(-0.506493\pi\)
−0.0203977 + 0.999792i \(0.506493\pi\)
\(240\) 4.31534 0.278554
\(241\) 2.80776 0.180864 0.0904320 0.995903i \(-0.471175\pi\)
0.0904320 + 0.995903i \(0.471175\pi\)
\(242\) −17.9309 −1.15264
\(243\) 1.00000 0.0641500
\(244\) 55.3002 3.54023
\(245\) 3.19224 0.203944
\(246\) 6.56155 0.418349
\(247\) 0 0
\(248\) −10.2462 −0.650635
\(249\) −9.12311 −0.578153
\(250\) −13.9309 −0.881066
\(251\) −30.7386 −1.94021 −0.970103 0.242695i \(-0.921969\pi\)
−0.970103 + 0.242695i \(0.921969\pi\)
\(252\) −16.2462 −1.02342
\(253\) −4.00000 −0.251478
\(254\) 24.4924 1.53679
\(255\) 1.43845 0.0900791
\(256\) −27.0540 −1.69087
\(257\) −16.1771 −1.00910 −0.504549 0.863383i \(-0.668341\pi\)
−0.504549 + 0.863383i \(0.668341\pi\)
\(258\) 1.12311 0.0699215
\(259\) −12.2462 −0.760943
\(260\) 0 0
\(261\) −5.68466 −0.351872
\(262\) −44.4924 −2.74875
\(263\) −15.3693 −0.947713 −0.473856 0.880602i \(-0.657138\pi\)
−0.473856 + 0.880602i \(0.657138\pi\)
\(264\) −13.1231 −0.807671
\(265\) 6.56155 0.403073
\(266\) 10.2462 0.628236
\(267\) 13.1231 0.803121
\(268\) 2.00000 0.122169
\(269\) 3.36932 0.205431 0.102715 0.994711i \(-0.467247\pi\)
0.102715 + 0.994711i \(0.467247\pi\)
\(270\) 1.43845 0.0875411
\(271\) −1.06913 −0.0649450 −0.0324725 0.999473i \(-0.510338\pi\)
−0.0324725 + 0.999473i \(0.510338\pi\)
\(272\) 19.6847 1.19356
\(273\) 0 0
\(274\) −3.68466 −0.222598
\(275\) 9.36932 0.564991
\(276\) 9.12311 0.549146
\(277\) 17.6847 1.06257 0.531284 0.847194i \(-0.321710\pi\)
0.531284 + 0.847194i \(0.321710\pi\)
\(278\) 28.0000 1.67933
\(279\) −1.56155 −0.0934877
\(280\) −13.1231 −0.784256
\(281\) −2.80776 −0.167497 −0.0837486 0.996487i \(-0.526689\pi\)
−0.0837486 + 0.996487i \(0.526689\pi\)
\(282\) −21.1231 −1.25786
\(283\) −1.31534 −0.0781889 −0.0390945 0.999236i \(-0.512447\pi\)
−0.0390945 + 0.999236i \(0.512447\pi\)
\(284\) 63.8617 3.78950
\(285\) −0.630683 −0.0373584
\(286\) 0 0
\(287\) −9.12311 −0.538520
\(288\) 6.56155 0.386643
\(289\) −10.4384 −0.614026
\(290\) −8.17708 −0.480175
\(291\) −4.43845 −0.260186
\(292\) −8.56155 −0.501027
\(293\) 24.5616 1.43490 0.717451 0.696609i \(-0.245309\pi\)
0.717451 + 0.696609i \(0.245309\pi\)
\(294\) 14.5616 0.849247
\(295\) −6.24621 −0.363668
\(296\) 22.5616 1.31136
\(297\) −2.00000 −0.116052
\(298\) −16.8078 −0.973648
\(299\) 0 0
\(300\) −21.3693 −1.23376
\(301\) −1.56155 −0.0900064
\(302\) 39.3693 2.26545
\(303\) −3.43845 −0.197534
\(304\) −8.63068 −0.495004
\(305\) 6.80776 0.389811
\(306\) 6.56155 0.375099
\(307\) 10.1922 0.581702 0.290851 0.956768i \(-0.406062\pi\)
0.290851 + 0.956768i \(0.406062\pi\)
\(308\) 32.4924 1.85143
\(309\) −7.56155 −0.430162
\(310\) −2.24621 −0.127576
\(311\) −10.8769 −0.616772 −0.308386 0.951261i \(-0.599789\pi\)
−0.308386 + 0.951261i \(0.599789\pi\)
\(312\) 0 0
\(313\) −1.31534 −0.0743475 −0.0371738 0.999309i \(-0.511835\pi\)
−0.0371738 + 0.999309i \(0.511835\pi\)
\(314\) −11.1922 −0.631614
\(315\) −2.00000 −0.112687
\(316\) 43.6155 2.45357
\(317\) −23.0540 −1.29484 −0.647420 0.762133i \(-0.724152\pi\)
−0.647420 + 0.762133i \(0.724152\pi\)
\(318\) 29.9309 1.67844
\(319\) 11.3693 0.636560
\(320\) 0.807764 0.0451554
\(321\) 8.24621 0.460259
\(322\) −18.2462 −1.01682
\(323\) −2.87689 −0.160075
\(324\) 4.56155 0.253420
\(325\) 0 0
\(326\) 40.4924 2.24267
\(327\) 17.8078 0.984772
\(328\) 16.8078 0.928054
\(329\) 29.3693 1.61918
\(330\) −2.87689 −0.158368
\(331\) −23.8078 −1.30859 −0.654297 0.756238i \(-0.727035\pi\)
−0.654297 + 0.756238i \(0.727035\pi\)
\(332\) −41.6155 −2.28395
\(333\) 3.43845 0.188426
\(334\) −16.0000 −0.875481
\(335\) 0.246211 0.0134520
\(336\) −27.3693 −1.49312
\(337\) 2.12311 0.115653 0.0578265 0.998327i \(-0.481583\pi\)
