Properties

Label 3549.2.a.bg.1.8
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 1.8
Root \(0.182130\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.182130 q^{2} +1.00000 q^{3} -1.96683 q^{4} -1.12886 q^{5} -0.182130 q^{6} +1.00000 q^{7} +0.722479 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.182130 q^{2} +1.00000 q^{3} -1.96683 q^{4} -1.12886 q^{5} -0.182130 q^{6} +1.00000 q^{7} +0.722479 q^{8} +1.00000 q^{9} +0.205599 q^{10} -4.01616 q^{11} -1.96683 q^{12} -0.182130 q^{14} -1.12886 q^{15} +3.80207 q^{16} +3.02631 q^{17} -0.182130 q^{18} -5.30303 q^{19} +2.22027 q^{20} +1.00000 q^{21} +0.731463 q^{22} -2.17379 q^{23} +0.722479 q^{24} -3.72568 q^{25} +1.00000 q^{27} -1.96683 q^{28} +3.53344 q^{29} +0.205599 q^{30} +4.85911 q^{31} -2.13743 q^{32} -4.01616 q^{33} -0.551182 q^{34} -1.12886 q^{35} -1.96683 q^{36} +6.38019 q^{37} +0.965841 q^{38} -0.815576 q^{40} -4.76127 q^{41} -0.182130 q^{42} -2.34151 q^{43} +7.89910 q^{44} -1.12886 q^{45} +0.395913 q^{46} +9.81123 q^{47} +3.80207 q^{48} +1.00000 q^{49} +0.678558 q^{50} +3.02631 q^{51} +2.98129 q^{53} -0.182130 q^{54} +4.53367 q^{55} +0.722479 q^{56} -5.30303 q^{57} -0.643546 q^{58} -11.4726 q^{59} +2.22027 q^{60} +4.70310 q^{61} -0.884989 q^{62} +1.00000 q^{63} -7.21485 q^{64} +0.731463 q^{66} +1.77088 q^{67} -5.95223 q^{68} -2.17379 q^{69} +0.205599 q^{70} -3.24776 q^{71} +0.722479 q^{72} +7.80784 q^{73} -1.16202 q^{74} -3.72568 q^{75} +10.4302 q^{76} -4.01616 q^{77} +17.4983 q^{79} -4.29200 q^{80} +1.00000 q^{81} +0.867171 q^{82} +0.705610 q^{83} -1.96683 q^{84} -3.41627 q^{85} +0.426459 q^{86} +3.53344 q^{87} -2.90159 q^{88} -13.0567 q^{89} +0.205599 q^{90} +4.27548 q^{92} +4.85911 q^{93} -1.78692 q^{94} +5.98637 q^{95} -2.13743 q^{96} -1.76727 q^{97} -0.182130 q^{98} -4.01616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 2q^{2} + 15q^{3} + 28q^{4} + 9q^{5} - 2q^{6} + 15q^{7} - 9q^{8} + 15q^{9} + O(q^{10}) \) \( 15q - 2q^{2} + 15q^{3} + 28q^{4} + 9q^{5} - 2q^{6} + 15q^{7} - 9q^{8} + 15q^{9} + 21q^{10} - 5q^{11} + 28q^{12} - 2q^{14} + 9q^{15} + 50q^{16} - q^{17} - 2q^{18} - 3q^{19} + 23q^{20} + 15q^{21} + 21q^{22} + 4q^{23} - 9q^{24} + 50q^{25} + 15q^{27} + 28q^{28} + 9q^{29} + 21q^{30} - 7q^{31} - 35q^{32} - 5q^{33} + 2q^{34} + 9q^{35} + 28q^{36} - 17q^{37} - 12q^{38} + 46q^{40} + 22q^{41} - 2q^{42} + 36q^{43} - 29q^{44} + 9q^{45} + q^{46} + 12q^{47} + 50q^{48} + 15q^{49} - 53q^{50} - q^{51} - 5q^{53} - 2q^{54} + 43q^{55} - 9q^{56} - 3q^{57} - 29q^{58} + 29q^{59} + 23q^{60} + 12q^{61} + 14q^{62} + 15q^{63} + 95q^{64} + 21q^{66} + 12q^{67} - 16q^{68} + 4q^{69} + 21q^{70} - 36q^{71} - 9q^{72} + 29q^{73} - 5q^{74} + 50q^{75} - 25q^{76} - 5q^{77} + 35q^{79} + 89q^{80} + 15q^{81} + 51q^{82} + 10q^{83} + 28q^{84} - 23q^{85} - 19q^{86} + 9q^{87} + 73q^{88} - 25q^{89} + 21q^{90} - 31q^{92} - 7q^{93} - 19q^{94} - 7q^{95} - 35q^{96} + 26q^{97} - 2q^{98} - 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.182130 −0.128785 −0.0643927 0.997925i \(-0.520511\pi\)
−0.0643927 + 0.997925i \(0.520511\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.96683 −0.983414
\(5\) −1.12886 −0.504841 −0.252420 0.967618i \(-0.581227\pi\)
−0.252420 + 0.967618i \(0.581227\pi\)
\(6\) −0.182130 −0.0743543
\(7\) 1.00000 0.377964
\(8\) 0.722479 0.255435
\(9\) 1.00000 0.333333
\(10\) 0.205599 0.0650161
\(11\) −4.01616 −1.21092 −0.605459 0.795877i \(-0.707010\pi\)
−0.605459 + 0.795877i \(0.707010\pi\)
\(12\) −1.96683 −0.567775
\(13\) 0 0
\(14\) −0.182130 −0.0486763
\(15\) −1.12886 −0.291470
\(16\) 3.80207 0.950518
\(17\) 3.02631 0.733988 0.366994 0.930223i \(-0.380387\pi\)
0.366994 + 0.930223i \(0.380387\pi\)
\(18\) −0.182130 −0.0429285
\(19\) −5.30303 −1.21660 −0.608300 0.793708i \(-0.708148\pi\)
−0.608300 + 0.793708i \(0.708148\pi\)
\(20\) 2.22027 0.496468
\(21\) 1.00000 0.218218
\(22\) 0.731463 0.155948
\(23\) −2.17379 −0.453267 −0.226634 0.973980i \(-0.572772\pi\)
−0.226634 + 0.973980i \(0.572772\pi\)
\(24\) 0.722479 0.147475
\(25\) −3.72568 −0.745136
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.96683 −0.371696
\(29\) 3.53344 0.656143 0.328072 0.944653i \(-0.393601\pi\)
0.328072 + 0.944653i \(0.393601\pi\)
\(30\) 0.205599 0.0375371
\(31\) 4.85911 0.872721 0.436361 0.899772i \(-0.356267\pi\)
0.436361 + 0.899772i \(0.356267\pi\)
\(32\) −2.13743 −0.377848
