Properties

Label 3549.2.a.bh.1.6
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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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 1.6
Root \(-1.08127\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.08127 q^{2} +1.00000 q^{3} -0.830866 q^{4} -1.66920 q^{5} -1.08127 q^{6} -1.00000 q^{7} +3.06092 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.08127 q^{2} +1.00000 q^{3} -0.830866 q^{4} -1.66920 q^{5} -1.08127 q^{6} -1.00000 q^{7} +3.06092 q^{8} +1.00000 q^{9} +1.80485 q^{10} +4.63545 q^{11} -0.830866 q^{12} +1.08127 q^{14} -1.66920 q^{15} -1.64793 q^{16} +4.03225 q^{17} -1.08127 q^{18} +0.793678 q^{19} +1.38688 q^{20} -1.00000 q^{21} -5.01215 q^{22} +4.31595 q^{23} +3.06092 q^{24} -2.21376 q^{25} +1.00000 q^{27} +0.830866 q^{28} +4.61741 q^{29} +1.80485 q^{30} -5.75711 q^{31} -4.33998 q^{32} +4.63545 q^{33} -4.35994 q^{34} +1.66920 q^{35} -0.830866 q^{36} -5.73497 q^{37} -0.858176 q^{38} -5.10929 q^{40} -7.33613 q^{41} +1.08127 q^{42} +11.4344 q^{43} -3.85144 q^{44} -1.66920 q^{45} -4.66668 q^{46} +5.10901 q^{47} -1.64793 q^{48} +1.00000 q^{49} +2.39366 q^{50} +4.03225 q^{51} -6.41614 q^{53} -1.08127 q^{54} -7.73752 q^{55} -3.06092 q^{56} +0.793678 q^{57} -4.99264 q^{58} -12.2304 q^{59} +1.38688 q^{60} +10.3412 q^{61} +6.22497 q^{62} -1.00000 q^{63} +7.98853 q^{64} -5.01215 q^{66} +1.86971 q^{67} -3.35026 q^{68} +4.31595 q^{69} -1.80485 q^{70} -6.08759 q^{71} +3.06092 q^{72} +3.11657 q^{73} +6.20102 q^{74} -2.21376 q^{75} -0.659440 q^{76} -4.63545 q^{77} +2.28592 q^{79} +2.75073 q^{80} +1.00000 q^{81} +7.93230 q^{82} -0.655507 q^{83} +0.830866 q^{84} -6.73066 q^{85} -12.3636 q^{86} +4.61741 q^{87} +14.1887 q^{88} +10.8262 q^{89} +1.80485 q^{90} -3.58597 q^{92} -5.75711 q^{93} -5.52419 q^{94} -1.32481 q^{95} -4.33998 q^{96} +14.7169 q^{97} -1.08127 q^{98} +4.63545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 2q^{2} + 15q^{3} + 28q^{4} - 9q^{5} + 2q^{6} - 15q^{7} + 9q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 2q^{2} + 15q^{3} + 28q^{4} - 9q^{5} + 2q^{6} - 15q^{7} + 9q^{8} + 15q^{9} + 21q^{10} + 5q^{11} + 28q^{12} - 2q^{14} - 9q^{15} + 50q^{16} - q^{17} + 2q^{18} + 3q^{19} - 23q^{20} - 15q^{21} + 21q^{22} + 4q^{23} + 9q^{24} + 50q^{25} + 15q^{27} - 28q^{28} + 9q^{29} + 21q^{30} + 7q^{31} + 35q^{32} + 5q^{33} - 2q^{34} + 9q^{35} + 28q^{36} + 17q^{37} - 12q^{38} + 46q^{40} - 22q^{41} - 2q^{42} + 36q^{43} + 29q^{44} - 9q^{45} - q^{46} - 12q^{47} + 50q^{48} + 15q^{49} + 53q^{50} - q^{51} - 5q^{53} + 2q^{54} + 43q^{55} - 9q^{56} + 3q^{57} + 29q^{58} - 29q^{59} - 23q^{60} + 12q^{61} + 14q^{62} - 15q^{63} + 95q^{64} + 21q^{66} - 12q^{67} - 16q^{68} + 4q^{69} - 21q^{70} + 36q^{71} + 9q^{72} - 29q^{73} - 5q^{74} + 50q^{75} + 25q^{76} - 5q^{77} + 35q^{79} - 89q^{80} + 15q^{81} + 51q^{82} - 10q^{83} - 28q^{84} + 23q^{85} + 19q^{86} + 9q^{87} + 73q^{88} + 25q^{89} + 21q^{90} - 31q^{92} + 7q^{93} - 19q^{94} - 7q^{95} + 35q^{96} - 26q^{97} + 2q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08127 −0.764570 −0.382285 0.924044i \(-0.624863\pi\)
−0.382285 + 0.924044i \(0.624863\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.830866 −0.415433
\(5\) −1.66920 −0.746491 −0.373245 0.927733i \(-0.621755\pi\)
−0.373245 + 0.927733i \(0.621755\pi\)
\(6\) −1.08127 −0.441425
\(7\) −1.00000 −0.377964
\(8\) 3.06092 1.08220
\(9\) 1.00000 0.333333
\(10\) 1.80485 0.570744
\(11\) 4.63545 1.39764 0.698821 0.715297i \(-0.253709\pi\)
0.698821 + 0.715297i \(0.253709\pi\)
\(12\) −0.830866 −0.239850
\(13\) 0 0
\(14\) 1.08127 0.288980
\(15\) −1.66920 −0.430987
\(16\) −1.64793 −0.411983
\(17\) 4.03225 0.977965 0.488983 0.872294i \(-0.337368\pi\)
0.488983 + 0.872294i \(0.337368\pi\)
\(18\) −1.08127 −0.254857
\(19\) 0.793678 0.182082 0.0910411 0.995847i \(-0.470981\pi\)
0.0910411 + 0.995847i \(0.470981\pi\)
\(20\) 1.38688 0.310117
\(21\) −1.00000 −0.218218
\(22\) −5.01215 −1.06859
\(23\) 4.31595 0.899937 0.449969 0.893044i \(-0.351435\pi\)
0.449969 + 0.893044i \(0.351435\pi\)
\(24\) 3.06092 0.624807
\(25\) −2.21376 −0.442752
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0.830866 0.157019
\(29\) 4.61741 0.857431 0.428716 0.903439i \(-0.358966\pi\)
0.428716 + 0.903439i \(0.358966\pi\)
\(30\) 1.80485 0.329519
\(31\) −5.75711 −1.03401 −0.517004 0.855983i \(-0.672953\pi\)
−0.517004 + 0.855983i \(0.672953\pi\)
\(32\) −4.33998 −0.767208
\(33\) 4.63545 0.806929
\(34\) −4.35994 −0.747723
\(35\) 1.66920 0.282147
\(36\) −0.830866 −0.138478
\(37\) −5.73497 −0.942823 −0.471412 0.881913i \(-0.656255\pi\)
−0.471412 + 0.881913i \(0.656255\pi\)
\(38\) −0.858176 −0.139215
\(39\) 0 0
\(40\) −5.10929 −0.807850
\(41\) −7.33613 −1.14571 −0.572855 0.819657i \(-0.694164\pi\)
−0.572855 + 0.819657i \(0.694164\pi\)
\(42\) 1.08127 0.166843
\(43\) 11.4344 1.74373 0.871863 0.489750i \(-0.162912\pi\)
0.871863 + 0.489750i \(0.162912\pi\)
\(44\) −3.85144 −0.580626
\(45\) −1.66920 −0.248830
\(46\) −4.66668 −0.688065
\(47\) 5.10901 0.745226 0.372613 0.927987i \(-0.378462\pi\)
0.372613 + 0.927987i \(0.378462\pi\)
