Properties

Label 3549.2.a.bh.1.3
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + 4553 x^{7} - 6393 x^{6} - 6785 x^{5} + 7806 x^{4} + 4632 x^{3} - 3811 x^{2} - 1041 x + 281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.41695\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41695 q^{2} +1.00000 q^{3} +3.84166 q^{4} -1.98263 q^{5} -2.41695 q^{6} -1.00000 q^{7} -4.45120 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.41695 q^{2} +1.00000 q^{3} +3.84166 q^{4} -1.98263 q^{5} -2.41695 q^{6} -1.00000 q^{7} -4.45120 q^{8} +1.00000 q^{9} +4.79192 q^{10} -5.56184 q^{11} +3.84166 q^{12} +2.41695 q^{14} -1.98263 q^{15} +3.07502 q^{16} -5.66208 q^{17} -2.41695 q^{18} -2.77489 q^{19} -7.61658 q^{20} -1.00000 q^{21} +13.4427 q^{22} -2.20066 q^{23} -4.45120 q^{24} -1.06918 q^{25} +1.00000 q^{27} -3.84166 q^{28} -5.79053 q^{29} +4.79192 q^{30} -0.187633 q^{31} +1.47022 q^{32} -5.56184 q^{33} +13.6850 q^{34} +1.98263 q^{35} +3.84166 q^{36} +5.00098 q^{37} +6.70678 q^{38} +8.82508 q^{40} -8.39681 q^{41} +2.41695 q^{42} -6.93636 q^{43} -21.3667 q^{44} -1.98263 q^{45} +5.31889 q^{46} +11.6646 q^{47} +3.07502 q^{48} +1.00000 q^{49} +2.58417 q^{50} -5.66208 q^{51} -2.78207 q^{53} -2.41695 q^{54} +11.0271 q^{55} +4.45120 q^{56} -2.77489 q^{57} +13.9954 q^{58} +6.18664 q^{59} -7.61658 q^{60} +13.1933 q^{61} +0.453499 q^{62} -1.00000 q^{63} -9.70349 q^{64} +13.4427 q^{66} -10.0891 q^{67} -21.7518 q^{68} -2.20066 q^{69} -4.79192 q^{70} +13.0368 q^{71} -4.45120 q^{72} +3.12787 q^{73} -12.0871 q^{74} -1.06918 q^{75} -10.6602 q^{76} +5.56184 q^{77} -8.48573 q^{79} -6.09663 q^{80} +1.00000 q^{81} +20.2947 q^{82} -15.5582 q^{83} -3.84166 q^{84} +11.2258 q^{85} +16.7648 q^{86} -5.79053 q^{87} +24.7569 q^{88} -6.24519 q^{89} +4.79192 q^{90} -8.45418 q^{92} -0.187633 q^{93} -28.1929 q^{94} +5.50158 q^{95} +1.47022 q^{96} +0.650954 q^{97} -2.41695 q^{98} -5.56184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 2q^{2} + 15q^{3} + 28q^{4} - 9q^{5} + 2q^{6} - 15q^{7} + 9q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 2q^{2} + 15q^{3} + 28q^{4} - 9q^{5} + 2q^{6} - 15q^{7} + 9q^{8} + 15q^{9} + 21q^{10} + 5q^{11} + 28q^{12} - 2q^{14} - 9q^{15} + 50q^{16} - q^{17} + 2q^{18} + 3q^{19} - 23q^{20} - 15q^{21} + 21q^{22} + 4q^{23} + 9q^{24} + 50q^{25} + 15q^{27} - 28q^{28} + 9q^{29} + 21q^{30} + 7q^{31} + 35q^{32} + 5q^{33} - 2q^{34} + 9q^{35} + 28q^{36} + 17q^{37} - 12q^{38} + 46q^{40} - 22q^{41} - 2q^{42} + 36q^{43} + 29q^{44} - 9q^{45} - q^{46} - 12q^{47} + 50q^{48} + 15q^{49} + 53q^{50} - q^{51} - 5q^{53} + 2q^{54} + 43q^{55} - 9q^{56} + 3q^{57} + 29q^{58} - 29q^{59} - 23q^{60} + 12q^{61} + 14q^{62} - 15q^{63} + 95q^{64} + 21q^{66} - 12q^{67} - 16q^{68} + 4q^{69} - 21q^{70} + 36q^{71} + 9q^{72} - 29q^{73} - 5q^{74} + 50q^{75} + 25q^{76} - 5q^{77} + 35q^{79} - 89q^{80} + 15q^{81} + 51q^{82} - 10q^{83} - 28q^{84} + 23q^{85} + 19q^{86} + 9q^{87} + 73q^{88} + 25q^{89} + 21q^{90} - 31q^{92} + 7q^{93} - 19q^{94} - 7q^{95} + 35q^{96} - 26q^{97} + 2q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41695 −1.70904 −0.854522 0.519416i \(-0.826150\pi\)
−0.854522 + 0.519416i \(0.826150\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.84166 1.92083
\(5\) −1.98263 −0.886658 −0.443329 0.896359i \(-0.646203\pi\)
−0.443329 + 0.896359i \(0.646203\pi\)
\(6\) −2.41695 −0.986717
\(7\) −1.00000 −0.377964
\(8\) −4.45120 −1.57374
\(9\) 1.00000 0.333333
\(10\) 4.79192 1.51534
\(11\) −5.56184 −1.67696 −0.838479 0.544934i \(-0.816555\pi\)
−0.838479 + 0.544934i \(0.816555\pi\)
\(12\) 3.84166 1.10899
\(13\) 0 0
\(14\) 2.41695 0.645958
\(15\) −1.98263 −0.511912
\(16\) 3.07502 0.768755
\(17\) −5.66208 −1.37326 −0.686629 0.727008i \(-0.740910\pi\)
−0.686629 + 0.727008i \(0.740910\pi\)
\(18\) −2.41695 −0.569681
\(19\) −2.77489 −0.636603 −0.318302 0.947989i \(-0.603112\pi\)
−0.318302 + 0.947989i \(0.603112\pi\)
\(20\) −7.61658 −1.70312
\(21\) −1.00000 −0.218218
\(22\) 13.4427 2.86599
\(23\) −2.20066 −0.458869 −0.229434 0.973324i \(-0.573688\pi\)
−0.229434 + 0.973324i \(0.573688\pi\)
\(24\) −4.45120 −0.908597
\(25\) −1.06918 −0.213837
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −3.84166 −0.726005
\(29\) −5.79053 −1.07527 −0.537637 0.843177i \(-0.680683\pi\)
−0.537637 + 0.843177i \(0.680683\pi\)
\(30\) 4.79192 0.874881
\(31\) −0.187633 −0.0336998 −0.0168499 0.999858i \(-0.505364\pi\)
−0.0168499 + 0.999858i \(0.505364\pi\)
\(32\) 1.47022 0.259901
\(33\) −5.56184 −0.968192
\(34\) 13.6850 2.34696
\(35\) 1.98263 0.335125
\(36\) 3.84166 0.640276
\(37\) 5.00098 0.822157 0.411078 0.911600i \(-0.365152\pi\)
0.411078 + 0.911600i \(0.365152\pi\)
\(38\) 6.70678 1.08798
\(39\) 0 0
\(40\) 8.82508 1.39537
\(41\) −8.39681 −1.31136 −0.655681 0.755038i \(-0.727618\pi\)
−0.655681 + 0.755038i \(0.727618\pi\)
\(42\) 2.41695 0.372944
\(43\) −6.93636 −1.05778 −0.528892 0.848689i \(-0.677392\pi\)
−0.528892 + 0.848689i \(0.677392\pi\)
\(44\) −21.3667 −3.22115
\(45\) −1.98263 −0.295553
\(46\) 5.31889 0.784227
\(47\) 11.6646 1.70146 0.850732 0.525601i \(-0.176159\pi\)