\(33\) −4.01616 −0.699123
\(34\) −0.551182 −0.0945269
\(35\) −1.12886 −0.190812
\(36\) −1.96683 −0.327805
\(37\) 6.38019 1.04890 0.524448 0.851442i \(-0.324272\pi\)
0.524448 + 0.851442i \(0.324272\pi\)
\(38\) 0.965841 0.156680
\(39\) 0 0
\(40\) −0.815576 −0.128954
\(41\) −4.76127 −0.743586 −0.371793 0.928316i \(-0.621257\pi\)
−0.371793 + 0.928316i \(0.621257\pi\)
\(42\) −0.182130 −0.0281033
\(43\) −2.34151 −0.357077 −0.178538 0.983933i \(-0.557137\pi\)
−0.178538 + 0.983933i \(0.557137\pi\)
\(44\) 7.89910 1.19083
\(45\) −1.12886 −0.168280
\(46\) 0.395913 0.0583742
\(47\) 9.81123 1.43111 0.715557 0.698554i \(-0.246173\pi\)
0.715557 + 0.698554i \(0.246173\pi\)
\(48\) 3.80207 0.548782
\(49\) 1.00000 0.142857
\(50\) 0.678558 0.0959626
\(51\) 3.02631 0.423768
\(52\) 0 0
\(53\) 2.98129 0.409512 0.204756 0.978813i \(-0.434360\pi\)
0.204756 + 0.978813i \(0.434360\pi\)
\(54\) −0.182130 −0.0247848
\(55\) 4.53367 0.611321
\(56\) 0.722479 0.0965453
\(57\) −5.30303 −0.702404
\(58\) −0.643546 −0.0845017
\(59\) −11.4726 −1.49361 −0.746803 0.665046i \(-0.768412\pi\)
−0.746803 + 0.665046i \(0.768412\pi\)
\(60\) 2.22027 0.286636
\(61\) 4.70310 0.602170 0.301085 0.953597i \(-0.402651\pi\)
0.301085 + 0.953597i \(0.402651\pi\)
\(62\) −0.884989 −0.112394
\(63\) 1.00000 0.125988
\(64\) −7.21485 −0.901857
\(65\) 0 0
\(66\) 0.731463 0.0900369
\(67\) 1.77088 0.216347 0.108174 0.994132i \(-0.465500\pi\)
0.108174 + 0.994132i \(0.465500\pi\)
\(68\) −5.95223 −0.721814
\(69\) −2.17379 −0.261694
\(70\) 0.205599 0.0245738
\(71\) −3.24776 −0.385438 −0.192719 0.981254i \(-0.561731\pi\)
−0.192719 + 0.981254i \(0.561731\pi\)
\(72\) 0.722479 0.0851449
\(73\) 7.80784 0.913838 0.456919 0.889508i \(-0.348953\pi\)
0.456919 + 0.889508i \(0.348953\pi\)
\(74\) −1.16202 −0.135083
\(75\) −3.72568 −0.430204
\(76\) 10.4302 1.19642
\(77\) −4.01616 −0.457684
\(78\) 0 0
\(79\) 17.4983 1.96871 0.984354 0.176200i \(-0.0563805\pi\)
0.984354 + 0.176200i \(0.0563805\pi\)
\(80\) −4.29200 −0.479860
\(81\) 1.00000 0.111111
\(82\) 0.867171 0.0957630
\(83\) 0.705610 0.0774508 0.0387254 0.999250i \(-0.487670\pi\)
0.0387254 + 0.999250i \(0.487670\pi\)
\(84\) −1.96683 −0.214599
\(85\) −3.41627 −0.370547
\(86\) 0.426459 0.0459863
\(87\) 3.53344 0.378825
\(88\) −2.90159 −0.309310
\(89\) −13.0567 −1.38401 −0.692004 0.721894i \(-0.743272\pi\)
−0.692004 + 0.721894i \(0.743272\pi\)
\(90\) 0.205599 0.0216720
\(91\) 0 0
\(92\) 4.27548 0.445750
\(93\) 4.85911 0.503866
\(94\) −1.78692 −0.184307
\(95\) 5.98637 0.614189
\(96\) −2.13743 −0.218150
\(97\) −1.76727 −0.179439 −0.0897195 0.995967i \(-0.528597\pi\)
−0.0897195 + 0.995967i \(0.528597\pi\)
\(98\) −0.182130 −0.0183979
\(99\) −4.01616 −0.403639
\(100\) 7.32777 0.732777
\(101\) 15.3230 1.52469 0.762347 0.647169i \(-0.224047\pi\)
0.762347 + 0.647169i \(0.224047\pi\)
\(102\) −0.551182 −0.0545751
\(103\) 8.19962 0.807933 0.403966 0.914774i \(-0.367631\pi\)
0.403966 + 0.914774i \(0.367631\pi\)
\(104\) 0 0
\(105\) −1.12886 −0.110165
\(106\) −0.542982 −0.0527391
\(107\) 16.8953 1.63333 0.816664 0.577114i \(-0.195821\pi\)
0.816664 + 0.577114i \(0.195821\pi\)
\(108\) −1.96683 −0.189258
\(109\) −5.62135 −0.538427 −0.269214 0.963080i \(-0.586764\pi\)
−0.269214 + 0.963080i \(0.586764\pi\)
\(110\) −0.825718 −0.0787291
\(111\) 6.38019 0.605581
\(112\) 3.80207 0.359262
\(113\) −2.71753 −0.255644 −0.127822 0.991797i \(-0.540799\pi\)
−0.127822 + 0.991797i \(0.540799\pi\)
\(114\) 0.965841 0.0904593
\(115\) 2.45391 0.228828
\(116\) −6.94967 −0.645261
\(117\) 0 0
\(118\) 2.08950 0.192354
\(119\) 3.02631 0.277421
\(120\) −0.815576 −0.0744516
\(121\) 5.12953 0.466321
\(122\) −0.856575 −0.0775507
\(123\) −4.76127 −0.429310
\(124\) −9.55703 −0.858247
\(125\) 9.85006 0.881016
\(126\) −0.182130 −0.0162254
\(127\) 11.6456 1.03338 0.516691 0.856172i \(-0.327164\pi\)
0.516691 + 0.856172i \(0.327164\pi\)
\(128\) 5.58890 0.493994
\(129\) −2.34151 −0.206158
\(130\) 0 0
\(131\) 14.7907 1.29227 0.646133 0.763225i \(-0.276385\pi\)
0.646133 + 0.763225i \(0.276385\pi\)
\(132\) 7.89910 0.687528
\(133\) −5.30303 −0.459831
\(134\) −0.322530 −0.0278623
\(135\) −1.12886 −0.0971567
\(136\) 2.18644 0.187486
\(137\) −15.4499 −1.31998 −0.659989 0.751276i \(-0.729439\pi\)
−0.659989 + 0.751276i \(0.729439\pi\)
\(138\) 0.395913 0.0337024
\(139\) −11.4715 −0.973001 −0.486501 0.873680i \(-0.661727\pi\)