0.0578265 + 0.998327i \(0.481583\pi\)
\(338\) 0 0
\(339\) −14.8078 −0.804247
\(340\) 6.56155 0.355850
\(341\) 3.12311 0.169126
\(342\) −2.87689 −0.155565
\(343\) 4.68466 0.252948
\(344\) 2.87689 0.155112
\(345\) 1.12311 0.0604660
\(346\) 9.61553 0.516934
\(347\) −13.6155 −0.730920 −0.365460 0.930827i \(-0.619088\pi\)
−0.365460 + 0.930827i \(0.619088\pi\)
\(348\) −25.9309 −1.39004
\(349\) −13.8078 −0.739113 −0.369556 0.929208i \(-0.620490\pi\)
−0.369556 + 0.929208i \(0.620490\pi\)
\(350\) 42.7386 2.28448
\(351\) 0 0
\(352\) −13.1231 −0.699464
\(353\) 17.6847 0.941259 0.470630 0.882331i \(-0.344027\pi\)
0.470630 + 0.882331i \(0.344027\pi\)
\(354\) −28.4924 −1.51436
\(355\) 7.86174 0.417258
\(356\) 59.8617 3.17267
\(357\) −9.12311 −0.482846
\(358\) −33.6155 −1.77664
\(359\) 15.3693 0.811162 0.405581 0.914059i \(-0.367069\pi\)
0.405581 + 0.914059i \(0.367069\pi\)
\(360\) 3.68466 0.194199
\(361\) −17.7386 −0.933612
\(362\) 24.8078 1.30387
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) −1.05398 −0.0551676
\(366\) 31.0540 1.62322
\(367\) 20.0540 1.04681 0.523404 0.852084i \(-0.324662\pi\)
0.523404 + 0.852084i \(0.324662\pi\)
\(368\) 15.3693 0.801181
\(369\) 2.56155 0.133349
\(370\) 4.94602 0.257132
\(371\) −41.6155 −2.16057
\(372\) −7.12311 −0.369316
\(373\) 3.63068 0.187990 0.0939948 0.995573i \(-0.470036\pi\)
0.0939948 + 0.995573i \(0.470036\pi\)
\(374\) −13.1231 −0.678580
\(375\) −5.43845 −0.280840
\(376\) −54.1080 −2.79040
\(377\) 0 0
\(378\) −9.12311 −0.469242
\(379\) 11.3153 0.581230 0.290615 0.956840i \(-0.406140\pi\)
0.290615 + 0.956840i \(0.406140\pi\)
\(380\) −2.87689 −0.147582
\(381\) 9.56155 0.489853
\(382\) −2.24621 −0.114926
\(383\) 26.7386 1.36628 0.683140 0.730287i \(-0.260614\pi\)
0.683140 + 0.730287i \(0.260614\pi\)
\(384\) −9.43845 −0.481654
\(385\) 4.00000 0.203859
\(386\) 49.9309 2.54141
\(387\) 0.438447 0.0222875
\(388\) −20.2462 −1.02785
\(389\) −3.05398 −0.154843 −0.0774213 0.996998i \(-0.524669\pi\)
−0.0774213 + 0.996998i \(0.524669\pi\)
\(390\) 0 0
\(391\) 5.12311 0.259087
\(392\) 37.3002 1.88394
\(393\) −17.3693 −0.876166
\(394\) −29.1231 −1.46720
\(395\) 5.36932 0.270160
\(396\) −9.12311 −0.458453
\(397\) −12.0540 −0.604972 −0.302486 0.953154i \(-0.597816\pi\)
−0.302486 + 0.953154i \(0.597816\pi\)
\(398\) −59.3693 −2.97591
\(399\) 4.00000 0.200250
\(400\) −36.0000 −1.80000
\(401\) 18.5616 0.926920 0.463460 0.886118i \(-0.346608\pi\)
0.463460 + 0.886118i \(0.346608\pi\)
\(402\) 1.12311 0.0560154
\(403\) 0 0
\(404\) −15.6847 −0.780341
\(405\) 0.561553 0.0279038
\(406\) 51.8617 2.57385
\(407\) −6.87689 −0.340875
\(408\) 16.8078 0.832108
\(409\) −18.3693 −0.908304 −0.454152 0.890924i \(-0.650058\pi\)
−0.454152 + 0.890924i \(0.650058\pi\)
\(410\) 3.68466 0.181972
\(411\) −1.43845 −0.0709534
\(412\) −34.4924 −1.69932
\(413\) 39.6155 1.94935
\(414\) 5.12311 0.251787
\(415\) −5.12311 −0.251483
\(416\) 0 0
\(417\) 10.9309 0.535287
\(418\) 5.75379 0.281427
\(419\) −17.7538 −0.867329 −0.433665 0.901074i \(-0.642780\pi\)
−0.433665 + 0.901074i \(0.642780\pi\)
\(420\) −9.12311 −0.445162
\(421\) 14.7538 0.719056 0.359528 0.933134i \(-0.382938\pi\)
0.359528 + 0.933134i \(0.382938\pi\)
\(422\) 18.7386 0.912182
\(423\) −8.24621 −0.400945
\(424\) 76.6695 3.72340
\(425\) −12.0000 −0.582086
\(426\) 35.8617 1.73751
\(427\) −43.1771 −2.08949
\(428\) 37.6155 1.81822
\(429\) 0 0
\(430\) 0.630683 0.0304142
\(431\) −2.87689 −0.138575 −0.0692876 0.997597i \(-0.522073\pi\)
−0.0692876 + 0.997597i \(0.522073\pi\)
\(432\) 7.68466 0.369728
\(433\) 25.2462 1.21326 0.606628 0.794986i \(-0.292522\pi\)
0.606628 + 0.794986i \(0.292522\pi\)
\(434\) 14.2462 0.683840
\(435\) −3.19224 −0.153056
\(436\) 81.2311 3.89026
\(437\) −2.24621 −0.107451