\(48\) −1.64793 −0.237858
\(49\) 1.00000 0.142857
\(50\) 2.39366 0.338514
\(51\) 4.03225 0.564629
\(52\) 0 0
\(53\) −6.41614 −0.881325 −0.440663 0.897673i \(-0.645256\pi\)
−0.440663 + 0.897673i \(0.645256\pi\)
\(54\) −1.08127 −0.147142
\(55\) −7.73752 −1.04333
\(56\) −3.06092 −0.409032
\(57\) 0.793678 0.105125
\(58\) −4.99264 −0.655566
\(59\) −12.2304 −1.59227 −0.796133 0.605122i \(-0.793124\pi\)
−0.796133 + 0.605122i \(0.793124\pi\)
\(60\) 1.38688 0.179046
\(61\) 10.3412 1.32406 0.662029 0.749478i \(-0.269695\pi\)
0.662029 + 0.749478i \(0.269695\pi\)
\(62\) 6.22497 0.790571
\(63\) −1.00000 −0.125988
\(64\) 7.98853 0.998567
\(65\) 0 0
\(66\) −5.01215 −0.616953
\(67\) 1.86971 0.228422 0.114211 0.993457i \(-0.463566\pi\)
0.114211 + 0.993457i \(0.463566\pi\)
\(68\) −3.35026 −0.406279
\(69\) 4.31595 0.519579
\(70\) −1.80485 −0.215721
\(71\) −6.08759 −0.722464 −0.361232 0.932476i \(-0.617644\pi\)
−0.361232 + 0.932476i \(0.617644\pi\)
\(72\) 3.06092 0.360732
\(73\) 3.11657 0.364767 0.182384 0.983227i \(-0.441619\pi\)
0.182384 + 0.983227i \(0.441619\pi\)
\(74\) 6.20102 0.720854
\(75\) −2.21376 −0.255623
\(76\) −0.659440 −0.0756429
\(77\) −4.63545 −0.528259
\(78\) 0 0
\(79\) 2.28592 0.257186 0.128593 0.991697i \(-0.458954\pi\)
0.128593 + 0.991697i \(0.458954\pi\)
\(80\) 2.75073 0.307541
\(81\) 1.00000 0.111111
\(82\) 7.93230 0.875976
\(83\) −0.655507 −0.0719513 −0.0359756 0.999353i \(-0.511454\pi\)
−0.0359756 + 0.999353i \(0.511454\pi\)
\(84\) 0.830866 0.0906549
\(85\) −6.73066 −0.730042
\(86\) −12.3636 −1.33320
\(87\) 4.61741 0.495038
\(88\) 14.1887 1.51252
\(89\) 10.8262 1.14758 0.573788 0.819004i \(-0.305473\pi\)
0.573788 + 0.819004i \(0.305473\pi\)
\(90\) 1.80485 0.190248
\(91\) 0 0
\(92\) −3.58597 −0.373864
\(93\) −5.75711 −0.596985
\(94\) −5.52419 −0.569777
\(95\) −1.32481 −0.135923
\(96\) −4.33998 −0.442948
\(97\) 14.7169 1.49427 0.747136 0.664671i \(-0.231428\pi\)
0.747136 + 0.664671i \(0.231428\pi\)
\(98\) −1.08127 −0.109224
\(99\) 4.63545 0.465880
\(100\) 1.83934 0.183934
\(101\) −7.83185 −0.779298 −0.389649 0.920963i \(-0.627404\pi\)
−0.389649 + 0.920963i \(0.627404\pi\)
\(102\) −4.35994 −0.431698
\(103\) −14.2990 −1.40892 −0.704461 0.709743i \(-0.748811\pi\)
−0.704461 + 0.709743i \(0.748811\pi\)
\(104\) 0 0
\(105\) 1.66920 0.162898
\(106\) 6.93755 0.673835
\(107\) −13.6120 −1.31592 −0.657960 0.753053i \(-0.728580\pi\)
−0.657960 + 0.753053i \(0.728580\pi\)
\(108\) −0.830866 −0.0799501
\(109\) 1.08293 0.103726 0.0518631 0.998654i \(-0.483484\pi\)
0.0518631 + 0.998654i \(0.483484\pi\)
\(110\) 8.36630 0.797696
\(111\) −5.73497 −0.544339
\(112\) 1.64793 0.155715
\(113\) 14.7313 1.38581 0.692903 0.721031i \(-0.256331\pi\)
0.692903 + 0.721031i \(0.256331\pi\)
\(114\) −0.858176 −0.0803755
\(115\) −7.20420 −0.671795
\(116\) −3.83645 −0.356205
\(117\) 0 0
\(118\) 13.2243 1.21740
\(119\) −4.03225 −0.369636
\(120\) −5.10929 −0.466413
\(121\) 10.4874 0.953401
\(122\) −11.1816 −1.01234
\(123\) −7.33613 −0.661476
\(124\) 4.78339 0.429561
\(125\) 12.0412 1.07700
\(126\) 1.08127 0.0963267
\(127\) 19.2838 1.71116 0.855582 0.517667i \(-0.173199\pi\)
0.855582 + 0.517667i \(0.173199\pi\)
\(128\) 0.0422458 0.00373404
\(129\) 11.4344 1.00674
\(130\) 0 0
\(131\) 0.626566 0.0547433 0.0273717 0.999625i \(-0.491286\pi\)
0.0273717 + 0.999625i \(0.491286\pi\)
\(132\) −3.85144 −0.335225
\(133\) −0.793678 −0.0688206
\(134\) −2.02166 −0.174644
\(135\) −1.66920 −0.143662
\(136\) 12.3424 1.05835
\(137\) −1.27827 −0.109210 −0.0546052 0.998508i \(-0.517390\pi\)
−0.0546052 + 0.998508i \(0.517390\pi\)
\(138\) −4.66668 −0.397254
\(139\) 17.3701 1.47331 0.736656 0.676268i \(-0.236404\pi\)
0.736656 + 0.676268i \(0.236404\pi\)
\(140\) −1.38688 −0.117213
\(141\) 5.10901 0.430256
\(142\) 6.58230 0.552375
\(143\) 0 0
\(144\) −1.64793 −0.137328
\(145\) −7.70740 −0.640065
\(146\) −3.36984 −0.278890
\(147\) 1.00000 0.0824786
\(148\) 4.76499 0.391680
\(149\) 2.06250 0.168967 0.0844834 0.996425i \(-0.473076\pi\)
0.0844834 + 0.996425i \(0.473076\pi\)
\(150\) 2.39366 0.195441
\(151\) −10.6932 −0.870199 −0.435100 0.900382i \(-0.643287\pi\)
−0.435100 + 0.900382i \(0.643287\pi\)
\(152\) 2.42938 0.197049
\(153\) 4.03225 0.325988
\(154\) 5.01215 0.403891
\(155\) 9.60980 0.771878
\(156\) 0 0
\(157\) −9.91978 −0.791685 −0.395842 0.918318i \(-0.629547\pi\)
−0.395842 + 0.918318i \(0.629547\pi\)
\(158\) −2.47169 −0.196637
\(159\) −6.41614 −0.508833
\(160\) 7.24432 0.572714
\(161\) −4.31595 −0.340144
\(162\) −1.08127 −0.0849522
\(163\) 20.9575 1.64152 0.820759 0.571275i \(-0.193551\pi\)
0.820759 + 0.571275i \(0.193551\pi\)
\(164\) 6.09534 0.475966
\(165\) −7.73752 −0.602365
\(166\) 0.708777 0.0550118
\(167\) 8.36605 0.647384 0.323692 0.946162i \(-0.395076\pi\)
0.323692 + 0.946162i \(0.395076\pi\)
\(168\) −3.06092 −0.236155
\(169\) 0 0
\(170\) 7.27762 0.558168
\(171\) 0.793678 0.0606940
\(172\) −9.50043 −0.724401
\(173\) 5.93501 0.451230 0.225615 0.974217i \(-0.427561\pi\)
0.225615 + 0.974217i \(0.427561\pi\)
\(174\) −4.99264 −0.378491
\(175\) 2.21376 0.167344
\(176\) −7.63890 −0.575804
\(177\) −12.2304 −0.919295