−0.486501 + 0.873680i \(0.661727\pi\)
\(140\) 2.22027 0.187647
\(141\) 9.81123 0.826254
\(142\) 0.591514 0.0496388
\(143\) 0 0
\(144\) 3.80207 0.316839
\(145\) −3.98875 −0.331248
\(146\) −1.42204 −0.117689
\(147\) 1.00000 0.0824786
\(148\) −12.5487 −1.03150
\(149\) 2.99238 0.245146 0.122573 0.992460i \(-0.460886\pi\)
0.122573 + 0.992460i \(0.460886\pi\)
\(150\) 0.678558 0.0554040
\(151\) −0.860674 −0.0700407 −0.0350203 0.999387i \(-0.511150\pi\)
−0.0350203 + 0.999387i \(0.511150\pi\)
\(152\) −3.83133 −0.310762
\(153\) 3.02631 0.244663
\(154\) 0.731463 0.0589430
\(155\) −5.48525 −0.440585
\(156\) 0 0
\(157\) −4.62432 −0.369061 −0.184530 0.982827i \(-0.559076\pi\)
−0.184530 + 0.982827i \(0.559076\pi\)
\(158\) −3.18696 −0.253541
\(159\) 2.98129 0.236432
\(160\) 2.41285 0.190753
\(161\) −2.17379 −0.171319
\(162\) −0.182130 −0.0143095
\(163\) 23.0446 1.80499 0.902496 0.430698i \(-0.141733\pi\)
0.902496 + 0.430698i \(0.141733\pi\)
\(164\) 9.36461 0.731253
\(165\) 4.53367 0.352946
\(166\) −0.128513 −0.00997453
\(167\) −3.47296 −0.268745 −0.134373 0.990931i \(-0.542902\pi\)
−0.134373 + 0.990931i \(0.542902\pi\)
\(168\) 0.722479 0.0557404
\(169\) 0 0
\(170\) 0.622206 0.0477210
\(171\) −5.30303 −0.405533
\(172\) 4.60535 0.351154
\(173\) 19.7442 1.50112 0.750560 0.660802i \(-0.229784\pi\)
0.750560 + 0.660802i \(0.229784\pi\)
\(174\) −0.643546 −0.0487871
\(175\) −3.72568 −0.281635
\(176\) −15.2697 −1.15100
\(177\) −11.4726 −0.862333
\(178\) 2.37802 0.178240
\(179\) 19.0553 1.42426 0.712129 0.702049i \(-0.247731\pi\)
0.712129 + 0.702049i \(0.247731\pi\)
\(180\) 2.22027 0.165489
\(181\) −3.45717 −0.256969 −0.128485 0.991712i \(-0.541011\pi\)
−0.128485 + 0.991712i \(0.541011\pi\)
\(182\) 0 0
\(183\) 4.70310 0.347663
\(184\) −1.57052 −0.115780
\(185\) −7.20233 −0.529526
\(186\) −0.884989 −0.0648906
\(187\) −12.1541 −0.888798
\(188\) −19.2970 −1.40738
\(189\) 1.00000 0.0727393
\(190\) −1.09030 −0.0790986
\(191\) −21.0189 −1.52087 −0.760437 0.649411i \(-0.775015\pi\)
−0.760437 + 0.649411i \(0.775015\pi\)
\(192\) −7.21485 −0.520687
\(193\) 18.9411 1.36341 0.681704 0.731628i \(-0.261239\pi\)
0.681704 + 0.731628i \(0.261239\pi\)
\(194\) 0.321873 0.0231091
\(195\) 0 0
\(196\) −1.96683 −0.140488
\(197\) 8.43497 0.600966 0.300483 0.953787i \(-0.402852\pi\)
0.300483 + 0.953787i \(0.402852\pi\)
\(198\) 0.731463 0.0519828
\(199\) 20.3783 1.44458 0.722291 0.691590i \(-0.243089\pi\)
0.722291 + 0.691590i \(0.243089\pi\)
\(200\) −2.69172 −0.190334
\(201\) 1.77088 0.124908
\(202\) −2.79077 −0.196358
\(203\) 3.53344 0.247999
\(204\) −5.95223 −0.416740
\(205\) 5.37481 0.375393
\(206\) −1.49340 −0.104050
\(207\) −2.17379 −0.151089
\(208\) 0 0
\(209\) 21.2978 1.47320
\(210\) 0.205599 0.0141877
\(211\) 24.7518 1.70398 0.851992 0.523555i \(-0.175394\pi\)
0.851992 + 0.523555i \(0.175394\pi\)
\(212\) −5.86369 −0.402720
\(213\) −3.24776 −0.222533
\(214\) −3.07714 −0.210349
\(215\) 2.64323 0.180267
\(216\) 0.722479 0.0491584
\(217\) 4.85911 0.329858
\(218\) 1.02382 0.0693416
\(219\) 7.80784 0.527605
\(220\) −8.91696 −0.601181
\(221\) 0 0
\(222\) −1.16202 −0.0779899
\(223\) 24.1397 1.61652 0.808258 0.588829i \(-0.200411\pi\)
0.808258 + 0.588829i \(0.200411\pi\)
\(224\) −2.13743 −0.142813
\(225\) −3.72568 −0.248379
\(226\) 0.494944 0.0329232
\(227\) 20.6751 1.37226 0.686128 0.727480i \(-0.259309\pi\)
0.686128 + 0.727480i \(0.259309\pi\)
\(228\) 10.4302 0.690754
\(229\) 7.17578 0.474189 0.237094 0.971487i \(-0.423805\pi\)
0.237094 + 0.971487i \(0.423805\pi\)
\(230\) −0.446930 −0.0294697
\(231\) −4.01616 −0.264244
\(232\) 2.55284 0.167602
\(233\) −15.9753 −1.04658 −0.523288 0.852156i \(-0.675295\pi\)
−0.523288 + 0.852156i \(0.675295\pi\)
\(234\) 0 0
\(235\) −11.0755 −0.722485
\(236\) 22.5646 1.46883
\(237\) 17.4983 1.13663
\(238\) −0.551182 −0.0357278
\(239\) 16.2796 1.05304 0.526519 0.850164i \(-0.323497\pi\)
0.526519 + 0.850164i \(0.323497\pi\)
\(240\) −4.29200 −0.277048
\(241\) −23.2533 −1.49788 −0.748939 0.662639i \(-0.769437\pi\)
−0.748939 + 0.662639i \(0.769437\pi\)
\(242\) −0.934241 −0.0600553
\(243\) 1.00000 0.0641500
\(244\) −9.25018 −0.592182
\(245\) −1.12886 −0.0721201
\(246\) 0.867171 0.0552888
\(247\) 0 0
\(248\) 3.51060 0.222923
\(249\) 0.705610 0.0447162
\(250\) −1.79399 −0.113462
\(251\) 3.48547 0.220001 0.110000 0.993932i \(-0.464915\pi\)