\(438\) −4.80776 −0.229724
\(439\) −1.31534 −0.0627778 −0.0313889 0.999507i \(-0.509993\pi\)
−0.0313889 + 0.999507i \(0.509993\pi\)
\(440\) −7.36932 −0.351318
\(441\) 5.68466 0.270698
\(442\) 0 0
\(443\) 14.7386 0.700254 0.350127 0.936702i \(-0.386138\pi\)
0.350127 + 0.936702i \(0.386138\pi\)
\(444\) 15.6847 0.744361
\(445\) 7.36932 0.349339
\(446\) 20.4924 0.970344
\(447\) −6.56155 −0.310351
\(448\) −5.12311 −0.242044
\(449\) 8.24621 0.389163 0.194581 0.980886i \(-0.437665\pi\)
0.194581 + 0.980886i \(0.437665\pi\)
\(450\) −12.0000 −0.565685
\(451\) −5.12311 −0.241238
\(452\) −67.5464 −3.17712
\(453\) 15.3693 0.722113
\(454\) −2.87689 −0.135019
\(455\) 0 0
\(456\) −7.36932 −0.345100
\(457\) −28.6155 −1.33858 −0.669289 0.743002i \(-0.733401\pi\)
−0.669289 + 0.743002i \(0.733401\pi\)
\(458\) 0.630683 0.0294699
\(459\) 2.56155 0.119563
\(460\) 5.12311 0.238866
\(461\) −36.8078 −1.71431 −0.857154 0.515060i \(-0.827770\pi\)
−0.857154 + 0.515060i \(0.827770\pi\)
\(462\) 18.2462 0.848891
\(463\) −26.6847 −1.24014 −0.620071 0.784546i \(-0.712896\pi\)
−0.620071 + 0.784546i \(0.712896\pi\)
\(464\) −43.6847 −2.02801
\(465\) −0.876894 −0.0406650
\(466\) 66.6004 3.08520
\(467\) −26.0000 −1.20314 −0.601568 0.798821i \(-0.705457\pi\)
−0.601568 + 0.798821i \(0.705457\pi\)
\(468\) 0 0
\(469\) −1.56155 −0.0721058
\(470\) −11.8617 −0.547141
\(471\) −4.36932 −0.201327
\(472\) −72.9848 −3.35940
\(473\) −0.876894 −0.0403196
\(474\) 24.4924 1.12497
\(475\) 5.26137 0.241408
\(476\) −41.6155 −1.90744
\(477\) 11.6847 0.535004
\(478\) −1.61553 −0.0738925
\(479\) −6.24621 −0.285397 −0.142698 0.989766i \(-0.545578\pi\)
−0.142698 + 0.989766i \(0.545578\pi\)
\(480\) 3.68466 0.168181
\(481\) 0 0
\(482\) 7.19224 0.327597
\(483\) −7.12311 −0.324113
\(484\) −31.9309 −1.45140
\(485\) −2.49242 −0.113175
\(486\) 2.56155 0.116194
\(487\) −1.12311 −0.0508928 −0.0254464 0.999676i \(-0.508101\pi\)
−0.0254464 + 0.999676i \(0.508101\pi\)
\(488\) 79.5464 3.60090
\(489\) 15.8078 0.714852
\(490\) 8.17708 0.369403
\(491\) −19.7538 −0.891476 −0.445738 0.895163i \(-0.647059\pi\)
−0.445738 + 0.895163i \(0.647059\pi\)
\(492\) 11.6847 0.526785
\(493\) −14.5616 −0.655819
\(494\) 0 0
\(495\) −1.12311 −0.0504798
\(496\) −12.0000 −0.538816
\(497\) −49.8617 −2.23660
\(498\) −23.3693 −1.04720
\(499\) −28.4924 −1.27550 −0.637748 0.770245i \(-0.720134\pi\)
−0.637748 + 0.770245i \(0.720134\pi\)
\(500\) −24.8078 −1.10944
\(501\) −6.24621 −0.279060
\(502\) −78.7386 −3.51428
\(503\) −11.7538 −0.524076 −0.262038 0.965058i \(-0.584395\pi\)
−0.262038 + 0.965058i \(0.584395\pi\)
\(504\) −23.3693 −1.04095
\(505\) −1.93087 −0.0859226
\(506\) −10.2462 −0.455500
\(507\) 0 0
\(508\) 43.6155 1.93513
\(509\) 6.80776 0.301749 0.150874 0.988553i \(-0.451791\pi\)
0.150874 + 0.988553i \(0.451791\pi\)
\(510\) 3.68466 0.163159
\(511\) 6.68466 0.295712
\(512\) −50.4233 −2.22842
\(513\) −1.12311 −0.0495863
\(514\) −41.4384 −1.82777
\(515\) −4.24621 −0.187110
\(516\) 2.00000 0.0880451
\(517\) 16.4924 0.725336
\(518\) −31.3693 −1.37829
\(519\) 3.75379 0.164773
\(520\) 0 0
\(521\) −37.9309 −1.66178 −0.830891 0.556436i \(-0.812169\pi\)
−0.830891 + 0.556436i \(0.812169\pi\)
\(522\) −14.5616 −0.637342
\(523\) −23.8617 −1.04340 −0.521701 0.853129i \(-0.674702\pi\)
−0.521701 + 0.853129i \(0.674702\pi\)
\(524\) −79.2311 −3.46122
\(525\) 16.6847 0.728178
\(526\) −39.3693 −1.71658
\(527\) −4.00000 −0.174243
\(528\) −15.3693 −0.668864
\(529\) −19.0000 −0.826087
\(530\) 16.8078 0.730083
\(531\) −11.1231 −0.482702
\(532\) 18.2462 0.791074
\(533\) 0 0
\(534\) 33.6155 1.45469
\(535\) 4.63068 0.200202
\(536\) 2.87689 0.124263
\(537\) −13.1231 −0.566304
\(538\) 8.63068 0.372095
\(539\) −11.3693 −0.489711