\(178\) −11.7060 −0.877403
\(179\) 14.5930 1.09073 0.545365 0.838199i \(-0.316391\pi\)
0.545365 + 0.838199i \(0.316391\pi\)
\(180\) 1.38688 0.103372
\(181\) −2.33617 −0.173646 −0.0868232 0.996224i \(-0.527672\pi\)
−0.0868232 + 0.996224i \(0.527672\pi\)
\(182\) 0 0
\(183\) 10.3412 0.764446
\(184\) 13.2108 0.973910
\(185\) 9.57283 0.703809
\(186\) 6.22497 0.456437
\(187\) 18.6913 1.36684
\(188\) −4.24490 −0.309591
\(189\) −1.00000 −0.0727393
\(190\) 1.43247 0.103922
\(191\) 2.92459 0.211616 0.105808 0.994387i \(-0.466257\pi\)
0.105808 + 0.994387i \(0.466257\pi\)
\(192\) 7.98853 0.576523
\(193\) 24.0430 1.73065 0.865327 0.501207i \(-0.167111\pi\)
0.865327 + 0.501207i \(0.167111\pi\)
\(194\) −15.9128 −1.14248
\(195\) 0 0
\(196\) −0.830866 −0.0593476
\(197\) 14.6055 1.04060 0.520299 0.853984i \(-0.325820\pi\)
0.520299 + 0.853984i \(0.325820\pi\)
\(198\) −5.01215 −0.356198
\(199\) −15.4843 −1.09765 −0.548827 0.835936i \(-0.684925\pi\)
−0.548827 + 0.835936i \(0.684925\pi\)
\(200\) −6.77613 −0.479145
\(201\) 1.86971 0.131879
\(202\) 8.46830 0.595828
\(203\) −4.61741 −0.324079
\(204\) −3.35026 −0.234565
\(205\) 12.2455 0.855262
\(206\) 15.4610 1.07722
\(207\) 4.31595 0.299979
\(208\) 0 0
\(209\) 3.67905 0.254486
\(210\) −1.80485 −0.124547
\(211\) 19.8268 1.36493 0.682467 0.730917i \(-0.260907\pi\)
0.682467 + 0.730917i \(0.260907\pi\)
\(212\) 5.33095 0.366131
\(213\) −6.08759 −0.417115
\(214\) 14.7182 1.00611
\(215\) −19.0863 −1.30168
\(216\) 3.06092 0.208269
\(217\) 5.75711 0.390818
\(218\) −1.17094 −0.0793059
\(219\) 3.11657 0.210599
\(220\) 6.42884 0.433432
\(221\) 0 0
\(222\) 6.20102 0.416185
\(223\) −27.3639 −1.83242 −0.916211 0.400695i \(-0.868769\pi\)
−0.916211 + 0.400695i \(0.868769\pi\)
\(224\) 4.33998 0.289977
\(225\) −2.21376 −0.147584
\(226\) −15.9285 −1.05955
\(227\) 15.1297 1.00419 0.502096 0.864812i \(-0.332562\pi\)
0.502096 + 0.864812i \(0.332562\pi\)
\(228\) −0.659440 −0.0436725
\(229\) 10.6735 0.705322 0.352661 0.935751i \(-0.385277\pi\)
0.352661 + 0.935751i \(0.385277\pi\)
\(230\) 7.78965 0.513634
\(231\) −4.63545 −0.304990
\(232\) 14.1335 0.927910
\(233\) −10.6341 −0.696660 −0.348330 0.937372i \(-0.613251\pi\)
−0.348330 + 0.937372i \(0.613251\pi\)
\(234\) 0 0
\(235\) −8.52798 −0.556304
\(236\) 10.1618 0.661479
\(237\) 2.28592 0.148487
\(238\) 4.35994 0.282613
\(239\) 14.5524 0.941315 0.470658 0.882316i \(-0.344017\pi\)
0.470658 + 0.882316i \(0.344017\pi\)
\(240\) 2.75073 0.177559
\(241\) 21.0486 1.35586 0.677929 0.735127i \(-0.262877\pi\)
0.677929 + 0.735127i \(0.262877\pi\)
\(242\) −11.3397 −0.728942
\(243\) 1.00000 0.0641500
\(244\) −8.59218 −0.550058
\(245\) −1.66920 −0.106642
\(246\) 7.93230 0.505745
\(247\) 0 0
\(248\) −17.6220 −1.11900
\(249\) −0.655507 −0.0415411
\(250\) −13.0198 −0.823442
\(251\) −16.3957 −1.03489 −0.517443 0.855718i \(-0.673116\pi\)
−0.517443 + 0.855718i \(0.673116\pi\)
\(252\) 0.830866 0.0523396
\(253\) 20.0064 1.25779
\(254\) −20.8509 −1.30830
\(255\) −6.73066 −0.421490
\(256\) −16.0227 −1.00142
\(257\) −14.2261 −0.887400 −0.443700 0.896175i \(-0.646334\pi\)
−0.443700 + 0.896175i \(0.646334\pi\)
\(258\) −12.3636 −0.769724
\(259\) 5.73497 0.356354
\(260\) 0 0
\(261\) 4.61741 0.285810
\(262\) −0.677484 −0.0418551
\(263\) 17.0566 1.05175 0.525876 0.850561i \(-0.323737\pi\)
0.525876 + 0.850561i \(0.323737\pi\)
\(264\) 14.1887 0.873256
\(265\) 10.7099 0.657901
\(266\) 0.858176 0.0526181
\(267\) 10.8262 0.662554
\(268\) −1.55348 −0.0948940
\(269\) 17.2732 1.05317 0.526583 0.850124i \(-0.323473\pi\)
0.526583 + 0.850124i \(0.323473\pi\)
\(270\) 1.80485 0.109840
\(271\) −7.90154 −0.479985 −0.239992 0.970775i \(-0.577145\pi\)
−0.239992 + 0.970775i \(0.577145\pi\)
\(272\) −6.64487 −0.402905
\(273\) 0 0
\(274\) 1.38215 0.0834990
\(275\) −10.2618 −0.618808
\(276\) −3.58597 −0.215850
\(277\) 4.50995 0.270977 0.135488 0.990779i \(-0.456740\pi\)
0.135488 + 0.990779i \(0.456740\pi\)
\(278\) −18.7817 −1.12645
\(279\) −5.75711 −0.344669
\(280\) 5.10929 0.305339
\(281\) −20.7726 −1.23919 −0.619594 0.784922i \(-0.712703\pi\)
−0.619594 + 0.784922i \(0.712703\pi\)
\(282\) −5.52419 −0.328961
\(283\) −2.02320 −0.120267 −0.0601335 0.998190i \(-0.519153\pi\)
−0.0601335 + 0.998190i \(0.519153\pi\)
\(284\) 5.05797 0.300136
\(285\) −1.32481 −0.0784750
\(286\) 0 0
\(287\) 7.33613 0.433038
\(288\) −4.33998 −0.255736
\(289\) −0.740925 −0.0435838
\(290\) 8.33374 0.489374
\(291\) 14.7169 0.862719
\(292\) −2.58946 −0.151536
\(293\) −25.9848 −1.51805 −0.759024 0.651063i \(-0.774323\pi\)
−0.759024 + 0.651063i \(0.774323\pi\)
\(294\) −1.08127 −0.0630607
\(295\) 20.4151 1.18861
\(296\) −17.5543 −1.02032
\(297\) 4.63545 0.268976
\(298\) −2.23011 −0.129187
\(299\) 0 0
\(300\) 1.83934 0.106194
\(301\) −11.4344 −0.659067
\(302\) 11.5622 0.665328
\(303\) −7.83185 −0.449928
\(304\) −1.30793 −0.0750147
\(305\) −17.2616 −0.988398
\(306\) −4.35994 −0.249241
\(307\) −24.6185 −1.40505 −0.702526 0.711658i \(-0.747945\pi\)
−0.702526 + 0.711658i \(0.747945\pi\)
\(308\) 3.85144 0.219456
\(309\) −14.2990 −0.813441
\(310\) −10.3907 −0.590154