0.110000 + 0.993932i \(0.464915\pi\)
\(252\) −1.96683 −0.123899
\(253\) 8.73030 0.548869
\(254\) −2.12102 −0.133084
\(255\) −3.41627 −0.213935
\(256\) 13.4118 0.838238
\(257\) −12.3642 −0.771260 −0.385630 0.922654i \(-0.626016\pi\)
−0.385630 + 0.922654i \(0.626016\pi\)
\(258\) 0.426459 0.0265502
\(259\) 6.38019 0.396446
\(260\) 0 0
\(261\) 3.53344 0.218714
\(262\) −2.69382 −0.166425
\(263\) −24.7086 −1.52360 −0.761800 0.647813i \(-0.775684\pi\)
−0.761800 + 0.647813i \(0.775684\pi\)
\(264\) −2.90159 −0.178580
\(265\) −3.36545 −0.206738
\(266\) 0.965841 0.0592195
\(267\) −13.0567 −0.799057
\(268\) −3.48301 −0.212759
\(269\) −12.5615 −0.765887 −0.382944 0.923772i \(-0.625090\pi\)
−0.382944 + 0.923772i \(0.625090\pi\)
\(270\) 0.205599 0.0125124
\(271\) −3.82063 −0.232087 −0.116044 0.993244i \(-0.537021\pi\)
−0.116044 + 0.993244i \(0.537021\pi\)
\(272\) 11.5062 0.697669
\(273\) 0 0
\(274\) 2.81390 0.169994
\(275\) 14.9629 0.902298
\(276\) 4.27548 0.257354
\(277\) 20.1591 1.21124 0.605620 0.795754i \(-0.292925\pi\)
0.605620 + 0.795754i \(0.292925\pi\)
\(278\) 2.08931 0.125308
\(279\) 4.85911 0.290907
\(280\) −0.815576 −0.0487400
\(281\) −7.17004 −0.427729 −0.213864 0.976863i \(-0.568605\pi\)
−0.213864 + 0.976863i \(0.568605\pi\)
\(282\) −1.78692 −0.106409
\(283\) −9.74389 −0.579214 −0.289607 0.957146i \(-0.593525\pi\)
−0.289607 + 0.957146i \(0.593525\pi\)
\(284\) 6.38778 0.379045
\(285\) 5.98637 0.354602
\(286\) 0 0
\(287\) −4.76127 −0.281049
\(288\) −2.13743 −0.125949
\(289\) −7.84145 −0.461262
\(290\) 0.726472 0.0426599
\(291\) −1.76727 −0.103599
\(292\) −15.3567 −0.898681
\(293\) −0.348391 −0.0203532 −0.0101766 0.999948i \(-0.503239\pi\)
−0.0101766 + 0.999948i \(0.503239\pi\)
\(294\) −0.182130 −0.0106220
\(295\) 12.9509 0.754033
\(296\) 4.60955 0.267925
\(297\) −4.01616 −0.233041
\(298\) −0.545003 −0.0315712
\(299\) 0 0
\(300\) 7.32777 0.423069
\(301\) −2.34151 −0.134962
\(302\) 0.156755 0.00902021
\(303\) 15.3230 0.880282
\(304\) −20.1625 −1.15640
\(305\) −5.30913 −0.304000
\(306\) −0.551182 −0.0315090
\(307\) −5.97146 −0.340809 −0.170405 0.985374i \(-0.554507\pi\)
−0.170405 + 0.985374i \(0.554507\pi\)
\(308\) 7.89910 0.450093
\(309\) 8.19962 0.466460
\(310\) 0.999028 0.0567410
\(311\) −6.16012 −0.349308 −0.174654 0.984630i \(-0.555881\pi\)
−0.174654 + 0.984630i \(0.555881\pi\)
\(312\) 0 0
\(313\) 22.5210 1.27296 0.636482 0.771291i \(-0.280389\pi\)
0.636482 + 0.771291i \(0.280389\pi\)
\(314\) 0.842228 0.0475296
\(315\) −1.12886 −0.0636040
\(316\) −34.4161 −1.93606
\(317\) 30.4700 1.71136 0.855682 0.517501i \(-0.173138\pi\)
0.855682 + 0.517501i \(0.173138\pi\)
\(318\) −0.542982 −0.0304489
\(319\) −14.1909 −0.794535
\(320\) 8.14455 0.455294
\(321\) 16.8953 0.943002
\(322\) 0.395913 0.0220634
\(323\) −16.0486 −0.892969
\(324\) −1.96683 −0.109268
\(325\) 0 0
\(326\) −4.19711 −0.232457
\(327\) −5.62135 −0.310861
\(328\) −3.43992 −0.189938
\(329\) 9.81123 0.540910
\(330\) −0.825718 −0.0454543
\(331\) 1.62675 0.0894142 0.0447071 0.999000i \(-0.485765\pi\)
0.0447071 + 0.999000i \(0.485765\pi\)
\(332\) −1.38781 −0.0761662
\(333\) 6.38019 0.349632
\(334\) 0.632530 0.0346105
\(335\) −1.99907 −0.109221
\(336\) 3.80207 0.207420
\(337\) 3.76808 0.205261 0.102630 0.994720i \(-0.467274\pi\)
0.102630 + 0.994720i \(0.467274\pi\)
\(338\) 0 0
\(339\) −2.71753 −0.147596
\(340\) 6.71923 0.364401
\(341\) −19.5149 −1.05679
\(342\) 0.965841 0.0522267
\(343\) 1.00000 0.0539949
\(344\) −1.69169 −0.0912098
\(345\) 2.45391 0.132114
\(346\) −3.59600 −0.193322
\(347\) 11.2926 0.606216 0.303108 0.952956i \(-0.401976\pi\)
0.303108 + 0.952956i \(0.401976\pi\)
\(348\) −6.94967 −0.372542
\(349\) −2.74933 −0.147168 −0.0735840 0.997289i \(-0.523444\pi\)
−0.0735840 + 0.997289i \(0.523444\pi\)
\(350\) 0.678558 0.0362704
\(351\) 0 0
\(352\) 8.58425 0.457542
\(353\) −21.6903 −1.15446 −0.577229 0.816582i \(-0.695866\pi\)
−0.577229 + 0.816582i \(0.695866\pi\)
\(354\) 2.08950 0.111056
\(355\) 3.66626 0.194585
\(356\) 25.6803 1.36105
\(357\) 3.02631 0.160169
\(358\) −3.47054 −0.183424
\(359\) 25.2837 1.33442 0.667212 0.744868i \(-0.267488\pi\)
0.667212 + 0.744868i \(0.267488\pi\)
\(360\) −0.815576 −0.0429846
\(361\) 9.12216 0.480114
\(362\) 0.629654 0.0330939
\(363\) 5.12953 0.269230
\(364\) 0 0
\(365\) −8.81394 −0.461343
\(366\) −0.856575 −0.0447739