\(540\) 2.56155 0.110232
\(541\) −29.7386 −1.27856 −0.639282 0.768972i \(-0.720768\pi\)
−0.639282 + 0.768972i \(0.720768\pi\)
\(542\) −2.73863 −0.117634
\(543\) 9.68466 0.415608
\(544\) 16.8078 0.720627
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) −24.9309 −1.06597 −0.532984 0.846126i \(-0.678929\pi\)
−0.532984 + 0.846126i \(0.678929\pi\)
\(548\) −6.56155 −0.280296
\(549\) 12.1231 0.517402
\(550\) 24.0000 1.02336
\(551\) 6.38447 0.271988
\(552\) 13.1231 0.558556
\(553\) −34.0540 −1.44812
\(554\) 45.3002 1.92462
\(555\) 1.93087 0.0819609
\(556\) 49.8617 2.11461
\(557\) −14.0691 −0.596128 −0.298064 0.954546i \(-0.596341\pi\)
−0.298064 + 0.954546i \(0.596341\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) −15.3693 −0.649472
\(561\) −5.12311 −0.216298
\(562\) −7.19224 −0.303386
\(563\) 1.36932 0.0577098 0.0288549 0.999584i \(-0.490814\pi\)
0.0288549 + 0.999584i \(0.490814\pi\)
\(564\) −37.6155 −1.58390
\(565\) −8.31534 −0.349829
\(566\) −3.36932 −0.141623
\(567\) −3.56155 −0.149571
\(568\) 91.8617 3.85443
\(569\) 40.7386 1.70785 0.853926 0.520394i \(-0.174215\pi\)
0.853926 + 0.520394i \(0.174215\pi\)
\(570\) −1.61553 −0.0676670
\(571\) 19.3693 0.810581 0.405290 0.914188i \(-0.367170\pi\)
0.405290 + 0.914188i \(0.367170\pi\)
\(572\) 0 0
\(573\) −0.876894 −0.0366328
\(574\) −23.3693 −0.975416
\(575\) −9.36932 −0.390728
\(576\) 1.43845 0.0599353
\(577\) −29.6847 −1.23579 −0.617894 0.786261i \(-0.712014\pi\)
−0.617894 + 0.786261i \(0.712014\pi\)
\(578\) −26.7386 −1.11218
\(579\) 19.4924 0.810077
\(580\) −14.5616 −0.604636
\(581\) 32.4924 1.34801
\(582\) −11.3693 −0.471273
\(583\) −23.3693 −0.967858
\(584\) −12.3153 −0.509612
\(585\) 0 0
\(586\) 62.9157 2.59902
\(587\) 14.6307 0.603873 0.301936 0.953328i \(-0.402367\pi\)
0.301936 + 0.953328i \(0.402367\pi\)
\(588\) 25.9309 1.06937
\(589\) 1.75379 0.0722636
\(590\) −16.0000 −0.658710
\(591\) −11.3693 −0.467671
\(592\) 26.4233 1.08599
\(593\) 44.4233 1.82425 0.912123 0.409917i \(-0.134442\pi\)
0.912123 + 0.409917i \(0.134442\pi\)
\(594\) −5.12311 −0.210204
\(595\) −5.12311 −0.210027
\(596\) −29.9309 −1.22602
\(597\) −23.1771 −0.948575
\(598\) 0 0
\(599\) −0.384472 −0.0157091 −0.00785455 0.999969i \(-0.502500\pi\)
−0.00785455 + 0.999969i \(0.502500\pi\)
\(600\) −30.7386 −1.25490
\(601\) −35.9309 −1.46565 −0.732825 0.680417i \(-0.761799\pi\)
−0.732825 + 0.680417i \(0.761799\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0.438447 0.0178549
\(604\) 70.1080 2.85265
\(605\) −3.93087 −0.159813
\(606\) −8.80776 −0.357791
\(607\) −16.0000 −0.649420 −0.324710 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) −7.36932 −0.298865
\(609\) 20.2462 0.820418
\(610\) 17.4384 0.706062
\(611\) 0 0
\(612\) 11.6847 0.472324
\(613\) 22.8617 0.923377 0.461688 0.887042i \(-0.347244\pi\)
0.461688 + 0.887042i \(0.347244\pi\)
\(614\) 26.1080 1.05363
\(615\) 1.43845 0.0580038
\(616\) 46.7386 1.88315
\(617\) −10.8078 −0.435104 −0.217552 0.976049i \(-0.569807\pi\)
−0.217552 + 0.976049i \(0.569807\pi\)
\(618\) −19.3693 −0.779148
\(619\) −24.3002 −0.976707 −0.488353 0.872646i \(-0.662402\pi\)
−0.488353 + 0.872646i \(0.662402\pi\)
\(620\) −4.00000 −0.160644
\(621\) 2.00000 0.0802572
\(622\) −27.8617 −1.11715
\(623\) −46.7386 −1.87254
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) −3.36932 −0.134665
\(627\) 2.24621 0.0897050
\(628\) −19.9309 −0.795328
\(629\) 8.80776 0.351189
\(630\) −5.12311 −0.204109
\(631\) −14.4384 −0.574786 −0.287393 0.957813i \(-0.592788\pi\)
−0.287393 + 0.957813i \(0.592788\pi\)
\(632\) 62.7386 2.49561
\(633\) 7.31534 0.290759
\(634\) −59.0540 −2.34533
\(635\) 5.36932 0.213075
\(636\) 53.3002 2.11349
\(637\) 0 0
\(638\) 29.1231 1.15299