0.850732 + 0.525601i \(0.176159\pi\)
\(48\) 3.07502 0.443841
\(49\) 1.00000 0.142857
\(50\) 2.58417 0.365456
\(51\) −5.66208 −0.792850
\(52\) 0 0
\(53\) −2.78207 −0.382147 −0.191073 0.981576i \(-0.561197\pi\)
−0.191073 + 0.981576i \(0.561197\pi\)
\(54\) −2.41695 −0.328906
\(55\) 11.0271 1.48689
\(56\) 4.45120 0.594817
\(57\) −2.77489 −0.367543
\(58\) 13.9954 1.83769
\(59\) 6.18664 0.805433 0.402716 0.915325i \(-0.368066\pi\)
0.402716 + 0.915325i \(0.368066\pi\)
\(60\) −7.61658 −0.983296
\(61\) 13.1933 1.68922 0.844612 0.535380i \(-0.179831\pi\)
0.844612 + 0.535380i \(0.179831\pi\)
\(62\) 0.453499 0.0575944
\(63\) −1.00000 −0.125988
\(64\) −9.70349 −1.21294
\(65\) 0 0
\(66\) 13.4427 1.65468
\(67\) −10.0891 −1.23258 −0.616291 0.787519i \(-0.711365\pi\)
−0.616291 + 0.787519i \(0.711365\pi\)
\(68\) −21.7518 −2.63779
\(69\) −2.20066 −0.264928
\(70\) −4.79192 −0.572744
\(71\) 13.0368 1.54718 0.773589 0.633688i \(-0.218460\pi\)
0.773589 + 0.633688i \(0.218460\pi\)
\(72\) −4.45120 −0.524579
\(73\) 3.12787 0.366090 0.183045 0.983105i \(-0.441405\pi\)
0.183045 + 0.983105i \(0.441405\pi\)
\(74\) −12.0871 −1.40510
\(75\) −1.06918 −0.123459
\(76\) −10.6602 −1.22281
\(77\) 5.56184 0.633831
\(78\) 0 0
\(79\) −8.48573 −0.954719 −0.477360 0.878708i \(-0.658406\pi\)
−0.477360 + 0.878708i \(0.658406\pi\)
\(80\) −6.09663 −0.681623
\(81\) 1.00000 0.111111
\(82\) 20.2947 2.24117
\(83\) −15.5582 −1.70773 −0.853865 0.520494i \(-0.825748\pi\)
−0.853865 + 0.520494i \(0.825748\pi\)
\(84\) −3.84166 −0.419159
\(85\) 11.2258 1.21761
\(86\) 16.7648 1.80780
\(87\) −5.79053 −0.620810
\(88\) 24.7569 2.63909
\(89\) −6.24519 −0.661988 −0.330994 0.943633i \(-0.607384\pi\)
−0.330994 + 0.943633i \(0.607384\pi\)
\(90\) 4.79192 0.505113
\(91\) 0 0
\(92\) −8.45418 −0.881409
\(93\) −0.187633 −0.0194566
\(94\) −28.1929 −2.90787
\(95\) 5.50158 0.564450
\(96\) 1.47022 0.150054
\(97\) 0.650954 0.0660944 0.0330472 0.999454i \(-0.489479\pi\)
0.0330472 + 0.999454i \(0.489479\pi\)
\(98\) −2.41695 −0.244149
\(99\) −5.56184 −0.558986
\(100\) −4.10744 −0.410744
\(101\) 1.72491 0.171635 0.0858174 0.996311i \(-0.472650\pi\)
0.0858174 + 0.996311i \(0.472650\pi\)
\(102\) 13.6850 1.35502
\(103\) 3.77565 0.372026 0.186013 0.982547i \(-0.440443\pi\)
0.186013 + 0.982547i \(0.440443\pi\)
\(104\) 0 0
\(105\) 1.98263 0.193485
\(106\) 6.72413 0.653105
\(107\) 15.7248 1.52017 0.760085 0.649824i \(-0.225157\pi\)
0.760085 + 0.649824i \(0.225157\pi\)
\(108\) 3.84166 0.369664
\(109\) 17.5694 1.68284 0.841421 0.540380i \(-0.181720\pi\)
0.841421 + 0.540380i \(0.181720\pi\)
\(110\) −26.6519 −2.54116
\(111\) 5.00098 0.474672
\(112\) −3.07502 −0.290562
\(113\) 14.2914 1.34442 0.672209 0.740362i \(-0.265346\pi\)
0.672209 + 0.740362i \(0.265346\pi\)
\(114\) 6.70678 0.628147
\(115\) 4.36309 0.406860
\(116\) −22.2452 −2.06542
\(117\) 0 0
\(118\) −14.9528 −1.37652
\(119\) 5.66208 0.519042
\(120\) 8.82508 0.805616
\(121\) 19.9341 1.81219
\(122\) −31.8875 −2.88696
\(123\) −8.39681 −0.757115
\(124\) −0.720820 −0.0647316
\(125\) 12.0329 1.07626
\(126\) 2.41695 0.215319
\(127\) −10.2530 −0.909807 −0.454903 0.890541i \(-0.650326\pi\)
−0.454903 + 0.890541i \(0.650326\pi\)
\(128\) 20.5124 1.81306
\(129\) −6.93636 −0.610712
\(130\) 0 0
\(131\) −11.7740 −1.02870 −0.514348 0.857582i \(-0.671966\pi\)
−0.514348 + 0.857582i \(0.671966\pi\)
\(132\) −21.3667 −1.85973
\(133\) 2.77489 0.240613
\(134\) 24.3849 2.10654
\(135\) −1.98263 −0.170637
\(136\) 25.2031 2.16115
\(137\) −19.6795 −1.68133 −0.840667 0.541552i \(-0.817837\pi\)
−0.840667 + 0.541552i \(0.817837\pi\)
\(138\) 5.31889 0.452774
\(139\) 12.5276 1.06258 0.531288 0.847192i \(-0.321708\pi\)
0.531288 + 0.847192i \(0.321708\pi\)
\(140\) 7.61658 0.643719
\(141\) 11.6646 0.982340
\(142\) −31.5092 −2.64419
\(143\) 0 0
\(144\) 3.07502 0.256252
\(145\) 11.4805 0.953400
\(146\) −7.55991 −0.625663
\(147\) 1.00000 0.0824786
\(148\) 19.2121 1.57922
\(149\) −8.66190 −0.709611 −0.354805 0.934940i \(-0.615453\pi\)
−0.354805 + 0.934940i \(0.615453\pi\)
\(150\) 2.58417 0.210996
\(151\) −13.5708 −1.10438 −0.552188 0.833720i \(-0.686207\pi\)
−0.552188 + 0.833720i \(0.686207\pi\)
\(152\) 12.3516 1.00185
\(153\) −5.66208 −0.457752
\(154\) −13.4427 −1.08324
\(155\) 0.372006 0.0298802
\(156\) 0 0
\(157\) 5.12450 0.408979 0.204490 0.978869i \(-0.434447\pi\)
0.204490 + 0.978869i \(0.434447\pi\)
\(158\) 20.5096 1.63166
\(159\) −2.78207 −0.220632
\(160\) −2.91490 −0.230443
\(161\) 2.20066 0.173436
\(162\) −2.41695 −0.189894
\(163\) 17.6790 1.38472 0.692362 0.721550i \(-0.256570\pi\)
0.692362 + 0.721550i \(0.256570\pi\)
\(164\) −32.2577 −2.51890
\(165\) 11.0271 0.858456
\(166\) 37.6034 2.91859
\(167\) 19.3998 1.50120 0.750600 0.660757i \(-0.229764\pi\)
0.750600 + 0.660757i \(0.229764\pi\)
\(168\) 4.45120 0.343418
\(169\) 0 0
\(170\) −27.1322 −2.08095
\(171\) −2.77489 −0.212201
\(172\) −26.6471 −2.03182
\(173\) 3.66425 0.278588 0.139294 0.990251i \(-0.455517\pi\)
0.139294 + 0.990251i \(0.455517\pi\)
\(174\) 13.9954 1.06099
\(175\) 1.06918 0.0808227
\(176\) −17.1028 −1.28917
\(177\) 6.18664 0.465017