\(367\) 38.1307 1.99041 0.995203 0.0978339i \(-0.0311914\pi\)
0.995203 + 0.0978339i \(0.0311914\pi\)
\(368\) −8.26492 −0.430839
\(369\) −4.76127 −0.247862
\(370\) 1.31176 0.0681952
\(371\) 2.98129 0.154781
\(372\) −9.55703 −0.495509
\(373\) −26.6379 −1.37926 −0.689630 0.724162i \(-0.742227\pi\)
−0.689630 + 0.724162i \(0.742227\pi\)
\(374\) 2.21363 0.114464
\(375\) 9.85006 0.508655
\(376\) 7.08840 0.365556
\(377\) 0 0
\(378\) −0.182130 −0.00936776
\(379\) −6.72882 −0.345636 −0.172818 0.984954i \(-0.555287\pi\)
−0.172818 + 0.984954i \(0.555287\pi\)
\(380\) −11.7742 −0.604002
\(381\) 11.6456 0.596623
\(382\) 3.82817 0.195866
\(383\) −1.17016 −0.0597924 −0.0298962 0.999553i \(-0.509518\pi\)
−0.0298962 + 0.999553i \(0.509518\pi\)
\(384\) 5.58890 0.285207
\(385\) 4.53367 0.231057
\(386\) −3.44974 −0.175587
\(387\) −2.34151 −0.119026
\(388\) 3.47592 0.176463
\(389\) −21.3343 −1.08169 −0.540847 0.841121i \(-0.681896\pi\)
−0.540847 + 0.841121i \(0.681896\pi\)
\(390\) 0 0
\(391\) −6.57857 −0.332693
\(392\) 0.722479 0.0364907
\(393\) 14.7907 0.746090
\(394\) −1.53626 −0.0773957
\(395\) −19.7531 −0.993885
\(396\) 7.89910 0.396944
\(397\) 26.3909 1.32452 0.662262 0.749273i \(-0.269597\pi\)
0.662262 + 0.749273i \(0.269597\pi\)
\(398\) −3.71151 −0.186041
\(399\) −5.30303 −0.265484
\(400\) −14.1653 −0.708265
\(401\) −11.6178 −0.580165 −0.290082 0.957002i \(-0.593683\pi\)
−0.290082 + 0.957002i \(0.593683\pi\)
\(402\) −0.322530 −0.0160863
\(403\) 0 0
\(404\) −30.1377 −1.49941
\(405\) −1.12886 −0.0560934
\(406\) −0.643546 −0.0319386
\(407\) −25.6239 −1.27013
\(408\) 2.18644 0.108245
\(409\) −10.8356 −0.535785 −0.267893 0.963449i \(-0.586327\pi\)
−0.267893 + 0.963449i \(0.586327\pi\)
\(410\) −0.978913 −0.0483451
\(411\) −15.4499 −0.762089
\(412\) −16.1273 −0.794533
\(413\) −11.4726 −0.564530
\(414\) 0.395913 0.0194581
\(415\) −0.796534 −0.0391003
\(416\) 0 0
\(417\) −11.4715 −0.561763
\(418\) −3.87897 −0.189727
\(419\) −37.6639 −1.84000 −0.920001 0.391917i \(-0.871812\pi\)
−0.920001 + 0.391917i \(0.871812\pi\)
\(420\) 2.22027 0.108338
\(421\) 2.59040 0.126249 0.0631243 0.998006i \(-0.479894\pi\)
0.0631243 + 0.998006i \(0.479894\pi\)
\(422\) −4.50805 −0.219448
\(423\) 9.81123 0.477038
\(424\) 2.15392 0.104603
\(425\) −11.2751 −0.546920
\(426\) 0.591514 0.0286590
\(427\) 4.70310 0.227599
\(428\) −33.2301 −1.60624
\(429\) 0 0
\(430\) −0.481412 −0.0232157
\(431\) −39.9546 −1.92455 −0.962273 0.272086i \(-0.912287\pi\)
−0.962273 + 0.272086i \(0.912287\pi\)
\(432\) 3.80207 0.182927
\(433\) 6.08994 0.292664 0.146332 0.989236i \(-0.453253\pi\)
0.146332 + 0.989236i \(0.453253\pi\)
\(434\) −0.884989 −0.0424808
\(435\) −3.98875 −0.191246
\(436\) 11.0562 0.529497
\(437\) 11.5277 0.551445
\(438\) −1.42204 −0.0679478
\(439\) −20.6796 −0.986984 −0.493492 0.869750i \(-0.664280\pi\)
−0.493492 + 0.869750i \(0.664280\pi\)
\(440\) 3.27548 0.156153
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −9.26524 −0.440205 −0.220102 0.975477i \(-0.570639\pi\)
−0.220102 + 0.975477i \(0.570639\pi\)
\(444\) −12.5487 −0.595537
\(445\) 14.7392 0.698704
\(446\) −4.39657 −0.208184
\(447\) 2.99238 0.141535
\(448\) −7.21485 −0.340870
\(449\) −36.2207 −1.70936 −0.854681 0.519154i \(-0.826247\pi\)
−0.854681 + 0.519154i \(0.826247\pi\)
\(450\) 0.678558 0.0319875
\(451\) 19.1220 0.900421
\(452\) 5.34492 0.251404
\(453\) −0.860674 −0.0404380
\(454\) −3.76556 −0.176727
\(455\) 0 0
\(456\) −3.83133 −0.179418
\(457\) −16.7049 −0.781422 −0.390711 0.920513i \(-0.627771\pi\)
−0.390711 + 0.920513i \(0.627771\pi\)
\(458\) −1.30693 −0.0610686
\(459\) 3.02631 0.141256
\(460\) −4.82641 −0.225033
\(461\) −24.5420 −1.14303 −0.571517 0.820590i \(-0.693645\pi\)
−0.571517 + 0.820590i \(0.693645\pi\)
\(462\) 0.731463 0.0340307
\(463\) −18.5726 −0.863143 −0.431572 0.902079i \(-0.642041\pi\)
−0.431572 + 0.902079i \(0.642041\pi\)
\(464\) 13.4344 0.623676
\(465\) −5.48525 −0.254372
\(466\) 2.90958 0.134784
\(467\) 11.8882 0.550119 0.275059 0.961427i \(-0.411302\pi\)
0.275059 + 0.961427i \(0.411302\pi\)
\(468\) 0 0
\(469\) 1.77088 0.0817715
\(470\) 2.01718 0.0930455
\(471\) −4.62432 −0.213077
\(472\) −8.28871 −0.381519
\(473\) 9.40387 0.432390
\(474\) −3.18696 −0.146382
\(475\) 19.7574 0.906531
\(476\) −5.95223 −0.272820
\(477\) 2.98129 0.136504
\(478\) −2.96500 −0.135616
\(479\) −11.8285 −0.540460 −0.270230 0.962796i \(-0.587100\pi\)