\(639\) 14.0000 0.553831
\(640\) −5.30019 −0.209508
\(641\) 26.1771 1.03393 0.516966 0.856006i \(-0.327061\pi\)
0.516966 + 0.856006i \(0.327061\pi\)
\(642\) 21.1231 0.833662
\(643\) 38.5464 1.52012 0.760061 0.649852i \(-0.225169\pi\)
0.760061 + 0.649852i \(0.225169\pi\)
\(644\) −32.4924 −1.28038
\(645\) 0.246211 0.00969456
\(646\) −7.36932 −0.289942
\(647\) −47.6155 −1.87196 −0.935980 0.352054i \(-0.885483\pi\)
−0.935980 + 0.352054i \(0.885483\pi\)
\(648\) 6.56155 0.257762
\(649\) 22.2462 0.873240
\(650\) 0 0
\(651\) 5.56155 0.217974
\(652\) 72.1080 2.82397
\(653\) 14.8769 0.582178 0.291089 0.956696i \(-0.405982\pi\)
0.291089 + 0.956696i \(0.405982\pi\)
\(654\) 45.6155 1.78371
\(655\) −9.75379 −0.381112
\(656\) 19.6847 0.768557
\(657\) −1.87689 −0.0732246
\(658\) 75.2311 2.93281
\(659\) 14.2462 0.554954 0.277477 0.960732i \(-0.410502\pi\)
0.277477 + 0.960732i \(0.410502\pi\)
\(660\) −5.12311 −0.199417
\(661\) 30.3693 1.18123 0.590615 0.806954i \(-0.298885\pi\)
0.590615 + 0.806954i \(0.298885\pi\)
\(662\) −60.9848 −2.37024
\(663\) 0 0
\(664\) −59.8617 −2.32309
\(665\) 2.24621 0.0871043
\(666\) 8.80776 0.341294
\(667\) −11.3693 −0.440222
\(668\) −28.4924 −1.10240
\(669\) 8.00000 0.309298
\(670\) 0.630683 0.0243654
\(671\) −24.2462 −0.936015
\(672\) −23.3693 −0.901491
\(673\) −6.75379 −0.260339 −0.130170 0.991492i \(-0.541552\pi\)
−0.130170 + 0.991492i \(0.541552\pi\)
\(674\) 5.43845 0.209481
\(675\) −4.68466 −0.180313
\(676\) 0 0
\(677\) 25.6155 0.984485 0.492242 0.870458i \(-0.336177\pi\)
0.492242 + 0.870458i \(0.336177\pi\)
\(678\) −37.9309 −1.45673
\(679\) 15.8078 0.606646
\(680\) 9.43845 0.361948
\(681\) −1.12311 −0.0430375
\(682\) 8.00000 0.306336
\(683\) 36.1080 1.38163 0.690816 0.723030i \(-0.257251\pi\)
0.690816 + 0.723030i \(0.257251\pi\)
\(684\) −5.12311 −0.195887
\(685\) −0.807764 −0.0308631
\(686\) 12.0000 0.458162
\(687\) 0.246211 0.00939355
\(688\) 3.36932 0.128454
\(689\) 0 0
\(690\) 2.87689 0.109521
\(691\) 2.30019 0.0875032 0.0437516 0.999042i \(-0.486069\pi\)
0.0437516 + 0.999042i \(0.486069\pi\)
\(692\) 17.1231 0.650923
\(693\) 7.12311 0.270584
\(694\) −34.8769 −1.32391
\(695\) 6.13826 0.232837
\(696\) −37.3002 −1.41386
\(697\) 6.56155 0.248537
\(698\) −35.3693 −1.33875
\(699\) 26.0000 0.983410
\(700\) 76.1080 2.87661
\(701\) 19.3693 0.731569 0.365785 0.930700i \(-0.380801\pi\)
0.365785 + 0.930700i \(0.380801\pi\)
\(702\) 0 0
\(703\) −3.86174 −0.145648
\(704\) −2.87689 −0.108427
\(705\) −4.63068 −0.174402
\(706\) 45.3002 1.70490
\(707\) 12.2462 0.460566
\(708\) −50.7386 −1.90687
\(709\) 25.4924 0.957388 0.478694 0.877982i \(-0.341110\pi\)
0.478694 + 0.877982i \(0.341110\pi\)
\(710\) 20.1383 0.755775
\(711\) 9.56155 0.358586
\(712\) 86.1080 3.22703
\(713\) −3.12311 −0.116961
\(714\) −23.3693 −0.874575
\(715\) 0 0
\(716\) −59.8617 −2.23714
\(717\) −0.630683 −0.0235533
\(718\) 39.3693 1.46925
\(719\) 1.36932 0.0510669 0.0255335 0.999674i \(-0.491872\pi\)
0.0255335 + 0.999674i \(0.491872\pi\)
\(720\) 4.31534 0.160823
\(721\) 26.9309 1.00296
\(722\) −45.4384 −1.69104
\(723\) 2.80776 0.104422
\(724\) 44.1771 1.64183
\(725\) 26.6307 0.989039
\(726\) −17.9309 −0.665477
\(727\) 39.6695 1.47126 0.735630 0.677383i \(-0.236886\pi\)
0.735630 + 0.677383i \(0.236886\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −2.69981 −0.0999246
\(731\) 1.12311 0.0415396
\(732\) 55.3002 2.04395
\(733\) 53.4924 1.97579 0.987894 0.155131i \(-0.0495801\pi\)
0.987894 + 0.155131i \(0.0495801\pi\)
\(734\) 51.3693 1.89608
\(735\) 3.19224 0.117747
\(736\) 13.1231 0.483724
\(737\) −0.876894 −0.0323008
\(738\) 6.56155 0.241534
\(739\) −6.24621 −0.229771 −0.114885 0.993379i \(-0.536650\pi\)
−0.114885 + 0.993379i \(0.536650\pi\)