\(178\) 15.0943 1.13137
\(179\) −5.59386 −0.418105 −0.209052 0.977904i \(-0.567038\pi\)
−0.209052 + 0.977904i \(0.567038\pi\)
\(180\) −7.61658 −0.567706
\(181\) −6.97271 −0.518277 −0.259139 0.965840i \(-0.583439\pi\)
−0.259139 + 0.965840i \(0.583439\pi\)
\(182\) 0 0
\(183\) 13.1933 0.975273
\(184\) 9.79557 0.722139
\(185\) −9.91510 −0.728972
\(186\) 0.453499 0.0332522
\(187\) 31.4916 2.30289
\(188\) 44.8116 3.26822
\(189\) −1.00000 −0.0727393
\(190\) −13.2970 −0.964669
\(191\) 6.07010 0.439217 0.219609 0.975588i \(-0.429522\pi\)
0.219609 + 0.975588i \(0.429522\pi\)
\(192\) −9.70349 −0.700289
\(193\) −10.9046 −0.784929 −0.392465 0.919767i \(-0.628377\pi\)
−0.392465 + 0.919767i \(0.628377\pi\)
\(194\) −1.57333 −0.112958
\(195\) 0 0
\(196\) 3.84166 0.274404
\(197\) −17.1724 −1.22348 −0.611742 0.791057i \(-0.709531\pi\)
−0.611742 + 0.791057i \(0.709531\pi\)
\(198\) 13.4427 0.955331
\(199\) 1.10806 0.0785482 0.0392741 0.999228i \(-0.487495\pi\)
0.0392741 + 0.999228i \(0.487495\pi\)
\(200\) 4.75915 0.336523
\(201\) −10.0891 −0.711632
\(202\) −4.16902 −0.293331
\(203\) 5.79053 0.406415
\(204\) −21.7518 −1.52293
\(205\) 16.6478 1.16273
\(206\) −9.12558 −0.635809
\(207\) −2.20066 −0.152956
\(208\) 0 0
\(209\) 15.4335 1.06756
\(210\) −4.79192 −0.330674
\(211\) 3.56849 0.245665 0.122832 0.992427i \(-0.460802\pi\)
0.122832 + 0.992427i \(0.460802\pi\)
\(212\) −10.6878 −0.734038
\(213\) 13.0368 0.893264
\(214\) −38.0060 −2.59804
\(215\) 13.7522 0.937894
\(216\) −4.45120 −0.302866
\(217\) 0.187633 0.0127373
\(218\) −42.4644 −2.87605
\(219\) 3.12787 0.211362
\(220\) 42.3622 2.85606
\(221\) 0 0
\(222\) −12.0871 −0.811236
\(223\) −12.2147 −0.817954 −0.408977 0.912545i \(-0.634114\pi\)
−0.408977 + 0.912545i \(0.634114\pi\)
\(224\) −1.47022 −0.0982332
\(225\) −1.06918 −0.0712789
\(226\) −34.5415 −2.29767
\(227\) −22.0874 −1.46599 −0.732995 0.680234i \(-0.761878\pi\)
−0.732995 + 0.680234i \(0.761878\pi\)
\(228\) −10.6602 −0.705987
\(229\) −21.2490 −1.40418 −0.702088 0.712090i \(-0.747749\pi\)
−0.702088 + 0.712090i \(0.747749\pi\)
\(230\) −10.5454 −0.695341
\(231\) 5.56184 0.365942
\(232\) 25.7748 1.69220
\(233\) −2.11080 −0.138283 −0.0691414 0.997607i \(-0.522026\pi\)
−0.0691414 + 0.997607i \(0.522026\pi\)
\(234\) 0 0
\(235\) −23.1267 −1.50862
\(236\) 23.7670 1.54710
\(237\) −8.48573 −0.551208
\(238\) −13.6850 −0.887066
\(239\) 16.9892 1.09894 0.549469 0.835514i \(-0.314830\pi\)
0.549469 + 0.835514i \(0.314830\pi\)
\(240\) −6.09663 −0.393535
\(241\) −19.1884 −1.23604 −0.618018 0.786164i \(-0.712064\pi\)
−0.618018 + 0.786164i \(0.712064\pi\)
\(242\) −48.1797 −3.09711
\(243\) 1.00000 0.0641500
\(244\) 50.6840 3.24471
\(245\) −1.98263 −0.126665
\(246\) 20.2947 1.29394
\(247\) 0 0
\(248\) 0.835190 0.0530346
\(249\) −15.5582 −0.985959
\(250\) −29.0830 −1.83937
\(251\) 12.1438 0.766509 0.383255 0.923643i \(-0.374803\pi\)
0.383255 + 0.923643i \(0.374803\pi\)
\(252\) −3.84166 −0.242002
\(253\) 12.2397 0.769504
\(254\) 24.7810 1.55490
\(255\) 11.2258 0.702987
\(256\) −30.1706 −1.88566
\(257\) −16.8396 −1.05042 −0.525212 0.850972i \(-0.676014\pi\)
−0.525212 + 0.850972i \(0.676014\pi\)
\(258\) 16.7648 1.04373
\(259\) −5.00098 −0.310746
\(260\) 0 0
\(261\) −5.79053 −0.358425
\(262\) 28.4571 1.75808
\(263\) −26.8348 −1.65471 −0.827353 0.561682i \(-0.810154\pi\)
−0.827353 + 0.561682i \(0.810154\pi\)
\(264\) 24.7569 1.52368
\(265\) 5.51581 0.338833
\(266\) −6.70678 −0.411219
\(267\) −6.24519 −0.382199
\(268\) −38.7589 −2.36758
\(269\) −6.86560 −0.418603 −0.209302 0.977851i \(-0.567119\pi\)
−0.209302 + 0.977851i \(0.567119\pi\)
\(270\) 4.79192 0.291627
\(271\) −7.27072 −0.441665 −0.220833 0.975312i \(-0.570877\pi\)
−0.220833 + 0.975312i \(0.570877\pi\)
\(272\) −17.4110 −1.05570
\(273\) 0 0
\(274\) 47.5644 2.87347
\(275\) 5.94663 0.358595
\(276\) −8.45418 −0.508882
\(277\) −8.01571 −0.481617 −0.240809 0.970573i \(-0.577413\pi\)
−0.240809 + 0.970573i \(0.577413\pi\)
\(278\) −30.2786 −1.81599
\(279\) −0.187633 −0.0112333
\(280\) −8.82508 −0.527399
\(281\) 7.76205 0.463045 0.231522 0.972830i \(-0.425629\pi\)
0.231522 + 0.972830i \(0.425629\pi\)
\(282\) −28.1929 −1.67886
\(283\) 9.99319 0.594033 0.297017 0.954872i \(-0.404008\pi\)
0.297017 + 0.954872i \(0.404008\pi\)
\(284\) 50.0828 2.97186
\(285\) 5.50158 0.325885
\(286\) 0 0
\(287\) 8.39681 0.495648
\(288\) 1.47022 0.0866335
\(289\) 15.0592 0.885835
\(290\) −27.7477 −1.62940
\(291\) 0.650954 0.0381596
\(292\) 12.0162 0.703195
\(293\) −7.05939 −0.412414 −0.206207 0.978508i \(-0.566112\pi\)
−0.206207 + 0.978508i \(0.566112\pi\)
\(294\) −2.41695 −0.140960
\(295\) −12.2658 −0.714144
\(296\) −22.2604 −1.29386
\(297\) −5.56184 −0.322731
\(298\) 20.9354 1.21276
\(299\) 0 0
\(300\) −4.10744 −0.237143
\(301\) 6.93636 0.399805
\(302\) 32.8000 1.88743
\(303\) 1.72491 0.0990934
\(304\) −8.53285 −0.489392
\(305\) −26.1573 −1.49776
\(306\) 13.6850 0.782319
\(307\) 25.6817 1.46573 0.732867 0.680372i \(-0.238182\pi\)
0.732867 + 0.680372i \(0.238182\pi\)
\(308\) 21.3667 1.21748
\(309\) 3.77565 0.214789
\(310\) −0.899120 −0.0510666