\(311\) 33.1216 1.87815 0.939076 0.343711i \(-0.111684\pi\)
0.939076 + 0.343711i \(0.111684\pi\)
\(312\) 0 0
\(313\) −7.05477 −0.398759 −0.199380 0.979922i \(-0.563893\pi\)
−0.199380 + 0.979922i \(0.563893\pi\)
\(314\) 10.7259 0.605298
\(315\) 1.66920 0.0940490
\(316\) −1.89930 −0.106844
\(317\) −13.0084 −0.730622 −0.365311 0.930886i \(-0.619037\pi\)
−0.365311 + 0.930886i \(0.619037\pi\)
\(318\) 6.93755 0.389039
\(319\) 21.4038 1.19838
\(320\) −13.3345 −0.745421
\(321\) −13.6120 −0.759747
\(322\) 4.66668 0.260064
\(323\) 3.20031 0.178070
\(324\) −0.830866 −0.0461592
\(325\) 0 0
\(326\) −22.6606 −1.25506
\(327\) 1.08293 0.0598863
\(328\) −22.4553 −1.23988
\(329\) −5.10901 −0.281669
\(330\) 8.36630 0.460550
\(331\) −17.9368 −0.985894 −0.492947 0.870059i \(-0.664080\pi\)
−0.492947 + 0.870059i \(0.664080\pi\)
\(332\) 0.544638 0.0298909
\(333\) −5.73497 −0.314274
\(334\) −9.04592 −0.494970
\(335\) −3.12093 −0.170515
\(336\) 1.64793 0.0899020
\(337\) 28.3388 1.54371 0.771856 0.635797i \(-0.219328\pi\)
0.771856 + 0.635797i \(0.219328\pi\)
\(338\) 0 0
\(339\) 14.7313 0.800095
\(340\) 5.59227 0.303284
\(341\) −26.6868 −1.44517
\(342\) −0.858176 −0.0464048
\(343\) −1.00000 −0.0539949
\(344\) 34.9997 1.88706
\(345\) −7.20420 −0.387861
\(346\) −6.41732 −0.344997
\(347\) −8.49524 −0.456048 −0.228024 0.973655i \(-0.573227\pi\)
−0.228024 + 0.973655i \(0.573227\pi\)
\(348\) −3.83645 −0.205655
\(349\) 11.0479 0.591381 0.295691 0.955284i \(-0.404450\pi\)
0.295691 + 0.955284i \(0.404450\pi\)
\(350\) −2.39366 −0.127946
\(351\) 0 0
\(352\) −20.1178 −1.07228
\(353\) −11.0968 −0.590621 −0.295310 0.955401i \(-0.595423\pi\)
−0.295310 + 0.955401i \(0.595423\pi\)
\(354\) 13.2243 0.702865
\(355\) 10.1614 0.539313
\(356\) −8.99514 −0.476741
\(357\) −4.03225 −0.213410
\(358\) −15.7789 −0.833940
\(359\) 16.3104 0.860830 0.430415 0.902631i \(-0.358367\pi\)
0.430415 + 0.902631i \(0.358367\pi\)
\(360\) −5.10929 −0.269283
\(361\) −18.3701 −0.966846
\(362\) 2.52602 0.132765
\(363\) 10.4874 0.550446
\(364\) 0 0
\(365\) −5.20220 −0.272296
\(366\) −11.1816 −0.584472
\(367\) 15.3843 0.803056 0.401528 0.915847i \(-0.368479\pi\)
0.401528 + 0.915847i \(0.368479\pi\)
\(368\) −7.11238 −0.370758
\(369\) −7.33613 −0.381903
\(370\) −10.3508 −0.538111
\(371\) 6.41614 0.333110
\(372\) 4.78339 0.248007
\(373\) 21.1348 1.09432 0.547160 0.837028i \(-0.315709\pi\)
0.547160 + 0.837028i \(0.315709\pi\)
\(374\) −20.2103 −1.04505
\(375\) 12.0412 0.621807
\(376\) 15.6382 0.806481
\(377\) 0 0
\(378\) 1.08127 0.0556143
\(379\) 27.2340 1.39892 0.699458 0.714674i \(-0.253425\pi\)
0.699458 + 0.714674i \(0.253425\pi\)
\(380\) 1.10074 0.0564667
\(381\) 19.2838 0.987941
\(382\) −3.16226 −0.161795
\(383\) −15.5635 −0.795257 −0.397629 0.917546i \(-0.630167\pi\)
−0.397629 + 0.917546i \(0.630167\pi\)
\(384\) 0.0422458 0.00215585
\(385\) 7.73752 0.394340
\(386\) −25.9969 −1.32321
\(387\) 11.4344 0.581242
\(388\) −12.2278 −0.620770
\(389\) −25.1109 −1.27317 −0.636586 0.771206i \(-0.719654\pi\)
−0.636586 + 0.771206i \(0.719654\pi\)
\(390\) 0 0
\(391\) 17.4030 0.880107
\(392\) 3.06092 0.154600
\(393\) 0.626566 0.0316061
\(394\) −15.7924 −0.795610
\(395\) −3.81567 −0.191987
\(396\) −3.85144 −0.193542
\(397\) −3.55580 −0.178461 −0.0892304 0.996011i \(-0.528441\pi\)
−0.0892304 + 0.996011i \(0.528441\pi\)
\(398\) 16.7426 0.839233
\(399\) −0.793678 −0.0397336
\(400\) 3.64812 0.182406
\(401\) 31.3198 1.56403 0.782017 0.623257i \(-0.214191\pi\)
0.782017 + 0.623257i \(0.214191\pi\)
\(402\) −2.02166 −0.100831
\(403\) 0 0
\(404\) 6.50721 0.323746
\(405\) −1.66920 −0.0829434
\(406\) 4.99264 0.247781
\(407\) −26.5842 −1.31773
\(408\) 12.3424 0.611040
\(409\) 10.2735 0.507991 0.253996 0.967205i \(-0.418255\pi\)
0.253996 + 0.967205i \(0.418255\pi\)
\(410\) −13.2406 −0.653908
\(411\) −1.27827 −0.0630527
\(412\) 11.8805 0.585312
\(413\) 12.2304 0.601820
\(414\) −4.66668 −0.229355
\(415\) 1.09418 0.0537109
\(416\) 0 0
\(417\) 17.3701 0.850617
\(418\) −3.97803 −0.194572
\(419\) 5.29913 0.258879 0.129440 0.991587i \(-0.458682\pi\)
0.129440 + 0.991587i \(0.458682\pi\)
\(420\) −1.38688 −0.0676730
\(421\) 17.2903 0.842677 0.421339 0.906903i \(-0.361560\pi\)
0.421339 + 0.906903i \(0.361560\pi\)
\(422\) −21.4380 −1.04359
\(423\) 5.10901 0.248409
\(424\) −19.6393 −0.953768
\(425\) −8.92643 −0.432996
\(426\) 6.58230 0.318914
\(427\) −10.3412 −0.500447
\(428\) 11.3097 0.546676
\(429\) 0 0
\(430\) 20.6374 0.995222
\(431\) 19.2244 0.926008 0.463004 0.886356i \(-0.346771\pi\)
0.463004 + 0.886356i \(0.346771\pi\)
\(432\) −1.64793 −0.0792861
\(433\) −9.23571 −0.443840 −0.221920 0.975065i \(-0.571232\pi\)
−0.221920 + 0.975065i \(0.571232\pi\)
\(434\) −6.22497 −0.298808
\(435\) −7.70740 −0.369542
\(436\) −0.899772 −0.0430912
\(437\) 3.42547 0.163862
\(438\) −3.36984 −0.161017
\(439\) 36.8646 1.75945 0.879726 0.475480i \(-0.157726\pi\)
0.879726 + 0.475480i \(0.157726\pi\)
\(440\) −23.6839 −1.12909
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −20.1394 −0.956850 −0.478425 0.878128i \(-0.658792\pi\)
−0.478425 + 0.878128i \(0.658792\pi\)
\(444\) 4.76499 0.226136