−0.270230 + 0.962796i \(0.587100\pi\)
\(480\) 2.41285 0.110131
\(481\) 0 0
\(482\) 4.23513 0.192905
\(483\) −2.17379 −0.0989110
\(484\) −10.0889 −0.458586
\(485\) 1.99500 0.0905882
\(486\) −0.182130 −0.00826158
\(487\) 30.3027 1.37315 0.686573 0.727061i \(-0.259114\pi\)
0.686573 + 0.727061i \(0.259114\pi\)
\(488\) 3.39789 0.153815
\(489\) 23.0446 1.04211
\(490\) 0.205599 0.00928802
\(491\) −5.28571 −0.238541 −0.119270 0.992862i \(-0.538056\pi\)
−0.119270 + 0.992862i \(0.538056\pi\)
\(492\) 9.36461 0.422189
\(493\) 10.6933 0.481601
\(494\) 0 0
\(495\) 4.53367 0.203774
\(496\) 18.4747 0.829538
\(497\) −3.24776 −0.145682
\(498\) −0.128513 −0.00575880
\(499\) 34.7232 1.55442 0.777211 0.629240i \(-0.216634\pi\)
0.777211 + 0.629240i \(0.216634\pi\)
\(500\) −19.3734 −0.866404
\(501\) −3.47296 −0.155160
\(502\) −0.634809 −0.0283329
\(503\) −25.8848 −1.15414 −0.577072 0.816693i \(-0.695805\pi\)
−0.577072 + 0.816693i \(0.695805\pi\)
\(504\) 0.722479 0.0321818
\(505\) −17.2975 −0.769728
\(506\) −1.59005 −0.0706863
\(507\) 0 0
\(508\) −22.9049 −1.01624
\(509\) 21.4431 0.950449 0.475224 0.879865i \(-0.342367\pi\)
0.475224 + 0.879865i \(0.342367\pi\)
\(510\) 0.622206 0.0275518
\(511\) 7.80784 0.345398
\(512\) −13.6205 −0.601946
\(513\) −5.30303 −0.234135
\(514\) 2.25190 0.0993270
\(515\) −9.25622 −0.407878
\(516\) 4.60535 0.202739
\(517\) −39.4034 −1.73296
\(518\) −1.16202 −0.0510564
\(519\) 19.7442 0.866672
\(520\) 0 0
\(521\) −25.1058 −1.09990 −0.549951 0.835197i \(-0.685354\pi\)
−0.549951 + 0.835197i \(0.685354\pi\)
\(522\) −0.643546 −0.0281672
\(523\) 11.4300 0.499797 0.249898 0.968272i \(-0.419603\pi\)
0.249898 + 0.968272i \(0.419603\pi\)
\(524\) −29.0907 −1.27083
\(525\) −3.72568 −0.162602
\(526\) 4.50018 0.196217
\(527\) 14.7052 0.640567
\(528\) −15.2697 −0.664529
\(529\) −18.2746 −0.794549
\(530\) 0.612950 0.0266249
\(531\) −11.4726 −0.497868
\(532\) 10.4302 0.452205
\(533\) 0 0
\(534\) 2.37802 0.102907
\(535\) −19.0724 −0.824571
\(536\) 1.27942 0.0552626
\(537\) 19.0553 0.822295
\(538\) 2.28782 0.0986351
\(539\) −4.01616 −0.172988
\(540\) 2.22027 0.0955453
\(541\) −1.47534 −0.0634297 −0.0317148 0.999497i \(-0.510097\pi\)
−0.0317148 + 0.999497i \(0.510097\pi\)
\(542\) 0.695852 0.0298894
\(543\) −3.45717 −0.148361
\(544\) −6.46852 −0.277335
\(545\) 6.34571 0.271820
\(546\) 0 0
\(547\) 9.19462 0.393134 0.196567 0.980490i \(-0.437021\pi\)
0.196567 + 0.980490i \(0.437021\pi\)
\(548\) 30.3874 1.29808
\(549\) 4.70310 0.200723
\(550\) −2.72520 −0.116203
\(551\) −18.7380 −0.798264
\(552\) −1.57052 −0.0668457
\(553\) 17.4983 0.744102
\(554\) −3.67157 −0.155990
\(555\) −7.20233 −0.305722
\(556\) 22.5625 0.956863
\(557\) −7.14740 −0.302845 −0.151423 0.988469i \(-0.548385\pi\)
−0.151423 + 0.988469i \(0.548385\pi\)
\(558\) −0.884989 −0.0374646
\(559\) 0 0
\(560\) −4.29200 −0.181370
\(561\) −12.1541 −0.513148
\(562\) 1.30588 0.0550852
\(563\) −39.3042 −1.65648 −0.828238 0.560376i \(-0.810657\pi\)
−0.828238 + 0.560376i \(0.810657\pi\)
\(564\) −19.2970 −0.812550
\(565\) 3.06771 0.129059
\(566\) 1.77465 0.0745943
\(567\) 1.00000 0.0419961
\(568\) −2.34644 −0.0984542
\(569\) 29.9610 1.25603 0.628015 0.778201i \(-0.283868\pi\)
0.628015 + 0.778201i \(0.283868\pi\)
\(570\) −1.09030 −0.0456676
\(571\) 39.1397 1.63795 0.818973 0.573833i \(-0.194544\pi\)
0.818973 + 0.573833i \(0.194544\pi\)
\(572\) 0 0
\(573\) −21.0189 −0.878078
\(574\) 0.867171 0.0361950
\(575\) 8.09886 0.337746
\(576\) −7.21485 −0.300619
\(577\) 23.2869 0.969445 0.484723 0.874668i \(-0.338921\pi\)
0.484723 + 0.874668i \(0.338921\pi\)
\(578\) 1.42816 0.0594038
\(579\) 18.9411 0.787164
\(580\) 7.84520 0.325754
\(581\) 0.705610 0.0292736
\(582\) 0.321873 0.0133421
\(583\) −11.9733 −0.495885
\(584\) 5.64099 0.233426
\(585\) 0 0
\(586\) 0.0634524 0.00262120
\(587\) −14.6189 −0.603387 −0.301694 0.953405i \(-0.597552\pi\)
−0.301694 + 0.953405i \(0.597552\pi\)
\(588\) −1.96683 −0.0811106
\(589\) −25.7680 −1.06175
\(590\) −2.35875 −0.0971084
\(591\) 8.43497 0.346968
\(592\) 24.2579 0.996995
\(593\) 28.8553 1.18495 0.592473 0.805590i \(-0.298152\pi\)
0.592473 + 0.805590i \(0.298152\pi\)
\(594\) 0.731463 0.0300123
\(595\) −3.41627 −0.140054
\(596\) −5.88550 −0.241080
\(597\) 20.3783 0.834029
\(598\) 0 0
\(599\) 18.1056 0.739773 0.369887 0.929077i \(-0.379397\pi\)