\(311\) 32.5563 1.84610 0.923050 0.384680i \(-0.125688\pi\)
0.923050 + 0.384680i \(0.125688\pi\)
\(312\) 0 0
\(313\) −8.19460 −0.463186 −0.231593 0.972813i \(-0.574394\pi\)
−0.231593 + 0.972813i \(0.574394\pi\)
\(314\) −12.3857 −0.698963
\(315\) 1.98263 0.111708
\(316\) −32.5993 −1.83385
\(317\) 29.1019 1.63453 0.817263 0.576264i \(-0.195490\pi\)
0.817263 + 0.576264i \(0.195490\pi\)
\(318\) 6.72413 0.377070
\(319\) 32.2060 1.80319
\(320\) 19.2384 1.07546
\(321\) 15.7248 0.877670
\(322\) −5.31889 −0.296410
\(323\) 15.7117 0.874220
\(324\) 3.84166 0.213425
\(325\) 0 0
\(326\) −42.7293 −2.36655
\(327\) 17.5694 0.971589
\(328\) 37.3759 2.06374
\(329\) −11.6646 −0.643093
\(330\) −26.6519 −1.46714
\(331\) 29.3311 1.61218 0.806091 0.591792i \(-0.201579\pi\)
0.806091 + 0.591792i \(0.201579\pi\)
\(332\) −59.7692 −3.28026
\(333\) 5.00098 0.274052
\(334\) −46.8884 −2.56562
\(335\) 20.0030 1.09288
\(336\) −3.07502 −0.167756
\(337\) 1.49519 0.0814479 0.0407240 0.999170i \(-0.487034\pi\)
0.0407240 + 0.999170i \(0.487034\pi\)
\(338\) 0 0
\(339\) 14.2914 0.776200
\(340\) 43.1257 2.33882
\(341\) 1.04358 0.0565131
\(342\) 6.70678 0.362661
\(343\) −1.00000 −0.0539949
\(344\) 30.8751 1.66467
\(345\) 4.36309 0.234901
\(346\) −8.85631 −0.476118
\(347\) −4.89127 −0.262577 −0.131289 0.991344i \(-0.541911\pi\)
−0.131289 + 0.991344i \(0.541911\pi\)
\(348\) −22.2452 −1.19247
\(349\) 13.7842 0.737850 0.368925 0.929459i \(-0.379726\pi\)
0.368925 + 0.929459i \(0.379726\pi\)
\(350\) −2.58417 −0.138130
\(351\) 0 0
\(352\) −8.17713 −0.435842
\(353\) 1.70434 0.0907128 0.0453564 0.998971i \(-0.485558\pi\)
0.0453564 + 0.998971i \(0.485558\pi\)
\(354\) −14.9528 −0.794734
\(355\) −25.8470 −1.37182
\(356\) −23.9919 −1.27157
\(357\) 5.66208 0.299669
\(358\) 13.5201 0.714559
\(359\) 24.0112 1.26726 0.633630 0.773636i \(-0.281564\pi\)
0.633630 + 0.773636i \(0.281564\pi\)
\(360\) 8.82508 0.465122
\(361\) −11.3000 −0.594736
\(362\) 16.8527 0.885759
\(363\) 19.9341 1.04627
\(364\) 0 0
\(365\) −6.20140 −0.324596
\(366\) −31.8875 −1.66678
\(367\) −5.26694 −0.274932 −0.137466 0.990506i \(-0.543896\pi\)
−0.137466 + 0.990506i \(0.543896\pi\)
\(368\) −6.76707 −0.352758
\(369\) −8.39681 −0.437120
\(370\) 23.9643 1.24585
\(371\) 2.78207 0.144438
\(372\) −0.720820 −0.0373728
\(373\) 18.4633 0.955992 0.477996 0.878362i \(-0.341363\pi\)
0.477996 + 0.878362i \(0.341363\pi\)
\(374\) −76.1137 −3.93575
\(375\) 12.0329 0.621378
\(376\) −51.9217 −2.67766
\(377\) 0 0
\(378\) 2.41695 0.124315
\(379\) −5.24807 −0.269575 −0.134788 0.990875i \(-0.543035\pi\)
−0.134788 + 0.990875i \(0.543035\pi\)
\(380\) 21.1352 1.08421
\(381\) −10.2530 −0.525277
\(382\) −14.6711 −0.750641
\(383\) −9.75424 −0.498418 −0.249209 0.968450i \(-0.580171\pi\)
−0.249209 + 0.968450i \(0.580171\pi\)
\(384\) 20.5124 1.04677
\(385\) −11.0271 −0.561991
\(386\) 26.3559 1.34148
\(387\) −6.93636 −0.352595
\(388\) 2.50074 0.126956
\(389\) 12.0796 0.612458 0.306229 0.951958i \(-0.400933\pi\)
0.306229 + 0.951958i \(0.400933\pi\)
\(390\) 0 0
\(391\) 12.4603 0.630145
\(392\) −4.45120 −0.224820
\(393\) −11.7740 −0.593917
\(394\) 41.5049 2.09099
\(395\) 16.8241 0.846510
\(396\) −21.3667 −1.07372
\(397\) −26.9213 −1.35114 −0.675571 0.737295i \(-0.736103\pi\)
−0.675571 + 0.737295i \(0.736103\pi\)
\(398\) −2.67812 −0.134242
\(399\) 2.77489 0.138918
\(400\) −3.28776 −0.164388
\(401\) 31.9449 1.59525 0.797626 0.603152i \(-0.206089\pi\)
0.797626 + 0.603152i \(0.206089\pi\)
\(402\) 24.3849 1.21621
\(403\) 0 0
\(404\) 6.62651 0.329681
\(405\) −1.98263 −0.0985176
\(406\) −13.9954 −0.694581
\(407\) −27.8147 −1.37872
\(408\) 25.2031 1.24774
\(409\) 29.9248 1.47969 0.739843 0.672779i \(-0.234900\pi\)
0.739843 + 0.672779i \(0.234900\pi\)
\(410\) −40.2368 −1.98715
\(411\) −19.6795 −0.970719
\(412\) 14.5048 0.714599
\(413\) −6.18664 −0.304425
\(414\) 5.31889 0.261409
\(415\) 30.8461 1.51417
\(416\) 0 0
\(417\) 12.5276 0.613478
\(418\) −37.3020 −1.82450
\(419\) 29.8715 1.45932 0.729660 0.683811i \(-0.239679\pi\)
0.729660 + 0.683811i \(0.239679\pi\)
\(420\) 7.61658 0.371651
\(421\) 23.7308 1.15657 0.578283 0.815836i \(-0.303723\pi\)
0.578283 + 0.815836i \(0.303723\pi\)
\(422\) −8.62486 −0.419852
\(423\) 11.6646 0.567154
\(424\) 12.3835 0.601398
\(425\) 6.05381 0.293653
\(426\) −31.5092 −1.52663
\(427\) −13.1933 −0.638466
\(428\) 60.4091 2.91999
\(429\) 0 0
\(430\) −33.2385 −1.60290
\(431\) −11.1024 −0.534782 −0.267391 0.963588i \(-0.586162\pi\)
−0.267391 + 0.963588i \(0.586162\pi\)
\(432\) 3.07502 0.147947
\(433\) 10.4048 0.500025 0.250012 0.968243i \(-0.419565\pi\)
0.250012 + 0.968243i \(0.419565\pi\)
\(434\) −0.453499 −0.0217686
\(435\) 11.4805 0.550446
\(436\) 67.4956 3.23245
\(437\) 6.10658 0.292118
\(438\) −7.55991 −0.361227
\(439\) 5.20672 0.248503 0.124252 0.992251i \(-0.460347\pi\)
0.124252 + 0.992251i \(0.460347\pi\)
\(440\) −49.0837 −2.33997
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 15.1894 0.721669 0.360835 0.932630i \(-0.382492\pi\)
0.360835 + 0.932630i \(0.382492\pi\)
\(444\) 19.2121 0.911765
\(445\) 12.3819 0.586958