\(445\) −18.0712 −0.856656
\(446\) 29.5876 1.40102
\(447\) 2.06250 0.0975530
\(448\) −7.98853 −0.377423
\(449\) −33.2197 −1.56774 −0.783868 0.620927i \(-0.786756\pi\)
−0.783868 + 0.620927i \(0.786756\pi\)
\(450\) 2.39366 0.112838
\(451\) −34.0063 −1.60129
\(452\) −12.2398 −0.575709
\(453\) −10.6932 −0.502410
\(454\) −16.3592 −0.767776
\(455\) 0 0
\(456\) 2.42938 0.113766
\(457\) 25.1540 1.17665 0.588327 0.808623i \(-0.299787\pi\)
0.588327 + 0.808623i \(0.299787\pi\)
\(458\) −11.5408 −0.539268
\(459\) 4.03225 0.188210
\(460\) 5.98572 0.279086
\(461\) 23.1849 1.07983 0.539915 0.841720i \(-0.318457\pi\)
0.539915 + 0.841720i \(0.318457\pi\)
\(462\) 5.01215 0.233186
\(463\) −12.8679 −0.598022 −0.299011 0.954250i \(-0.596657\pi\)
−0.299011 + 0.954250i \(0.596657\pi\)
\(464\) −7.60917 −0.353247
\(465\) 9.60980 0.445644
\(466\) 11.4982 0.532645
\(467\) −25.5865 −1.18400 −0.592000 0.805938i \(-0.701661\pi\)
−0.592000 + 0.805938i \(0.701661\pi\)
\(468\) 0 0
\(469\) −1.86971 −0.0863353
\(470\) 9.22101 0.425333
\(471\) −9.91978 −0.457079
\(472\) −37.4363 −1.72315
\(473\) 53.0035 2.43710
\(474\) −2.47169 −0.113528
\(475\) −1.75701 −0.0806171
\(476\) 3.35026 0.153559
\(477\) −6.41614 −0.293775
\(478\) −15.7350 −0.719701
\(479\) 25.2752 1.15485 0.577426 0.816443i \(-0.304057\pi\)
0.577426 + 0.816443i \(0.304057\pi\)
\(480\) 7.24432 0.330656
\(481\) 0 0
\(482\) −22.7591 −1.03665
\(483\) −4.31595 −0.196382
\(484\) −8.71363 −0.396074
\(485\) −24.5655 −1.11546
\(486\) −1.08127 −0.0490472
\(487\) −42.4961 −1.92568 −0.962841 0.270068i \(-0.912954\pi\)
−0.962841 + 0.270068i \(0.912954\pi\)
\(488\) 31.6536 1.43289
\(489\) 20.9575 0.947731
\(490\) 1.80485 0.0815349
\(491\) −40.3541 −1.82116 −0.910578 0.413337i \(-0.864363\pi\)
−0.910578 + 0.413337i \(0.864363\pi\)
\(492\) 6.09534 0.274799
\(493\) 18.6186 0.838538
\(494\) 0 0
\(495\) −7.73752 −0.347775
\(496\) 9.48732 0.425993
\(497\) 6.08759 0.273066
\(498\) 0.708777 0.0317611
\(499\) 5.56698 0.249212 0.124606 0.992206i \(-0.460233\pi\)
0.124606 + 0.992206i \(0.460233\pi\)
\(500\) −10.0047 −0.447422
\(501\) 8.36605 0.373767
\(502\) 17.7281 0.791242
\(503\) 18.0562 0.805085 0.402542 0.915401i \(-0.368127\pi\)
0.402542 + 0.915401i \(0.368127\pi\)
\(504\) −3.06092 −0.136344
\(505\) 13.0729 0.581739
\(506\) −21.6322 −0.961668
\(507\) 0 0
\(508\) −16.0223 −0.710874
\(509\) 9.76430 0.432795 0.216397 0.976305i \(-0.430569\pi\)
0.216397 + 0.976305i \(0.430569\pi\)
\(510\) 7.27762 0.322259
\(511\) −3.11657 −0.137869
\(512\) 17.2403 0.761923
\(513\) 0.793678 0.0350417
\(514\) 15.3822 0.678480
\(515\) 23.8679 1.05175
\(516\) −9.50043 −0.418233
\(517\) 23.6826 1.04156
\(518\) −6.20102 −0.272457
\(519\) 5.93501 0.260518
\(520\) 0 0
\(521\) 19.4225 0.850914 0.425457 0.904979i \(-0.360113\pi\)
0.425457 + 0.904979i \(0.360113\pi\)
\(522\) −4.99264 −0.218522
\(523\) 15.4560 0.675845 0.337923 0.941174i \(-0.390276\pi\)
0.337923 + 0.941174i \(0.390276\pi\)
\(524\) −0.520592 −0.0227422
\(525\) 2.21376 0.0966163
\(526\) −18.4427 −0.804138
\(527\) −23.2141 −1.01122
\(528\) −7.63890 −0.332440
\(529\) −4.37260 −0.190113
\(530\) −11.5802 −0.503011
\(531\) −12.2304 −0.530755
\(532\) 0.659440 0.0285903
\(533\) 0 0
\(534\) −11.7060 −0.506569
\(535\) 22.7212 0.982322
\(536\) 5.72304 0.247197
\(537\) 14.5930 0.629733
\(538\) −18.6769 −0.805219
\(539\) 4.63545 0.199663
\(540\) 1.38688 0.0596820
\(541\) 3.65437 0.157114 0.0785569 0.996910i \(-0.474969\pi\)
0.0785569 + 0.996910i \(0.474969\pi\)
\(542\) 8.54366 0.366982
\(543\) −2.33617 −0.100255
\(544\) −17.4999 −0.750303
\(545\) −1.80764 −0.0774306
\(546\) 0 0
\(547\) 41.0035 1.75318 0.876592 0.481235i \(-0.159812\pi\)
0.876592 + 0.481235i \(0.159812\pi\)
\(548\) 1.06208 0.0453696
\(549\) 10.3412 0.441353
\(550\) 11.0957 0.473122
\(551\) 3.66473 0.156123
\(552\) 13.2108 0.562287
\(553\) −2.28592 −0.0972074
\(554\) −4.87645 −0.207181
\(555\) 9.57283 0.406344
\(556\) −14.4322 −0.612062
\(557\) 6.55475 0.277734 0.138867 0.990311i \(-0.455654\pi\)
0.138867 + 0.990311i \(0.455654\pi\)
\(558\) 6.22497 0.263524
\(559\) 0 0
\(560\) −2.75073 −0.116240
\(561\) 18.6913 0.789148
\(562\) 22.4607 0.947446
\(563\) −25.4130 −1.07103 −0.535516 0.844525i \(-0.679883\pi\)
−0.535516 + 0.844525i \(0.679883\pi\)
\(564\) −4.24490 −0.178743
\(565\) −24.5896 −1.03449
\(566\) 2.18762 0.0919526
\(567\) −1.00000 −0.0419961
\(568\) −18.6336 −0.781849
\(569\) 21.8432 0.915713 0.457857 0.889026i \(-0.348617\pi\)
0.457857 + 0.889026i \(0.348617\pi\)
\(570\) 1.43247 0.0599996
\(571\) 28.7510 1.20319 0.601596 0.798800i \(-0.294532\pi\)
0.601596 + 0.798800i \(0.294532\pi\)
\(572\) 0 0
\(573\) 2.92459 0.122177
\(574\) −7.93230 −0.331088
\(575\) −9.55446 −0.398449
\(576\) 7.98853 0.332856
\(577\) 46.2811 1.92671 0.963353 0.268238i \(-0.0864414\pi\)
0.963353 + 0.268238i \(0.0864414\pi\)
\(578\) 0.801136 0.0333229
\(579\) 24.0430 0.999194
\(580\) 6.40382 0.265904
\(581\) 0.655507 0.0271950
\(582\) −15.9128 −0.659609
\(583\) −29.7417 −1.23178
\(584\) 9.53957 0.394750
\(585\) 0 0
\(586\) 28.0965 1.16065
\(587\) −1.61084 −0.0664866 −0.0332433 0.999447i \(-0.510584\pi\)