\(740\) 8.80776 0.323780
\(741\) 0 0
\(742\) −106.600 −3.91342
\(743\) −37.3693 −1.37095 −0.685474 0.728097i \(-0.740405\pi\)
−0.685474 + 0.728097i \(0.740405\pi\)
\(744\) −10.2462 −0.375644
\(745\) −3.68466 −0.134995
\(746\) 9.30019 0.340504
\(747\) −9.12311 −0.333797
\(748\) −23.3693 −0.854467
\(749\) −29.3693 −1.07313
\(750\) −13.9309 −0.508683
\(751\) −30.1080 −1.09865 −0.549327 0.835607i \(-0.685116\pi\)
−0.549327 + 0.835607i \(0.685116\pi\)
\(752\) −63.3693 −2.31084
\(753\) −30.7386 −1.12018
\(754\) 0 0
\(755\) 8.63068 0.314103
\(756\) −16.2462 −0.590869
\(757\) 30.0000 1.09037 0.545184 0.838316i \(-0.316460\pi\)
0.545184 + 0.838316i \(0.316460\pi\)
\(758\) 28.9848 1.05278
\(759\) −4.00000 −0.145191
\(760\) −4.13826 −0.150110
\(761\) −15.3693 −0.557137 −0.278569 0.960416i \(-0.589860\pi\)
−0.278569 + 0.960416i \(0.589860\pi\)
\(762\) 24.4924 0.887267
\(763\) −63.4233 −2.29608
\(764\) −4.00000 −0.144715
\(765\) 1.43845 0.0520072
\(766\) 68.4924 2.47473
\(767\) 0 0
\(768\) −27.0540 −0.976226
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 10.2462 0.369248
\(771\) −16.1771 −0.582603
\(772\) 88.9157 3.20015
\(773\) −7.75379 −0.278884 −0.139442 0.990230i \(-0.544531\pi\)
−0.139442 + 0.990230i \(0.544531\pi\)
\(774\) 1.12311 0.0403692
\(775\) 7.31534 0.262775
\(776\) −29.1231 −1.04546
\(777\) −12.2462 −0.439330
\(778\) −7.82292 −0.280465
\(779\) −2.87689 −0.103075
\(780\) 0 0
\(781\) −28.0000 −1.00192
\(782\) 13.1231 0.469281
\(783\) −5.68466 −0.203153
\(784\) 43.6847 1.56017
\(785\) −2.45360 −0.0875728
\(786\) −44.4924 −1.58699
\(787\) −1.17708 −0.0419584 −0.0209792 0.999780i \(-0.506678\pi\)
−0.0209792 + 0.999780i \(0.506678\pi\)
\(788\) −51.8617 −1.84750
\(789\) −15.3693 −0.547162
\(790\) 13.7538 0.489338
\(791\) 52.7386 1.87517
\(792\) −13.1231 −0.466309
\(793\) 0 0
\(794\) −30.8769 −1.09578
\(795\) 6.56155 0.232714
\(796\) −105.723 −3.74727
\(797\) −41.6155 −1.47410 −0.737049 0.675840i \(-0.763781\pi\)
−0.737049 + 0.675840i \(0.763781\pi\)
\(798\) 10.2462 0.362712
\(799\) −21.1231 −0.747282
\(800\) −30.7386 −1.08677
\(801\) 13.1231 0.463682
\(802\) 47.5464 1.67892
\(803\) 3.75379 0.132468
\(804\) 2.00000 0.0705346
\(805\) −4.00000 −0.140981
\(806\) 0 0
\(807\) 3.36932 0.118606
\(808\) −22.5616 −0.793713
\(809\) 37.3002 1.31140 0.655702 0.755019i \(-0.272373\pi\)
0.655702 + 0.755019i \(0.272373\pi\)
\(810\) 1.43845 0.0505419
\(811\) −1.56155 −0.0548335 −0.0274168 0.999624i \(-0.508728\pi\)
−0.0274168 + 0.999624i \(0.508728\pi\)
\(812\) 92.3542 3.24100
\(813\) −1.06913 −0.0374960
\(814\) −17.6155 −0.617424
\(815\) 8.87689 0.310944
\(816\) 19.6847 0.689101
\(817\) −0.492423 −0.0172277
\(818\) −47.0540 −1.64520
\(819\) 0 0
\(820\) 6.56155 0.229139
\(821\) −26.4924 −0.924592 −0.462296 0.886726i \(-0.652974\pi\)
−0.462296 + 0.886726i \(0.652974\pi\)
\(822\) −3.68466 −0.128517
\(823\) 8.00000 0.278862 0.139431 0.990232i \(-0.455473\pi\)
0.139431 + 0.990232i \(0.455473\pi\)
\(824\) −49.6155 −1.72844
\(825\) 9.36932 0.326198
\(826\) 101.477 3.53085
\(827\) 34.7386 1.20798 0.603990 0.796992i \(-0.293577\pi\)
0.603990 + 0.796992i \(0.293577\pi\)
\(828\) 9.12311 0.317050
\(829\) −19.4924 −0.677000 −0.338500 0.940966i \(-0.609919\pi\)
−0.338500 + 0.940966i \(0.609919\pi\)
\(830\) −13.1231 −0.455510
\(831\) 17.6847 0.613474
\(832\) 0 0
\(833\) 14.5616 0.504528
\(834\) 28.0000 0.969561
\(835\) −3.50758 −0.121385
\(836\) 10.2462 0.354373
\(837\) −1.56155 −0.0539752
\(838\) −45.4773 −1.57099
\(839\) −19.6155 −0.677203 −0.338602 0.940930i \(-0.609954\pi\)
−0.338602 + 0.940930i \(0.609954\pi\)
\(840\) −13.1231 −0.452790
\(841\) 3.31534 0.114322
\(842\) 37.7926 1.30242
\(843\) −2.80776 −0.0967045
\(844\) 33.3693 1.14862