\(446\) 29.5223 1.39792
\(447\) −8.66190 −0.409694
\(448\) 9.70349 0.458447
\(449\) −1.69273 −0.0798847 −0.0399423 0.999202i \(-0.512717\pi\)
−0.0399423 + 0.999202i \(0.512717\pi\)
\(450\) 2.58417 0.121819
\(451\) 46.7017 2.19910
\(452\) 54.9025 2.58240
\(453\) −13.5708 −0.637612
\(454\) 53.3841 2.50544
\(455\) 0 0
\(456\) 12.3516 0.578416
\(457\) −3.72412 −0.174207 −0.0871036 0.996199i \(-0.527761\pi\)
−0.0871036 + 0.996199i \(0.527761\pi\)
\(458\) 51.3579 2.39980
\(459\) −5.66208 −0.264283
\(460\) 16.7615 0.781509
\(461\) 26.3885 1.22904 0.614518 0.788903i \(-0.289350\pi\)
0.614518 + 0.788903i \(0.289350\pi\)
\(462\) −13.4427 −0.625411
\(463\) 2.25761 0.104920 0.0524600 0.998623i \(-0.483294\pi\)
0.0524600 + 0.998623i \(0.483294\pi\)
\(464\) −17.8060 −0.826622
\(465\) 0.372006 0.0172513
\(466\) 5.10169 0.236331
\(467\) −30.8908 −1.42945 −0.714727 0.699403i \(-0.753449\pi\)
−0.714727 + 0.699403i \(0.753449\pi\)
\(468\) 0 0
\(469\) 10.0891 0.465872
\(470\) 55.8960 2.57829
\(471\) 5.12450 0.236124
\(472\) −27.5380 −1.26754
\(473\) 38.5789 1.77386
\(474\) 20.5096 0.942038
\(475\) 2.96687 0.136129
\(476\) 21.7518 0.996992
\(477\) −2.78207 −0.127382
\(478\) −41.0620 −1.87813
\(479\) −38.9723 −1.78069 −0.890346 0.455285i \(-0.849537\pi\)
−0.890346 + 0.455285i \(0.849537\pi\)
\(480\) −2.91490 −0.133046
\(481\) 0 0
\(482\) 46.3776 2.11244
\(483\) 2.20066 0.100133
\(484\) 76.5799 3.48090
\(485\) −1.29060 −0.0586032
\(486\) −2.41695 −0.109635
\(487\) −26.8581 −1.21706 −0.608529 0.793532i \(-0.708240\pi\)
−0.608529 + 0.793532i \(0.708240\pi\)
\(488\) −58.7258 −2.65839
\(489\) 17.6790 0.799471
\(490\) 4.79192 0.216477
\(491\) 36.5076 1.64757 0.823783 0.566905i \(-0.191859\pi\)
0.823783 + 0.566905i \(0.191859\pi\)
\(492\) −32.2577 −1.45429
\(493\) 32.7864 1.47663
\(494\) 0 0
\(495\) 11.0271 0.495630
\(496\) −0.576974 −0.0259069
\(497\) −13.0368 −0.584778
\(498\) 37.6034 1.68505
\(499\) −12.7129 −0.569107 −0.284553 0.958660i \(-0.591845\pi\)
−0.284553 + 0.958660i \(0.591845\pi\)
\(500\) 46.2264 2.06731
\(501\) 19.3998 0.866718
\(502\) −29.3510 −1.31000
\(503\) −32.0950 −1.43105 −0.715523 0.698589i \(-0.753811\pi\)
−0.715523 + 0.698589i \(0.753811\pi\)
\(504\) 4.45120 0.198272
\(505\) −3.41985 −0.152181
\(506\) −29.5828 −1.31512
\(507\) 0 0
\(508\) −39.3885 −1.74758
\(509\) 21.8972 0.970577 0.485288 0.874354i \(-0.338715\pi\)
0.485288 + 0.874354i \(0.338715\pi\)
\(510\) −27.1322 −1.20144
\(511\) −3.12787 −0.138369
\(512\) 31.8960 1.40962
\(513\) −2.77489 −0.122514
\(514\) 40.7004 1.79522
\(515\) −7.48572 −0.329860
\(516\) −26.6471 −1.17307
\(517\) −64.8769 −2.85328
\(518\) 12.0871 0.531079
\(519\) 3.66425 0.160843
\(520\) 0 0
\(521\) 17.7708 0.778554 0.389277 0.921121i \(-0.372725\pi\)
0.389277 + 0.921121i \(0.372725\pi\)
\(522\) 13.9954 0.612563
\(523\) 17.1501 0.749923 0.374961 0.927040i \(-0.377656\pi\)
0.374961 + 0.927040i \(0.377656\pi\)
\(524\) −45.2315 −1.97595
\(525\) 1.06918 0.0466630
\(526\) 64.8585 2.82796
\(527\) 1.06239 0.0462785
\(528\) −17.1028 −0.744303
\(529\) −18.1571 −0.789439
\(530\) −13.3314 −0.579081
\(531\) 6.18664 0.268478
\(532\) 10.6602 0.462177
\(533\) 0 0
\(534\) 15.0943 0.653195
\(535\) −31.1763 −1.34787
\(536\) 44.9087 1.93976
\(537\) −5.59386 −0.241393
\(538\) 16.5938 0.715411
\(539\) −5.56184 −0.239565
\(540\) −7.61658 −0.327765
\(541\) −36.1332 −1.55349 −0.776744 0.629817i \(-0.783130\pi\)
−0.776744 + 0.629817i \(0.783130\pi\)
\(542\) 17.5730 0.754825
\(543\) −6.97271 −0.299228
\(544\) −8.32451 −0.356910
\(545\) −34.8336 −1.49211
\(546\) 0 0
\(547\) −9.66117 −0.413082 −0.206541 0.978438i \(-0.566221\pi\)
−0.206541 + 0.978438i \(0.566221\pi\)
\(548\) −75.6019 −3.22956
\(549\) 13.1933 0.563074
\(550\) −14.3727 −0.612855
\(551\) 16.0681 0.684523
\(552\) 9.79557 0.416927
\(553\) 8.48573 0.360850
\(554\) 19.3736 0.823105
\(555\) −9.91510 −0.420872
\(556\) 48.1267 2.04103
\(557\) 25.6174 1.08544 0.542722 0.839913i \(-0.317394\pi\)
0.542722 + 0.839913i \(0.317394\pi\)
\(558\) 0.453499 0.0191981
\(559\) 0 0
\(560\) 6.09663 0.257629
\(561\) 31.4916 1.32958
\(562\) −18.7605 −0.791364
\(563\) −22.4791 −0.947380 −0.473690 0.880692i \(-0.657078\pi\)
−0.473690 + 0.880692i \(0.657078\pi\)
\(564\) 44.8116 1.88691
\(565\) −28.3345 −1.19204
\(566\) −24.1531 −1.01523
\(567\) −1.00000 −0.0419961
\(568\) −58.0292 −2.43485
\(569\) 35.4558 1.48638 0.743192 0.669078i \(-0.233311\pi\)
0.743192 + 0.669078i \(0.233311\pi\)
\(570\) −13.2970 −0.556952
\(571\) −25.4474 −1.06494 −0.532470 0.846449i \(-0.678736\pi\)
−0.532470 + 0.846449i \(0.678736\pi\)
\(572\) 0 0
\(573\) 6.07010 0.253582
\(574\) −20.2947 −0.847084
\(575\) 2.35291 0.0981231
\(576\) −9.70349 −0.404312
\(577\) −27.6703 −1.15193 −0.575965 0.817474i \(-0.695374\pi\)
−0.575965 + 0.817474i \(0.695374\pi\)
\(578\) −36.3974 −1.51393
\(579\) −10.9046 −0.453179
\(580\) 44.1040 1.83132
\(581\) 15.5582 0.645462
\(582\) −1.57333 −0.0652164
\(583\) 15.4734 0.640844
\(584\) −13.9228 −0.576129
\(585\) 0 0
\(586\) 17.0622 0.704834
\(587\) −4.28089 −0.176691 −0.0883456 0.996090i \(-0.528158\pi\)