−0.0332433 + 0.999447i \(0.510584\pi\)
\(588\) −0.830866 −0.0342643
\(589\) −4.56929 −0.188274
\(590\) −22.0741 −0.908776
\(591\) 14.6055 0.600790
\(592\) 9.45083 0.388427
\(593\) 36.9127 1.51582 0.757912 0.652357i \(-0.226220\pi\)
0.757912 + 0.652357i \(0.226220\pi\)
\(594\) −5.01215 −0.205651
\(595\) 6.73066 0.275930
\(596\) −1.71366 −0.0701944
\(597\) −15.4843 −0.633730
\(598\) 0 0
\(599\) −13.4868 −0.551054 −0.275527 0.961293i \(-0.588852\pi\)
−0.275527 + 0.961293i \(0.588852\pi\)
\(600\) −6.77613 −0.276634
\(601\) 47.7912 1.94945 0.974723 0.223418i \(-0.0717214\pi\)
0.974723 + 0.223418i \(0.0717214\pi\)
\(602\) 12.3636 0.503902
\(603\) 1.86971 0.0761406
\(604\) 8.88460 0.361509
\(605\) −17.5056 −0.711705
\(606\) 8.46830 0.344001
\(607\) −31.8176 −1.29144 −0.645718 0.763576i \(-0.723442\pi\)
−0.645718 + 0.763576i \(0.723442\pi\)
\(608\) −3.44455 −0.139695
\(609\) −4.61741 −0.187107
\(610\) 18.6644 0.755699
\(611\) 0 0
\(612\) −3.35026 −0.135426
\(613\) −35.4524 −1.43191 −0.715955 0.698146i \(-0.754009\pi\)
−0.715955 + 0.698146i \(0.754009\pi\)
\(614\) 26.6191 1.07426
\(615\) 12.2455 0.493786
\(616\) −14.1887 −0.571680
\(617\) 49.4752 1.99180 0.995899 0.0904759i \(-0.0288388\pi\)
0.995899 + 0.0904759i \(0.0288388\pi\)
\(618\) 15.4610 0.621933
\(619\) −11.7993 −0.474253 −0.237126 0.971479i \(-0.576206\pi\)
−0.237126 + 0.971479i \(0.576206\pi\)
\(620\) −7.98445 −0.320663
\(621\) 4.31595 0.173193
\(622\) −35.8132 −1.43598
\(623\) −10.8262 −0.433743
\(624\) 0 0
\(625\) −9.03049 −0.361220
\(626\) 7.62807 0.304879
\(627\) 3.67905 0.146927
\(628\) 8.24201 0.328892
\(629\) −23.1249 −0.922048
\(630\) −1.80485 −0.0719070
\(631\) 5.14719 0.204906 0.102453 0.994738i \(-0.467331\pi\)
0.102453 + 0.994738i \(0.467331\pi\)
\(632\) 6.99702 0.278327
\(633\) 19.8268 0.788045
\(634\) 14.0655 0.558611
\(635\) −32.1887 −1.27737
\(636\) 5.33095 0.211386
\(637\) 0 0
\(638\) −23.1432 −0.916246
\(639\) −6.08759 −0.240821
\(640\) −0.0705168 −0.00278742
\(641\) −13.5685 −0.535924 −0.267962 0.963429i \(-0.586350\pi\)
−0.267962 + 0.963429i \(0.586350\pi\)
\(642\) 14.7182 0.580879
\(643\) 11.1073 0.438030 0.219015 0.975721i \(-0.429716\pi\)
0.219015 + 0.975721i \(0.429716\pi\)
\(644\) 3.58597 0.141307
\(645\) −19.0863 −0.751523
\(646\) −3.46038 −0.136147
\(647\) −9.09478 −0.357553 −0.178776 0.983890i \(-0.557214\pi\)
−0.178776 + 0.983890i \(0.557214\pi\)
\(648\) 3.06092 0.120244
\(649\) −56.6935 −2.22542
\(650\) 0 0
\(651\) 5.75711 0.225639
\(652\) −17.4129 −0.681941
\(653\) 3.37219 0.131964 0.0659819 0.997821i \(-0.478982\pi\)
0.0659819 + 0.997821i \(0.478982\pi\)
\(654\) −1.17094 −0.0457873
\(655\) −1.04587 −0.0408654
\(656\) 12.0894 0.472013
\(657\) 3.11657 0.121589
\(658\) 5.52419 0.215355
\(659\) −20.0281 −0.780185 −0.390092 0.920776i \(-0.627557\pi\)
−0.390092 + 0.920776i \(0.627557\pi\)
\(660\) 6.42884 0.250242
\(661\) −25.9859 −1.01073 −0.505367 0.862905i \(-0.668643\pi\)
−0.505367 + 0.862905i \(0.668643\pi\)
\(662\) 19.3944 0.753785
\(663\) 0 0
\(664\) −2.00645 −0.0778655
\(665\) 1.32481 0.0513739
\(666\) 6.20102 0.240285
\(667\) 19.9285 0.771634
\(668\) −6.95106 −0.268945
\(669\) −27.3639 −1.05795
\(670\) 3.37456 0.130370
\(671\) 47.9363 1.85056
\(672\) 4.33998 0.167418
\(673\) −26.8919 −1.03661 −0.518303 0.855197i \(-0.673436\pi\)
−0.518303 + 0.855197i \(0.673436\pi\)
\(674\) −30.6417 −1.18028
\(675\) −2.21376 −0.0852076
\(676\) 0 0
\(677\) −23.0492 −0.885853 −0.442927 0.896558i \(-0.646060\pi\)
−0.442927 + 0.896558i \(0.646060\pi\)
\(678\) −15.9285 −0.611729
\(679\) −14.7169 −0.564782
\(680\) −20.6020 −0.790050
\(681\) 15.1297 0.579771
\(682\) 28.8555 1.10494
\(683\) 35.3125 1.35119 0.675597 0.737271i \(-0.263886\pi\)
0.675597 + 0.737271i \(0.263886\pi\)
\(684\) −0.659440 −0.0252143
\(685\) 2.13370 0.0815246
\(686\) 1.08127 0.0412829
\(687\) 10.6735 0.407218
\(688\) −18.8431 −0.718385
\(689\) 0 0
\(690\) 7.78965 0.296547
\(691\) −29.9975 −1.14116 −0.570580 0.821242i \(-0.693282\pi\)
−0.570580 + 0.821242i \(0.693282\pi\)
\(692\) −4.93120 −0.187456
\(693\) −4.63545 −0.176086
\(694\) 9.18560 0.348681
\(695\) −28.9942 −1.09981
\(696\) 14.1335 0.535729
\(697\) −29.5811 −1.12046
\(698\) −11.9457 −0.452152
\(699\) −10.6341 −0.402217
\(700\) −1.83934 −0.0695204
\(701\) −32.8331 −1.24009 −0.620045 0.784566i \(-0.712886\pi\)
−0.620045 + 0.784566i \(0.712886\pi\)
\(702\) 0 0
\(703\) −4.55172 −0.171671
\(704\) 37.0305 1.39564
\(705\) −8.52798 −0.321182
\(706\) 11.9985 0.451571
\(707\) 7.83185 0.294547
\(708\) 10.1618 0.381905
\(709\) 0.735973 0.0276400 0.0138200 0.999904i \(-0.495601\pi\)
0.0138200 + 0.999904i \(0.495601\pi\)
\(710\) −10.9872 −0.412343
\(711\) 2.28592 0.0857288
\(712\) 33.1382 1.24190
\(713\) −24.8474 −0.930542
\(714\) 4.35994 0.163166
\(715\) 0 0
\(716\) −12.1248 −0.453125
\(717\) 14.5524 0.543469
\(718\) −17.6359 −0.658165
\(719\) −12.0005 −0.447544 −0.223772 0.974642i \(-0.571837\pi\)
−0.223772 + 0.974642i \(0.571837\pi\)
\(720\) 2.75073 0.102514
\(721\) 14.2990 0.532522
\(722\) 19.8629 0.739221
\(723\) 21.0486 0.782805
\(724\) 1.94105 0.0721384