0.369887 + 0.929077i \(0.379397\pi\)
\(600\) −2.69172 −0.109889
\(601\) 0.149746 0.00610829 0.00305414 0.999995i \(-0.499028\pi\)
0.00305414 + 0.999995i \(0.499028\pi\)
\(602\) 0.426459 0.0173812
\(603\) 1.77088 0.0721157
\(604\) 1.69280 0.0688790
\(605\) −5.79051 −0.235418
\(606\) −2.79077 −0.113367
\(607\) 14.2380 0.577904 0.288952 0.957344i \(-0.406693\pi\)
0.288952 + 0.957344i \(0.406693\pi\)
\(608\) 11.3349 0.459689
\(609\) 3.53344 0.143182
\(610\) 0.966952 0.0391507
\(611\) 0 0
\(612\) −5.95223 −0.240605
\(613\) −1.87784 −0.0758451 −0.0379226 0.999281i \(-0.512074\pi\)
−0.0379226 + 0.999281i \(0.512074\pi\)
\(614\) 1.08758 0.0438912
\(615\) 5.37481 0.216733
\(616\) −2.90159 −0.116908
\(617\) 38.8688 1.56480 0.782400 0.622776i \(-0.213995\pi\)
0.782400 + 0.622776i \(0.213995\pi\)
\(618\) −1.49340 −0.0600733
\(619\) −41.6563 −1.67431 −0.837153 0.546969i \(-0.815782\pi\)
−0.837153 + 0.546969i \(0.815782\pi\)
\(620\) 10.7885 0.433278
\(621\) −2.17379 −0.0872313
\(622\) 1.12194 0.0449858
\(623\) −13.0567 −0.523106
\(624\) 0 0
\(625\) 7.50907 0.300363
\(626\) −4.10176 −0.163939
\(627\) 21.2978 0.850553
\(628\) 9.09525 0.362940
\(629\) 19.3084 0.769877
\(630\) 0.205599 0.00819126
\(631\) −28.6876 −1.14203 −0.571017 0.820938i \(-0.693451\pi\)
−0.571017 + 0.820938i \(0.693451\pi\)
\(632\) 12.6421 0.502877
\(633\) 24.7518 0.983796
\(634\) −5.54950 −0.220399
\(635\) −13.1463 −0.521693
\(636\) −5.86369 −0.232510
\(637\) 0 0
\(638\) 2.58458 0.102325
\(639\) −3.24776 −0.128479
\(640\) −6.30908 −0.249388
\(641\) −18.5217 −0.731564 −0.365782 0.930701i \(-0.619198\pi\)
−0.365782 + 0.930701i \(0.619198\pi\)
\(642\) −3.07714 −0.121445
\(643\) 31.6909 1.24977 0.624884 0.780718i \(-0.285146\pi\)
0.624884 + 0.780718i \(0.285146\pi\)
\(644\) 4.27548 0.168477
\(645\) 2.64323 0.104077
\(646\) 2.92293 0.115001
\(647\) 22.6938 0.892184 0.446092 0.894987i \(-0.352815\pi\)
0.446092 + 0.894987i \(0.352815\pi\)
\(648\) 0.722479 0.0283816
\(649\) 46.0758 1.80863
\(650\) 0 0
\(651\) 4.85911 0.190443
\(652\) −45.3248 −1.77505
\(653\) −45.9806 −1.79936 −0.899680 0.436550i \(-0.856200\pi\)
−0.899680 + 0.436550i \(0.856200\pi\)
\(654\) 1.02382 0.0400344
\(655\) −16.6966 −0.652389
\(656\) −18.1027 −0.706792
\(657\) 7.80784 0.304613
\(658\) −1.78692 −0.0696613
\(659\) 26.6785 1.03925 0.519623 0.854396i \(-0.326072\pi\)
0.519623 + 0.854396i \(0.326072\pi\)
\(660\) −8.91696 −0.347092
\(661\) −8.73868 −0.339895 −0.169948 0.985453i \(-0.554360\pi\)
−0.169948 + 0.985453i \(0.554360\pi\)
\(662\) −0.296280 −0.0115152
\(663\) 0 0
\(664\) 0.509788 0.0197836
\(665\) 5.98637 0.232142
\(666\) −1.16202 −0.0450275
\(667\) −7.68097 −0.297408
\(668\) 6.83071 0.264288
\(669\) 24.1397 0.933296
\(670\) 0.364091 0.0140660
\(671\) −18.8884 −0.729178
\(672\) −2.13743 −0.0824531
\(673\) 45.5288 1.75501 0.877503 0.479570i \(-0.159207\pi\)
0.877503 + 0.479570i \(0.159207\pi\)
\(674\) −0.686281 −0.0264346
\(675\) −3.72568 −0.143401
\(676\) 0 0
\(677\) 0.826184 0.0317528 0.0158764 0.999874i \(-0.494946\pi\)
0.0158764 + 0.999874i \(0.494946\pi\)
\(678\) 0.494944 0.0190082
\(679\) −1.76727 −0.0678216
\(680\) −2.46819 −0.0946506
\(681\) 20.6751 0.792273
\(682\) 3.55426 0.136100
\(683\) 24.3274 0.930860 0.465430 0.885085i \(-0.345900\pi\)
0.465430 + 0.885085i \(0.345900\pi\)
\(684\) 10.4302 0.398807
\(685\) 17.4408 0.666378
\(686\) −0.182130 −0.00695376
\(687\) 7.17578 0.273773
\(688\) −8.90259 −0.339408
\(689\) 0 0
\(690\) −0.446930 −0.0170143
\(691\) 24.8245 0.944367 0.472184 0.881500i \(-0.343466\pi\)
0.472184 + 0.881500i \(0.343466\pi\)
\(692\) −38.8334 −1.47622
\(693\) −4.01616 −0.152561
\(694\) −2.05671 −0.0780718
\(695\) 12.9497 0.491211
\(696\) 2.55284 0.0967650
\(697\) −14.4091 −0.545783
\(698\) 0.500735 0.0189531
\(699\) −15.9753 −0.604241
\(700\) 7.32777 0.276964
\(701\) −51.0338 −1.92752 −0.963760 0.266770i \(-0.914044\pi\)
−0.963760 + 0.266770i \(0.914044\pi\)
\(702\) 0 0
\(703\) −33.8344 −1.27609
\(704\) 28.9760 1.09207
\(705\) −11.0755 −0.417127
\(706\) 3.95045 0.148677
\(707\) 15.3230 0.576280
\(708\) 22.5646 0.848031
\(709\) 12.6412 0.474752 0.237376 0.971418i \(-0.423713\pi\)
0.237376 + 0.971418i \(0.423713\pi\)
\(710\) −0.667736 −0.0250597
\(711\) 17.4983 0.656236
\(712\) −9.43319 −0.353524
\(713\) −10.5627 −0.395576
\(714\) −0.551182 −0.0206275
\(715\) 0 0
\(716\) −37.4784 −1.40064