−0.0883456 + 0.996090i \(0.528158\pi\)
\(588\) 3.84166 0.158427
\(589\) 0.520660 0.0214534
\(590\) 29.6459 1.22050
\(591\) −17.1724 −0.706379
\(592\) 15.3781 0.632038
\(593\) 22.7798 0.935454 0.467727 0.883873i \(-0.345073\pi\)
0.467727 + 0.883873i \(0.345073\pi\)
\(594\) 13.4427 0.551561
\(595\) −11.2258 −0.460213
\(596\) −33.2761 −1.36304
\(597\) 1.10806 0.0453498
\(598\) 0 0
\(599\) 37.9205 1.54939 0.774694 0.632336i \(-0.217904\pi\)
0.774694 + 0.632336i \(0.217904\pi\)
\(600\) 4.75915 0.194292
\(601\) −4.67311 −0.190620 −0.0953101 0.995448i \(-0.530384\pi\)
−0.0953101 + 0.995448i \(0.530384\pi\)
\(602\) −16.7648 −0.683284
\(603\) −10.0891 −0.410861
\(604\) −52.1344 −2.12132
\(605\) −39.5218 −1.60679
\(606\) −4.16902 −0.169355
\(607\) 40.2861 1.63516 0.817580 0.575814i \(-0.195315\pi\)
0.817580 + 0.575814i \(0.195315\pi\)
\(608\) −4.07970 −0.165454
\(609\) 5.79053 0.234644
\(610\) 63.2210 2.55974
\(611\) 0 0
\(612\) −21.7518 −0.879264
\(613\) 44.5102 1.79775 0.898874 0.438207i \(-0.144386\pi\)
0.898874 + 0.438207i \(0.144386\pi\)
\(614\) −62.0715 −2.50500
\(615\) 16.6478 0.671302
\(616\) −24.7569 −0.997482
\(617\) 2.06890 0.0832907 0.0416453 0.999132i \(-0.486740\pi\)
0.0416453 + 0.999132i \(0.486740\pi\)
\(618\) −9.12558 −0.367085
\(619\) 36.0887 1.45053 0.725264 0.688471i \(-0.241718\pi\)
0.725264 + 0.688471i \(0.241718\pi\)
\(620\) 1.42912 0.0573948
\(621\) −2.20066 −0.0883094
\(622\) −78.6871 −3.15506
\(623\) 6.24519 0.250208
\(624\) 0 0
\(625\) −18.5109 −0.740437
\(626\) 19.8060 0.791605
\(627\) 15.4335 0.616354
\(628\) 19.6866 0.785579
\(629\) −28.3160 −1.12903
\(630\) −4.79192 −0.190915
\(631\) −24.1496 −0.961380 −0.480690 0.876891i \(-0.659614\pi\)
−0.480690 + 0.876891i \(0.659614\pi\)
\(632\) 37.7717 1.50248
\(633\) 3.56849 0.141835
\(634\) −70.3380 −2.79348
\(635\) 20.3279 0.806688
\(636\) −10.6878 −0.423797
\(637\) 0 0
\(638\) −77.8403 −3.08173
\(639\) 13.0368 0.515726
\(640\) −40.6686 −1.60757
\(641\) 10.6004 0.418689 0.209344 0.977842i \(-0.432867\pi\)
0.209344 + 0.977842i \(0.432867\pi\)
\(642\) −38.0060 −1.49998
\(643\) −22.0644 −0.870137 −0.435068 0.900397i \(-0.643276\pi\)
−0.435068 + 0.900397i \(0.643276\pi\)
\(644\) 8.45418 0.333141
\(645\) 13.7522 0.541493
\(646\) −37.9743 −1.49408
\(647\) 14.9272 0.586850 0.293425 0.955982i \(-0.405205\pi\)
0.293425 + 0.955982i \(0.405205\pi\)
\(648\) −4.45120 −0.174860
\(649\) −34.4091 −1.35068
\(650\) 0 0
\(651\) 0.187633 0.00735390
\(652\) 67.9166 2.65982
\(653\) −43.8355 −1.71542 −0.857708 0.514137i \(-0.828112\pi\)
−0.857708 + 0.514137i \(0.828112\pi\)
\(654\) −42.4644 −1.66049
\(655\) 23.3434 0.912101
\(656\) −25.8204 −1.00812
\(657\) 3.12787 0.122030
\(658\) 28.1929 1.09907
\(659\) 8.12476 0.316496 0.158248 0.987399i \(-0.449416\pi\)
0.158248 + 0.987399i \(0.449416\pi\)
\(660\) 42.3622 1.64895
\(661\) −25.0158 −0.973001 −0.486501 0.873680i \(-0.661727\pi\)
−0.486501 + 0.873680i \(0.661727\pi\)
\(662\) −70.8918 −2.75529
\(663\) 0 0
\(664\) 69.2525 2.68752
\(665\) −5.50158 −0.213342
\(666\) −12.0871 −0.468367
\(667\) 12.7430 0.493410
\(668\) 74.5273 2.88355
\(669\) −12.2147 −0.472246
\(670\) −48.3462 −1.86778
\(671\) −73.3788 −2.83276
\(672\) −1.47022 −0.0567150
\(673\) 45.4238 1.75096 0.875480 0.483254i \(-0.160545\pi\)
0.875480 + 0.483254i \(0.160545\pi\)
\(674\) −3.61379 −0.139198
\(675\) −1.06918 −0.0411529
\(676\) 0 0
\(677\) −18.3095 −0.703691 −0.351846 0.936058i \(-0.614446\pi\)
−0.351846 + 0.936058i \(0.614446\pi\)
\(678\) −34.5415 −1.32656
\(679\) −0.650954 −0.0249813
\(680\) −49.9683 −1.91620
\(681\) −22.0874 −0.846390
\(682\) −2.52229 −0.0965834
\(683\) 17.1187 0.655029 0.327515 0.944846i \(-0.393789\pi\)
0.327515 + 0.944846i \(0.393789\pi\)
\(684\) −10.6602 −0.407602
\(685\) 39.0172 1.49077
\(686\) 2.41695 0.0922797
\(687\) −21.2490 −0.810701
\(688\) −21.3295 −0.813178
\(689\) 0 0
\(690\) −10.5454 −0.401456
\(691\) −17.1598 −0.652791 −0.326395 0.945233i \(-0.605834\pi\)
−0.326395 + 0.945233i \(0.605834\pi\)
\(692\) 14.0768 0.535119
\(693\) 5.56184 0.211277
\(694\) 11.8220 0.448756
\(695\) −24.8375 −0.942141
\(696\) 25.7748 0.976991
\(697\) 47.5434 1.80084
\(698\) −33.3157 −1.26102
\(699\) −2.11080 −0.0798377
\(700\) 4.10744 0.155247
\(701\) 1.44147 0.0544435 0.0272217 0.999629i \(-0.491334\pi\)
0.0272217 + 0.999629i \(0.491334\pi\)
\(702\) 0 0
\(703\) −13.8772 −0.523388
\(704\) 53.9693 2.03404
\(705\) −23.1267 −0.871000
\(706\) −4.11931 −0.155032
\(707\) −1.72491 −0.0648719
\(708\) 23.7670 0.893218
\(709\) 6.69692 0.251508 0.125754 0.992061i \(-0.459865\pi\)
0.125754 + 0.992061i \(0.459865\pi\)
\(710\) 62.4711 2.34450
\(711\) −8.48573 −0.318240
\(712\) 27.7986 1.04180
\(713\) 0.412915 0.0154638
\(714\) −13.6850 −0.512148
\(715\) 0 0
\(716\) −21.4897 −0.803108
\(717\) 16.9892 0.634472
\(718\) −58.0338 −2.16580
\(719\) −12.9791 −0.484039 −0.242020 0.970271i \(-0.577810\pi\)
−0.242020 + 0.970271i \(0.577810\pi\)
\(720\) −6.09663 −0.227208
\(721\) −3.77565 −0.140613
\(722\) 27.3115 1.01643
\(723\) −19.1884 −0.713626
\(724\) −26.7868 −0.995522