\(725\) −10.2218 −0.379629
\(726\) −11.3397 −0.420855
\(727\) −37.0876 −1.37550 −0.687751 0.725947i \(-0.741402\pi\)
−0.687751 + 0.725947i \(0.741402\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 5.62496 0.208189
\(731\) 46.1063 1.70530
\(732\) −8.59218 −0.317576
\(733\) 29.3473 1.08397 0.541983 0.840390i \(-0.317674\pi\)
0.541983 + 0.840390i \(0.317674\pi\)
\(734\) −16.6345 −0.613992
\(735\) −1.66920 −0.0615695
\(736\) −18.7311 −0.690439
\(737\) 8.66697 0.319252
\(738\) 7.93230 0.291992
\(739\) −14.5291 −0.534463 −0.267231 0.963632i \(-0.586109\pi\)
−0.267231 + 0.963632i \(0.586109\pi\)
\(740\) −7.95374 −0.292385
\(741\) 0 0
\(742\) −6.93755 −0.254686
\(743\) 17.9329 0.657896 0.328948 0.944348i \(-0.393306\pi\)
0.328948 + 0.944348i \(0.393306\pi\)
\(744\) −17.6220 −0.646055
\(745\) −3.44274 −0.126132
\(746\) −22.8523 −0.836684
\(747\) −0.655507 −0.0239838
\(748\) −15.5300 −0.567832
\(749\) 13.6120 0.497371
\(750\) −13.0198 −0.475415
\(751\) −21.4043 −0.781055 −0.390527 0.920591i \(-0.627707\pi\)
−0.390527 + 0.920591i \(0.627707\pi\)
\(752\) −8.41929 −0.307020
\(753\) −16.3957 −0.597491
\(754\) 0 0
\(755\) 17.8491 0.649596
\(756\) 0.830866 0.0302183
\(757\) 43.6768 1.58746 0.793730 0.608270i \(-0.208136\pi\)
0.793730 + 0.608270i \(0.208136\pi\)
\(758\) −29.4472 −1.06957
\(759\) 20.0064 0.726185
\(760\) −4.05513 −0.147095
\(761\) −31.4196 −1.13896 −0.569479 0.822006i \(-0.692855\pi\)
−0.569479 + 0.822006i \(0.692855\pi\)
\(762\) −20.8509 −0.755350
\(763\) −1.08293 −0.0392048
\(764\) −2.42994 −0.0879122
\(765\) −6.73066 −0.243347
\(766\) 16.8283 0.608030
\(767\) 0 0
\(768\) −16.0227 −0.578171
\(769\) 5.48923 0.197947 0.0989733 0.995090i \(-0.468444\pi\)
0.0989733 + 0.995090i \(0.468444\pi\)
\(770\) −8.36630 −0.301501
\(771\) −14.2261 −0.512341
\(772\) −19.9765 −0.718971
\(773\) −31.3794 −1.12864 −0.564319 0.825557i \(-0.690861\pi\)
−0.564319 + 0.825557i \(0.690861\pi\)
\(774\) −12.3636 −0.444400
\(775\) 12.7449 0.457809
\(776\) 45.0471 1.61710
\(777\) 5.73497 0.205741
\(778\) 27.1515 0.973429
\(779\) −5.82252 −0.208613
\(780\) 0 0
\(781\) −28.2187 −1.00975
\(782\) −18.8173 −0.672904
\(783\) 4.61741 0.165013
\(784\) −1.64793 −0.0588546
\(785\) 16.5581 0.590985
\(786\) −0.677484 −0.0241651
\(787\) 7.84245 0.279553 0.139777 0.990183i \(-0.455362\pi\)
0.139777 + 0.990183i \(0.455362\pi\)
\(788\) −12.1352 −0.432299
\(789\) 17.0566 0.607230
\(790\) 4.12575 0.146788
\(791\) −14.7313 −0.523785
\(792\) 14.1887 0.504175
\(793\) 0 0
\(794\) 3.84477 0.136446
\(795\) 10.7099 0.379839
\(796\) 12.8654 0.456001
\(797\) 5.01412 0.177609 0.0888047 0.996049i \(-0.471695\pi\)
0.0888047 + 0.996049i \(0.471695\pi\)
\(798\) 0.858176 0.0303791
\(799\) 20.6008 0.728805
\(800\) 9.60767 0.339682
\(801\) 10.8262 0.382526
\(802\) −33.8650 −1.19581
\(803\) 14.4467 0.509814
\(804\) −1.55348 −0.0547870
\(805\) 7.20420 0.253915
\(806\) 0 0
\(807\) 17.2732 0.608046
\(808\) −23.9726 −0.843354
\(809\) −30.4281 −1.06980 −0.534898 0.844917i \(-0.679650\pi\)
−0.534898 + 0.844917i \(0.679650\pi\)
\(810\) 1.80485 0.0634160
\(811\) −49.4527 −1.73652 −0.868260 0.496109i \(-0.834762\pi\)
−0.868260 + 0.496109i \(0.834762\pi\)
\(812\) 3.83645 0.134633
\(813\) −7.90154 −0.277119
\(814\) 28.7445 1.00750
\(815\) −34.9823 −1.22538
\(816\) −6.64487 −0.232617
\(817\) 9.07521 0.317501
\(818\) −11.1084 −0.388395
\(819\) 0 0
\(820\) −10.1744 −0.355304
\(821\) −15.7133 −0.548398 −0.274199 0.961673i \(-0.588413\pi\)
−0.274199 + 0.961673i \(0.588413\pi\)
\(822\) 1.38215 0.0482082
\(823\) 9.09400 0.316997 0.158499 0.987359i \(-0.449335\pi\)
0.158499 + 0.987359i \(0.449335\pi\)
\(824\) −43.7680 −1.52473
\(825\) −10.2618 −0.357269
\(826\) −13.2243 −0.460133
\(827\) −8.35927 −0.290680 −0.145340 0.989382i \(-0.546428\pi\)
−0.145340 + 0.989382i \(0.546428\pi\)
\(828\) −3.58597 −0.124621
\(829\) −6.13514 −0.213082 −0.106541 0.994308i \(-0.533978\pi\)
−0.106541 + 0.994308i \(0.533978\pi\)
\(830\) −1.18309 −0.0410658
\(831\) 4.50995 0.156449
\(832\) 0 0
\(833\) 4.03225 0.139709
\(834\) −18.7817 −0.650356
\(835\) −13.9646 −0.483266
\(836\) −3.05680 −0.105722
\(837\) −5.75711 −0.198995
\(838\) −5.72976 −0.197931
\(839\) 38.5294 1.33018 0.665091 0.746762i \(-0.268393\pi\)
0.665091 + 0.746762i \(0.268393\pi\)
\(840\) 5.10929 0.176287
\(841\) −7.67953 −0.264811
\(842\) −18.6954 −0.644286
\(843\) −20.7726 −0.715446
\(844\) −16.4734 −0.567038
\(845\) 0 0
\(846\) −5.52419 −0.189926
\(847\) −10.4874 −0.360352
\(848\) 10.5734 0.363091
\(849\) −2.02320 −0.0694362
\(850\) 9.65184 0.331055
\(851\) −24.7518 −0.848482
\(852\) 5.05797 0.173283
\(853\) 27.8326 0.952969 0.476485 0.879183i \(-0.341911\pi\)
0.476485 + 0.879183i \(0.341911\pi\)
\(854\) 11.1816 0.382627
\(855\) −1.32481 −0.0453075
\(856\) −41.6651 −1.42408
\(857\) −24.6902 −0.843400 −0.421700 0.906735i \(-0.638566\pi\)
−0.421700 + 0.906735i \(0.638566\pi\)
\(858\) 0 0
\(859\) 31.2546 1.06639 0.533197 0.845991i \(-0.320990\pi\)
0.533197 + 0.845991i \(0.320990\pi\)
\(860\) 15.8582 0.540759
\(861\) 7.33613 0.250014
\(862\) −20.7867 −0.707998