\(845\) 0 0
\(846\) −21.1231 −0.726227
\(847\) 24.9309 0.856635
\(848\) 89.7926 3.08349
\(849\) −1.31534 −0.0451424
\(850\) −30.7386 −1.05433
\(851\) 6.87689 0.235737
\(852\) 63.8617 2.18787
\(853\) −6.12311 −0.209651 −0.104826 0.994491i \(-0.533428\pi\)
−0.104826 + 0.994491i \(0.533428\pi\)
\(854\) −110.600 −3.78467
\(855\) −0.630683 −0.0215689
\(856\) 54.1080 1.84937
\(857\) 31.4384 1.07392 0.536958 0.843609i \(-0.319573\pi\)
0.536958 + 0.843609i \(0.319573\pi\)
\(858\) 0 0
\(859\) 20.4384 0.697351 0.348675 0.937244i \(-0.386632\pi\)
0.348675 + 0.937244i \(0.386632\pi\)
\(860\) 1.12311 0.0382976
\(861\) −9.12311 −0.310915
\(862\) −7.36932 −0.251000
\(863\) −2.49242 −0.0848430 −0.0424215 0.999100i \(-0.513507\pi\)
−0.0424215 + 0.999100i \(0.513507\pi\)
\(864\) 6.56155 0.223229
\(865\) 2.10795 0.0716725
\(866\) 64.6695 2.19756
\(867\) −10.4384 −0.354508
\(868\) 25.3693 0.861091
\(869\) −19.1231 −0.648707
\(870\) −8.17708 −0.277229
\(871\) 0 0
\(872\) 116.847 3.95692
\(873\) −4.43845 −0.150219
\(874\) −5.75379 −0.194625
\(875\) 19.3693 0.654802
\(876\) −8.56155 −0.289268
\(877\) 19.4384 0.656390 0.328195 0.944610i \(-0.393560\pi\)
0.328195 + 0.944610i \(0.393560\pi\)
\(878\) −3.36932 −0.113709
\(879\) 24.5616 0.828441
\(880\) −8.63068 −0.290940
\(881\) −37.9309 −1.27792 −0.638962 0.769239i \(-0.720636\pi\)
−0.638962 + 0.769239i \(0.720636\pi\)
\(882\) 14.5616 0.490313
\(883\) −11.8078 −0.397363 −0.198681 0.980064i \(-0.563666\pi\)
−0.198681 + 0.980064i \(0.563666\pi\)
\(884\) 0 0
\(885\) −6.24621 −0.209964
\(886\) 37.7538 1.26836
\(887\) 49.3693 1.65766 0.828830 0.559501i \(-0.189007\pi\)
0.828830 + 0.559501i \(0.189007\pi\)
\(888\) 22.5616 0.757116
\(889\) −34.0540 −1.14213
\(890\) 18.8769 0.632755
\(891\) −2.00000 −0.0670025
\(892\) 36.4924 1.22186
\(893\) 9.26137 0.309920
\(894\) −16.8078 −0.562136
\(895\) −7.36932 −0.246329
\(896\) 33.6155 1.12302
\(897\) 0 0
\(898\) 21.1231 0.704887
\(899\) 8.87689 0.296061
\(900\) −21.3693 −0.712311
\(901\) 29.9309 0.997142
\(902\) −13.1231 −0.436952
\(903\) −1.56155 −0.0519652
\(904\) −97.1619 −3.23156
\(905\) 5.43845 0.180780
\(906\) 39.3693 1.30796
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) −5.12311 −0.170016
\(909\) −3.43845 −0.114046
\(910\) 0 0
\(911\) −10.7386 −0.355787 −0.177893 0.984050i \(-0.556928\pi\)
−0.177893 + 0.984050i \(0.556928\pi\)
\(912\) −8.63068 −0.285790
\(913\) 18.2462 0.603861
\(914\) −73.3002 −2.42455
\(915\) 6.80776 0.225058
\(916\) 1.12311 0.0371085
\(917\) 61.8617 2.04285
\(918\) 6.56155 0.216564
\(919\) 44.4924 1.46767 0.733835 0.679328i \(-0.237729\pi\)
0.733835 + 0.679328i \(0.237729\pi\)
\(920\) 7.36932 0.242959
\(921\) 10.1922 0.335846
\(922\) −94.2850 −3.10511
\(923\) 0 0
\(924\) 32.4924 1.06892
\(925\) −16.1080 −0.529626
\(926\) −68.3542 −2.24626
\(927\) −7.56155 −0.248354
\(928\) −37.3002 −1.22444
\(929\) 12.8078 0.420209 0.210105 0.977679i \(-0.432620\pi\)
0.210105 + 0.977679i \(0.432620\pi\)
\(930\) −2.24621 −0.0736562
\(931\) −6.38447 −0.209243
\(932\) 118.600 3.88488
\(933\) −10.8769 −0.356094
\(934\) −66.6004 −2.17923
\(935\) −2.87689 −0.0940845
\(936\) 0 0
\(937\) −3.43845 −0.112329 −0.0561646 0.998422i \(-0.517887\pi\)
−0.0561646 + 0.998422i \(0.517887\pi\)
\(938\) −4.00000 −0.130605
\(939\) −1.31534 −0.0429245
\(940\) −21.1231 −0.688960
\(941\) −2.49242 −0.0812507 −0.0406253 0.999174i \(-0.512935\pi\)
−0.0406253 + 0.999174i \(0.512935\pi\)
\(942\) −11.1922 −0.364663
\(943\) 5.12311 0.166831
\(944\) −85.4773 −2.78205
\(945\) −2.00000 −0.0650600
\(946\) −2.24621 −0.0730306
\(947\) 10.7386 0.348959 0.174479 0.984661i \(-0.444176\pi\)
0.174479 + 0.984661i \(0.444176\pi\)
\(948\) 43.6155 1.41657
\(949\) 0 0