\(725\) 6.19114 0.229933
\(726\) −48.1797 −1.78812
\(727\) −19.0926 −0.708106 −0.354053 0.935225i \(-0.615197\pi\)
−0.354053 + 0.935225i \(0.615197\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 14.9885 0.554749
\(731\) 39.2742 1.45261
\(732\) 50.6840 1.87333
\(733\) 3.15192 0.116419 0.0582094 0.998304i \(-0.481461\pi\)
0.0582094 + 0.998304i \(0.481461\pi\)
\(734\) 12.7299 0.469871
\(735\) −1.98263 −0.0731304
\(736\) −3.23545 −0.119260
\(737\) 56.1141 2.06699
\(738\) 20.2947 0.747058
\(739\) 7.52007 0.276630 0.138315 0.990388i \(-0.455831\pi\)
0.138315 + 0.990388i \(0.455831\pi\)
\(740\) −38.0904 −1.40023
\(741\) 0 0
\(742\) −6.72413 −0.246850
\(743\) −21.2016 −0.777812 −0.388906 0.921277i \(-0.627147\pi\)
−0.388906 + 0.921277i \(0.627147\pi\)
\(744\) 0.835190 0.0306195
\(745\) 17.1733 0.629182
\(746\) −44.6248 −1.63383
\(747\) −15.5582 −0.569244
\(748\) 120.980 4.42347
\(749\) −15.7248 −0.574570
\(750\) −29.0830 −1.06196
\(751\) 15.1412 0.552510 0.276255 0.961084i \(-0.410907\pi\)
0.276255 + 0.961084i \(0.410907\pi\)
\(752\) 35.8690 1.30801
\(753\) 12.1438 0.442544
\(754\) 0 0
\(755\) 26.9059 0.979204
\(756\) −3.84166 −0.139720
\(757\) −1.49444 −0.0543162 −0.0271581 0.999631i \(-0.508646\pi\)
−0.0271581 + 0.999631i \(0.508646\pi\)
\(758\) 12.6843 0.460716
\(759\) 12.2397 0.444273
\(760\) −24.4886 −0.888295
\(761\) 19.6602 0.712682 0.356341 0.934356i \(-0.384024\pi\)
0.356341 + 0.934356i \(0.384024\pi\)
\(762\) 24.7810 0.897722
\(763\) −17.5694 −0.636055
\(764\) 23.3193 0.843661
\(765\) 11.2258 0.405870
\(766\) 23.5755 0.851819
\(767\) 0 0
\(768\) −30.1706 −1.08869
\(769\) −30.8663 −1.11307 −0.556534 0.830825i \(-0.687869\pi\)
−0.556534 + 0.830825i \(0.687869\pi\)
\(770\) 26.6519 0.960467
\(771\) −16.8396 −0.606462
\(772\) −41.8917 −1.50772
\(773\) 5.18435 0.186468 0.0932341 0.995644i \(-0.470279\pi\)
0.0932341 + 0.995644i \(0.470279\pi\)
\(774\) 16.7648 0.602600
\(775\) 0.200614 0.00720626
\(776\) −2.89753 −0.104015
\(777\) −5.00098 −0.179409
\(778\) −29.1957 −1.04672
\(779\) 23.3002 0.834817
\(780\) 0 0
\(781\) −72.5083 −2.59455
\(782\) −30.1160 −1.07695
\(783\) −5.79053 −0.206937
\(784\) 3.07502 0.109822
\(785\) −10.1600 −0.362625
\(786\) 28.4571 1.01503
\(787\) −2.19799 −0.0783499 −0.0391749 0.999232i \(-0.512473\pi\)
−0.0391749 + 0.999232i \(0.512473\pi\)
\(788\) −65.9706 −2.35011
\(789\) −26.8348 −0.955345
\(790\) −40.6629 −1.44672
\(791\) −14.2914 −0.508142
\(792\) 24.7569 0.879697
\(793\) 0 0
\(794\) 65.0676 2.30916
\(795\) 5.51581 0.195626
\(796\) 4.25678 0.150878
\(797\) −43.7794 −1.55074 −0.775372 0.631505i \(-0.782438\pi\)
−0.775372 + 0.631505i \(0.782438\pi\)
\(798\) −6.70678 −0.237417
\(799\) −66.0462 −2.33655
\(800\) −1.57194 −0.0555763
\(801\) −6.24519 −0.220663
\(802\) −77.2093 −2.72635
\(803\) −17.3967 −0.613917
\(804\) −38.7589 −1.36692
\(805\) −4.36309 −0.153779
\(806\) 0 0
\(807\) −6.86560 −0.241681
\(808\) −7.67791 −0.270108
\(809\) −17.6845 −0.621755 −0.310877 0.950450i \(-0.600623\pi\)
−0.310877 + 0.950450i \(0.600623\pi\)
\(810\) 4.79192 0.168371
\(811\) 11.9195 0.418549 0.209275 0.977857i \(-0.432890\pi\)
0.209275 + 0.977857i \(0.432890\pi\)
\(812\) 22.2452 0.780654
\(813\) −7.27072 −0.254995
\(814\) 67.2267 2.35630
\(815\) −35.0509 −1.22778
\(816\) −17.4110 −0.609508
\(817\) 19.2476 0.673389
\(818\) −72.3268 −2.52885
\(819\) 0 0
\(820\) 63.9550 2.23340
\(821\) 3.20092 0.111713 0.0558564 0.998439i \(-0.482211\pi\)
0.0558564 + 0.998439i \(0.482211\pi\)
\(822\) 47.5644 1.65900
\(823\) 8.84236 0.308225 0.154113 0.988053i \(-0.450748\pi\)
0.154113 + 0.988053i \(0.450748\pi\)
\(824\) −16.8062 −0.585471
\(825\) 5.94663 0.207035
\(826\) 14.9528 0.520275
\(827\) 0.901661 0.0313538 0.0156769 0.999877i \(-0.495010\pi\)
0.0156769 + 0.999877i \(0.495010\pi\)
\(828\) −8.45418 −0.293803
\(829\) −12.5125 −0.434578 −0.217289 0.976107i \(-0.569721\pi\)
−0.217289 + 0.976107i \(0.569721\pi\)
\(830\) −74.5535 −2.58779
\(831\) −8.01571 −0.278062
\(832\) 0 0
\(833\) −5.66208 −0.196180
\(834\) −30.2786 −1.04846
\(835\) −38.4626 −1.33105
\(836\) 59.2902 2.05059
\(837\) −0.187633 −0.00648553
\(838\) −72.1980 −2.49404
\(839\) −24.4528 −0.844205 −0.422103 0.906548i \(-0.638708\pi\)
−0.422103 + 0.906548i \(0.638708\pi\)
\(840\) −8.82508 −0.304494
\(841\) 4.53019 0.156213
\(842\) −57.3561 −1.97662
\(843\) 7.76205 0.267339
\(844\) 13.7089 0.471880
\(845\) 0 0
\(846\) −28.1929 −0.969291
\(847\) −19.9341 −0.684943
\(848\) −8.55492 −0.293777
\(849\) 9.99319 0.342965
\(850\) −14.6318 −0.501866
\(851\) −11.0055 −0.377262
\(852\) 50.0828 1.71581
\(853\) −45.3747 −1.55360 −0.776800 0.629747i \(-0.783158\pi\)
−0.776800 + 0.629747i \(0.783158\pi\)
\(854\) 31.8875 1.09117
\(855\) 5.50158 0.188150
\(856\) −69.9940 −2.39235
\(857\) −52.7495 −1.80189 −0.900944 0.433935i \(-0.857125\pi\)
−0.900944 + 0.433935i \(0.857125\pi\)
\(858\) 0 0
\(859\) 39.1342 1.33524 0.667621 0.744501i \(-0.267313\pi\)
0.667621 + 0.744501i \(0.267313\pi\)
\(860\) 52.8313 1.80153
\(861\) 8.39681 0.286162
\(862\) 26.8339 0.913966