\(863\) 4.20143 0.143018 0.0715091 0.997440i \(-0.477218\pi\)
0.0715091 + 0.997440i \(0.477218\pi\)
\(864\) −4.33998 −0.147649
\(865\) −9.90674 −0.336839
\(866\) 9.98625 0.339346
\(867\) −0.740925 −0.0251631
\(868\) −4.78339 −0.162359
\(869\) 10.5963 0.359454
\(870\) 8.33374 0.282540
\(871\) 0 0
\(872\) 3.31477 0.112252
\(873\) 14.7169 0.498091
\(874\) −3.70384 −0.125284
\(875\) −12.0412 −0.407068
\(876\) −2.58946 −0.0874896
\(877\) −46.1765 −1.55927 −0.779635 0.626234i \(-0.784596\pi\)
−0.779635 + 0.626234i \(0.784596\pi\)
\(878\) −39.8604 −1.34522
\(879\) −25.9848 −0.876445
\(880\) 12.7509 0.429832
\(881\) 23.6571 0.797028 0.398514 0.917162i \(-0.369526\pi\)
0.398514 + 0.917162i \(0.369526\pi\)
\(882\) −1.08127 −0.0364081
\(883\) 28.4713 0.958134 0.479067 0.877778i \(-0.340975\pi\)
0.479067 + 0.877778i \(0.340975\pi\)
\(884\) 0 0
\(885\) 20.4151 0.686245
\(886\) 21.7760 0.731579
\(887\) −9.79667 −0.328940 −0.164470 0.986382i \(-0.552591\pi\)
−0.164470 + 0.986382i \(0.552591\pi\)
\(888\) −17.5543 −0.589082
\(889\) −19.2838 −0.646759
\(890\) 19.5397 0.654973
\(891\) 4.63545 0.155293
\(892\) 22.7357 0.761249
\(893\) 4.05491 0.135692
\(894\) −2.23011 −0.0745861
\(895\) −24.3587 −0.814220
\(896\) −0.0422458 −0.00141133
\(897\) 0 0
\(898\) 35.9194 1.19864
\(899\) −26.5830 −0.886591
\(900\) 1.83934 0.0613112
\(901\) −25.8715 −0.861905
\(902\) 36.7698 1.22430
\(903\) −11.4344 −0.380512
\(904\) 45.0913 1.49972
\(905\) 3.89955 0.129625
\(906\) 11.5622 0.384127
\(907\) −32.9274 −1.09334 −0.546669 0.837349i \(-0.684104\pi\)
−0.546669 + 0.837349i \(0.684104\pi\)
\(908\) −12.5707 −0.417175
\(909\) −7.83185 −0.259766
\(910\) 0 0
\(911\) −19.3453 −0.640940 −0.320470 0.947259i \(-0.603841\pi\)
−0.320470 + 0.947259i \(0.603841\pi\)
\(912\) −1.30793 −0.0433097
\(913\) −3.03857 −0.100562
\(914\) −27.1981 −0.899635
\(915\) −17.2616 −0.570652
\(916\) −8.86822 −0.293014
\(917\) −0.626566 −0.0206910
\(918\) −4.35994 −0.143899
\(919\) −26.9673 −0.889568 −0.444784 0.895638i \(-0.646720\pi\)
−0.444784 + 0.895638i \(0.646720\pi\)
\(920\) −22.0514 −0.727015
\(921\) −24.6185 −0.811208
\(922\) −25.0690 −0.825605
\(923\) 0 0
\(924\) 3.85144 0.126703
\(925\) 12.6958 0.417436
\(926\) 13.9136 0.457229
\(927\) −14.2990 −0.469640
\(928\) −20.0395 −0.657828
\(929\) −37.9175 −1.24403 −0.622017 0.783004i \(-0.713687\pi\)
−0.622017 + 0.783004i \(0.713687\pi\)
\(930\) −10.3907 −0.340726
\(931\) 0.793678 0.0260117
\(932\) 8.83547 0.289415
\(933\) 33.1216 1.08435
\(934\) 27.6657 0.905251
\(935\) −31.1996 −1.02034
\(936\) 0 0
\(937\) 33.3091 1.08816 0.544080 0.839033i \(-0.316879\pi\)
0.544080 + 0.839033i \(0.316879\pi\)
\(938\) 2.02166 0.0660094
\(939\) −7.05477 −0.230224
\(940\) 7.08561 0.231107
\(941\) −9.43451 −0.307556 −0.153778 0.988105i \(-0.549144\pi\)
−0.153778 + 0.988105i \(0.549144\pi\)
\(942\) 10.7259 0.349469
\(943\) −31.6623 −1.03107
\(944\) 20.1549 0.655985
\(945\) 1.66920 0.0542992
\(946\) −57.3108 −1.86334
\(947\) 8.18143 0.265861 0.132930 0.991125i \(-0.457561\pi\)
0.132930 + 0.991125i \(0.457561\pi\)
\(948\) −1.89930 −0.0616863
\(949\) 0 0
\(950\) 1.89979 0.0616374
\(951\) −13.0084 −0.421825
\(952\) −12.3424 −0.400019
\(953\) −35.7424 −1.15781 −0.578905 0.815395i \(-0.696520\pi\)
−0.578905 + 0.815395i \(0.696520\pi\)
\(954\) 6.93755 0.224612
\(955\) −4.88174 −0.157969
\(956\) −12.0911 −0.391053
\(957\) 21.4038 0.691886
\(958\) −27.3291 −0.882964
\(959\) 1.27827 0.0412777
\(960\) −13.3345 −0.430369
\(961\) 2.14436 0.0691728
\(962\) 0 0
\(963\) −13.6120 −0.438640
\(964\) −17.4886 −0.563268
\(965\) −40.1327 −1.29192
\(966\) 4.66668 0.150148
\(967\) 33.9787 1.09268 0.546341 0.837563i \(-0.316020\pi\)
0.546341 + 0.837563i \(0.316020\pi\)
\(968\) 32.1011 1.03177
\(969\) 3.20031 0.102809
\(970\) 26.5618 0.852848
\(971\) 37.6727 1.20897 0.604487 0.796615i \(-0.293378\pi\)
0.604487 + 0.796615i \(0.293378\pi\)
\(972\) −0.830866 −0.0266500
\(973\) −17.3701 −0.556860
\(974\) 45.9496 1.47232
\(975\) 0 0
\(976\) −17.0416 −0.545489
\(977\) 25.7786 0.824730 0.412365 0.911019i \(-0.364703\pi\)
0.412365 + 0.911019i \(0.364703\pi\)
\(978\) −22.6606 −0.724606
\(979\) 50.1844 1.60390
\(980\) 1.38688 0.0443024
\(981\) 1.08293 0.0345754
\(982\) 43.6335 1.39240
\(983\) −31.8264 −1.01510 −0.507552 0.861621i \(-0.669450\pi\)
−0.507552 + 0.861621i \(0.669450\pi\)
\(984\) −22.4553 −0.715848
\(985\) −24.3795 −0.776797
\(986\) −20.1316 −0.641121
\(987\) −5.10901 −0.162622
\(988\) 0 0
\(989\) 49.3502 1.56924
\(990\) 8.36630 0.265899
\(991\) 22.3388 0.709616 0.354808 0.934939i \(-0.384546\pi\)
0.354808 + 0.934939i \(0.384546\pi\)
\(992\) 24.9858 0.793299
\(993\) −17.9368 −0.569206
\(994\) −6.58230 −0.208778
\(995\) 25.8465 0.819388
\(996\) 0.544638 0.0172575
\(997\) 22.7194 0.719532 0.359766 0.933043i \(-0.382857\pi\)
0.359766 + 0.933043i \(0.382857\pi\)
\(998\) −6.01938 −0.190540
\(999\) −5.73497 −0.181446
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.1.6 yes 15
13.12 even 2 3549.2.a.bg.1.10 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.bg.1.10 15 13.12 even 2
3549.2.a.bh.1.6 yes 15 1.1 even 1 trivial