\(950\) 13.4773 0.437260
\(951\) −23.0540 −0.747576
\(952\) −59.8617 −1.94013
\(953\) −34.9848 −1.13327 −0.566635 0.823969i \(-0.691755\pi\)
−0.566635 + 0.823969i \(0.691755\pi\)
\(954\) 29.9309 0.969048
\(955\) −0.492423 −0.0159344
\(956\) −2.87689 −0.0930454
\(957\) 11.3693 0.367518
\(958\) −16.0000 −0.516937
\(959\) 5.12311 0.165434
\(960\) 0.807764 0.0260705
\(961\) −28.5616 −0.921340
\(962\) 0 0
\(963\) 8.24621 0.265730
\(964\) 12.8078 0.412510
\(965\) 10.9460 0.352365
\(966\) −18.2462 −0.587062
\(967\) −9.12311 −0.293379 −0.146690 0.989183i \(-0.546862\pi\)
−0.146690 + 0.989183i \(0.546862\pi\)
\(968\) −45.9309 −1.47627
\(969\) −2.87689 −0.0924192
\(970\) −6.38447 −0.204993
\(971\) 52.9848 1.70036 0.850182 0.526488i \(-0.176492\pi\)
0.850182 + 0.526488i \(0.176492\pi\)
\(972\) 4.56155 0.146312
\(973\) −38.9309 −1.24807
\(974\) −2.87689 −0.0921816
\(975\) 0 0
\(976\) 93.1619 2.98204
\(977\) 15.8229 0.506220 0.253110 0.967438i \(-0.418547\pi\)
0.253110 + 0.967438i \(0.418547\pi\)
\(978\) 40.4924 1.29480
\(979\) −26.2462 −0.838833
\(980\) 14.5616 0.465152
\(981\) 17.8078 0.568558
\(982\) −50.6004 −1.61472
\(983\) −27.6155 −0.880799 −0.440399 0.897802i \(-0.645163\pi\)
−0.440399 + 0.897802i \(0.645163\pi\)
\(984\) 16.8078 0.535812
\(985\) −6.38447 −0.203426
\(986\) −37.3002 −1.18788
\(987\) 29.3693 0.934836
\(988\) 0 0
\(989\) 0.876894 0.0278836
\(990\) −2.87689 −0.0914337
\(991\) 40.3542 1.28189 0.640946 0.767586i \(-0.278542\pi\)
0.640946 + 0.767586i \(0.278542\pi\)
\(992\) −10.2462 −0.325318
\(993\) −23.8078 −0.755517
\(994\) −127.723 −4.05114
\(995\) −13.0152 −0.412608
\(996\) −41.6155 −1.31864
\(997\) 20.6155 0.652900 0.326450 0.945214i \(-0.394147\pi\)
0.326450 + 0.945214i \(0.394147\pi\)
\(998\) −72.9848 −2.31029
\(999\) 3.43845 0.108788
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 507.2.a.g.1.2 2
3.2 odd 2 1521.2.a.g.1.1 2
4.3 odd 2 8112.2.a.bk.1.2 2
13.2 odd 12 507.2.j.g.316.4 8
13.3 even 3 39.2.e.b.22.1 yes 4
13.4 even 6 507.2.e.g.484.2 4
13.5 odd 4 507.2.b.d.337.1 4
13.6 odd 12 507.2.j.g.361.1 8
13.7 odd 12 507.2.j.g.361.4 8
13.8 odd 4 507.2.b.d.337.4 4
13.9 even 3 39.2.e.b.16.1 4
13.10 even 6 507.2.e.g.22.2 4
13.11 odd 12 507.2.j.g.316.1 8
13.12 even 2 507.2.a.d.1.1 2
39.5 even 4 1521.2.b.h.1351.4 4
39.8 even 4 1521.2.b.h.1351.1 4
39.29 odd 6 117.2.g.c.100.2 4
39.35 odd 6 117.2.g.c.55.2 4
39.38 odd 2 1521.2.a.m.1.2 2
52.3 odd 6 624.2.q.h.529.2 4
52.35 odd 6 624.2.q.h.289.2 4
52.51 odd 2 8112.2.a.bo.1.1 2
65.3 odd 12 975.2.bb.i.724.4 8
65.9 even 6 975.2.i.k.601.2 4
65.22 odd 12 975.2.bb.i.874.4 8
65.29 even 6 975.2.i.k.451.2 4
65.42 odd 12 975.2.bb.i.724.1 8
65.48 odd 12 975.2.bb.i.874.1 8
156.35 even 6 1872.2.t.r.289.1 4
156.107 even 6 1872.2.t.r.1153.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.e.b.16.1 4 13.9 even 3
39.2.e.b.22.1 yes 4 13.3 even 3
117.2.g.c.55.2 4 39.35 odd 6
117.2.g.c.100.2 4 39.29 odd 6
507.2.a.d.1.1 2 13.12 even 2
507.2.a.g.1.2 2 1.1 even 1 trivial
507.2.b.d.337.1 4 13.5 odd 4
507.2.b.d.337.4 4 13.8 odd 4
507.2.e.g.22.2 4 13.10 even 6
507.2.e.g.484.2 4 13.4 even 6
507.2.j.g.316.1 8 13.11 odd 12
507.2.j.g.316.4 8 13.2 odd 12
507.2.j.g.361.1 8 13.6 odd 12
507.2.j.g.361.4 8 13.7 odd 12
624.2.q.h.289.2 4 52.35 odd 6
624.2.q.h.529.2 4 52.3 odd 6
975.2.i.k.451.2 4 65.29 even 6
975.2.i.k.601.2 4 65.9 even 6
975.2.bb.i.724.1 8 65.42 odd 12
975.2.bb.i.724.4 8 65.3 odd 12
975.2.bb.i.874.1 8 65.48 odd 12
975.2.bb.i.874.4 8 65.22 odd 12
1521.2.a.g.1.1 2 3.2 odd 2
1521.2.a.m.1.2 2 39.38 odd 2
1521.2.b.h.1351.1 4 39.8 even 4
1521.2.b.h.1351.4 4 39.5 even 4
1872.2.t.r.289.1 4 156.35 even 6
1872.2.t.r.1153.1 4 156.107 even 6
8112.2.a.bk.1.2 2 4.3 odd 2
8112.2.a.bo.1.1 2 52.51 odd 2