\(863\) 30.7592 1.04706 0.523528 0.852008i \(-0.324615\pi\)
0.523528 + 0.852008i \(0.324615\pi\)
\(864\) 1.47022 0.0500179
\(865\) −7.26484 −0.247012
\(866\) −25.1480 −0.854564
\(867\) 15.0592 0.511437
\(868\) 0.720820 0.0244662
\(869\) 47.1963 1.60102
\(870\) −27.7477 −0.940736
\(871\) 0 0
\(872\) −78.2049 −2.64835
\(873\) 0.650954 0.0220315
\(874\) −14.7593 −0.499241
\(875\) −12.0329 −0.406788
\(876\) 12.0162 0.405990
\(877\) 20.5689 0.694562 0.347281 0.937761i \(-0.387105\pi\)
0.347281 + 0.937761i \(0.387105\pi\)
\(878\) −12.5844 −0.424703
\(879\) −7.05939 −0.238107
\(880\) 33.9085 1.14305
\(881\) 36.8744 1.24233 0.621166 0.783679i \(-0.286659\pi\)
0.621166 + 0.783679i \(0.286659\pi\)
\(882\) −2.41695 −0.0813830
\(883\) −19.6942 −0.662761 −0.331381 0.943497i \(-0.607514\pi\)
−0.331381 + 0.943497i \(0.607514\pi\)
\(884\) 0 0
\(885\) −12.2658 −0.412311
\(886\) −36.7120 −1.23336
\(887\) 10.1145 0.339612 0.169806 0.985478i \(-0.445686\pi\)
0.169806 + 0.985478i \(0.445686\pi\)
\(888\) −22.2604 −0.747010
\(889\) 10.2530 0.343875
\(890\) −29.9264 −1.00314
\(891\) −5.56184 −0.186329
\(892\) −46.9246 −1.57115
\(893\) −32.3681 −1.08316
\(894\) 20.9354 0.700185
\(895\) 11.0905 0.370716
\(896\) −20.5124 −0.685273
\(897\) 0 0
\(898\) 4.09124 0.136526
\(899\) 1.08649 0.0362365
\(900\) −4.10744 −0.136915
\(901\) 15.7523 0.524785
\(902\) −112.876 −3.75835
\(903\) 6.93636 0.230828
\(904\) −63.6137 −2.11576
\(905\) 13.8243 0.459535
\(906\) 32.8000 1.08971
\(907\) −21.9042 −0.727316 −0.363658 0.931533i \(-0.618472\pi\)
−0.363658 + 0.931533i \(0.618472\pi\)
\(908\) −84.8521 −2.81592
\(909\) 1.72491 0.0572116
\(910\) 0 0
\(911\) −15.5338 −0.514656 −0.257328 0.966324i \(-0.582842\pi\)
−0.257328 + 0.966324i \(0.582842\pi\)
\(912\) −8.53285 −0.282551
\(913\) 86.5321 2.86379
\(914\) 9.00103 0.297728
\(915\) −26.1573 −0.864734
\(916\) −81.6316 −2.69718
\(917\) 11.7740 0.388810
\(918\) 13.6850 0.451672
\(919\) −8.81313 −0.290718 −0.145359 0.989379i \(-0.546434\pi\)
−0.145359 + 0.989379i \(0.546434\pi\)
\(920\) −19.4210 −0.640291
\(921\) 25.6817 0.846242
\(922\) −63.7798 −2.10048
\(923\) 0 0
\(924\) 21.3667 0.702912
\(925\) −5.34697 −0.175807
\(926\) −5.45653 −0.179313
\(927\) 3.77565 0.124009
\(928\) −8.51334 −0.279464
\(929\) 30.1854 0.990352 0.495176 0.868793i \(-0.335104\pi\)
0.495176 + 0.868793i \(0.335104\pi\)
\(930\) −0.899120 −0.0294833
\(931\) −2.77489 −0.0909433
\(932\) −8.10896 −0.265618
\(933\) 32.5563 1.06585
\(934\) 74.6615 2.44300
\(935\) −62.4362 −2.04188
\(936\) 0 0
\(937\) −21.7393 −0.710192 −0.355096 0.934830i \(-0.615552\pi\)
−0.355096 + 0.934830i \(0.615552\pi\)
\(938\) −24.3849 −0.796196
\(939\) −8.19460 −0.267421
\(940\) −88.8447 −2.89779
\(941\) −23.8691 −0.778109 −0.389055 0.921215i \(-0.627198\pi\)
−0.389055 + 0.921215i \(0.627198\pi\)
\(942\) −12.3857 −0.403547
\(943\) 18.4785 0.601743
\(944\) 19.0241 0.619181
\(945\) 1.98263 0.0644949
\(946\) −93.2434 −3.03160
\(947\) 33.0288 1.07329 0.536645 0.843808i \(-0.319691\pi\)
0.536645 + 0.843808i \(0.319691\pi\)
\(948\) −32.5993 −1.05878
\(949\) 0 0
\(950\) −7.17078 −0.232651
\(951\) 29.1019 0.943694
\(952\) −25.2031 −0.816836
\(953\) −6.54439 −0.211994 −0.105997 0.994366i \(-0.533803\pi\)
−0.105997 + 0.994366i \(0.533803\pi\)
\(954\) 6.72413 0.217702
\(955\) −12.0348 −0.389436
\(956\) 65.2666 2.11087
\(957\) 32.2060 1.04107
\(958\) 94.1943 3.04328
\(959\) 19.6795 0.635485
\(960\) 19.2384 0.620917
\(961\) −30.9648 −0.998864
\(962\) 0 0
\(963\) 15.7248 0.506723
\(964\) −73.7154 −2.37421
\(965\) 21.6197 0.695964
\(966\) −5.31889 −0.171132
\(967\) 6.58745 0.211838 0.105919 0.994375i \(-0.466222\pi\)
0.105919 + 0.994375i \(0.466222\pi\)
\(968\) −88.7305 −2.85191
\(969\) 15.7117 0.504731
\(970\) 3.11932 0.100155
\(971\) −32.8858 −1.05535 −0.527677 0.849445i \(-0.676937\pi\)
−0.527677 + 0.849445i \(0.676937\pi\)
\(972\) 3.84166 0.123221
\(973\) −12.5276 −0.401616
\(974\) 64.9148 2.08000
\(975\) 0 0
\(976\) 40.5695 1.29860
\(977\) −7.48594 −0.239496 −0.119748 0.992804i \(-0.538209\pi\)
−0.119748 + 0.992804i \(0.538209\pi\)
\(978\) −42.7293 −1.36633
\(979\) 34.7347 1.11013
\(980\) −7.61658 −0.243303
\(981\) 17.5694 0.560947
\(982\) −88.2372 −2.81576
\(983\) 7.84023 0.250065 0.125032 0.992153i \(-0.460097\pi\)
0.125032 + 0.992153i \(0.460097\pi\)
\(984\) 37.3759 1.19150
\(985\) 34.0465 1.08481
\(986\) −79.2433 −2.52362
\(987\) −11.6646 −0.371290
\(988\) 0 0
\(989\) 15.2646 0.485385
\(990\) −26.6519 −0.847053
\(991\) −17.0937 −0.542999 −0.271500 0.962439i \(-0.587520\pi\)
−0.271500 + 0.962439i \(0.587520\pi\)
\(992\) −0.275861 −0.00875860
\(993\) 29.3311 0.930794
\(994\) 31.5092 0.999412
\(995\) −2.19687 −0.0696454
\(996\) −59.7692 −1.89386
\(997\) −58.6850 −1.85857 −0.929287 0.369358i \(-0.879577\pi\)
−0.929287 + 0.369358i \(0.879577\pi\)
\(998\) 30.7264 0.972628
\(999\) 5.00098 0.158224
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 3549.2.a.bh.1.3 yes 15
13.12 even 2 3549.2.a.bg.1.13 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.bg.1.13 15 13.12 even 2
3549.2.a.bh.1.3 yes 15 1.1 even 1 trivial