\(717\) 16.2796 0.607971
\(718\) −4.60493 −0.171854
\(719\) −7.25090 −0.270413 −0.135206 0.990817i \(-0.543170\pi\)
−0.135206 + 0.990817i \(0.543170\pi\)
\(720\) −4.29200 −0.159953
\(721\) 8.19962 0.305370
\(722\) −1.66142 −0.0618316
\(723\) −23.2533 −0.864801
\(724\) 6.79966 0.252707
\(725\) −13.1645 −0.488916
\(726\) −0.934241 −0.0346729
\(727\) 18.1643 0.673677 0.336838 0.941563i \(-0.390642\pi\)
0.336838 + 0.941563i \(0.390642\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 1.60528 0.0594142
\(731\) −7.08613 −0.262090
\(732\) −9.25018 −0.341897
\(733\) 13.6504 0.504189 0.252094 0.967703i \(-0.418881\pi\)
0.252094 + 0.967703i \(0.418881\pi\)
\(734\) −6.94474 −0.256335
\(735\) −1.12886 −0.0416386
\(736\) 4.64633 0.171266
\(737\) −7.11212 −0.261978
\(738\) 0.867171 0.0319210
\(739\) −24.7742 −0.911333 −0.455667 0.890150i \(-0.650599\pi\)
−0.455667 + 0.890150i \(0.650599\pi\)
\(740\) 14.1658 0.520743
\(741\) 0 0
\(742\) −0.542982 −0.0199335
\(743\) −29.4286 −1.07963 −0.539815 0.841783i \(-0.681506\pi\)
−0.539815 + 0.841783i \(0.681506\pi\)
\(744\) 3.51060 0.128705
\(745\) −3.37798 −0.123760
\(746\) 4.85157 0.177629
\(747\) 0.705610 0.0258169
\(748\) 23.9051 0.874057
\(749\) 16.8953 0.617340
\(750\) −1.79399 −0.0655073
\(751\) −23.6648 −0.863543 −0.431771 0.901983i \(-0.642111\pi\)
−0.431771 + 0.901983i \(0.642111\pi\)
\(752\) 37.3030 1.36030
\(753\) 3.48547 0.127018
\(754\) 0 0
\(755\) 0.971579 0.0353594
\(756\) −1.96683 −0.0715329
\(757\) 18.5355 0.673683 0.336841 0.941561i \(-0.390641\pi\)
0.336841 + 0.941561i \(0.390641\pi\)
\(758\) 1.22552 0.0445129
\(759\) 8.73030 0.316890
\(760\) 4.32503 0.156885
\(761\) 9.30977 0.337479 0.168739 0.985661i \(-0.446030\pi\)
0.168739 + 0.985661i \(0.446030\pi\)
\(762\) −2.12102 −0.0768363
\(763\) −5.62135 −0.203506
\(764\) 41.3406 1.49565
\(765\) −3.41627 −0.123516
\(766\) 0.213121 0.00770039
\(767\) 0 0
\(768\) 13.4118 0.483957
\(769\) −10.5730 −0.381272 −0.190636 0.981661i \(-0.561055\pi\)
−0.190636 + 0.981661i \(0.561055\pi\)
\(770\) −0.825718 −0.0297568
\(771\) −12.3642 −0.445287
\(772\) −37.2539 −1.34080
\(773\) 35.7370 1.28537 0.642686 0.766130i \(-0.277820\pi\)
0.642686 + 0.766130i \(0.277820\pi\)
\(774\) 0.426459 0.0153288
\(775\) −18.1035 −0.650296
\(776\) −1.27681 −0.0458350
\(777\) 6.38019 0.228888
\(778\) 3.88562 0.139306
\(779\) 25.2492 0.904646
\(780\) 0 0
\(781\) 13.0435 0.466733
\(782\) 1.19816 0.0428459
\(783\) 3.53344 0.126275
\(784\) 3.80207 0.135788
\(785\) 5.22020 0.186317
\(786\) −2.69382 −0.0960855
\(787\) −15.8102 −0.563574 −0.281787 0.959477i \(-0.590927\pi\)
−0.281787 + 0.959477i \(0.590927\pi\)
\(788\) −16.5901 −0.590999
\(789\) −24.7086 −0.879650
\(790\) 3.59763 0.127998
\(791\) −2.71753 −0.0966243
\(792\) −2.90159 −0.103103
\(793\) 0 0
\(794\) −4.80658 −0.170579
\(795\) −3.36545 −0.119360
\(796\) −40.0807 −1.42062
\(797\) 42.4880 1.50500 0.752502 0.658590i \(-0.228847\pi\)
0.752502 + 0.658590i \(0.228847\pi\)
\(798\) 0.965841 0.0341904
\(799\) 29.6918 1.05042
\(800\) 7.96337 0.281548
\(801\) −13.0567 −0.461336
\(802\) 2.11595 0.0747167
\(803\) −31.3575 −1.10658
\(804\) −3.48301 −0.122836
\(805\) 2.45391 0.0864888
\(806\) 0 0
\(807\) −12.5615 −0.442185
\(808\) 11.0705 0.389460
\(809\) −18.9303 −0.665553 −0.332776 0.943006i \(-0.607985\pi\)
−0.332776 + 0.943006i \(0.607985\pi\)
\(810\) 0.205599 0.00722401
\(811\) −18.5956 −0.652978 −0.326489 0.945201i \(-0.605866\pi\)
−0.326489 + 0.945201i \(0.605866\pi\)
\(812\) −6.94967 −0.243886
\(813\) −3.82063 −0.133996
\(814\) 4.66687 0.163574
\(815\) −26.0141 −0.911234
\(816\) 11.5062 0.402799
\(817\) 12.4171 0.434419
\(818\) 1.97349 0.0690013
\(819\) 0 0
\(820\) −10.5713 −0.369167
\(821\) 46.8163 1.63390 0.816951 0.576707i \(-0.195663\pi\)
0.816951 + 0.576707i \(0.195663\pi\)
\(822\) 2.81390 0.0981459
\(823\) −36.5999 −1.27579 −0.637896 0.770123i \(-0.720195\pi\)
−0.637896 + 0.770123i \(0.720195\pi\)
\(824\) 5.92405 0.206374
\(825\) 14.9629 0.520942
\(826\) 2.08950 0.0727032
\(827\) −37.5007 −1.30403 −0.652013 0.758208i \(-0.726075\pi\)
−0.652013 + 0.758208i \(0.726075\pi\)
\(828\) 4.27548 0.148583
\(829\) 3.74445 0.130050 0.0650250 0.997884i \(-0.479287\pi\)
0.0650250 + 0.997884i \(0.479287\pi\)
\(830\) 0.145073 0.00503555
\(831\) 20.1591 0.699310
\(832\) 0 0
\(833\) 3.02631 0.104855
\(834\) 2.08931 0.0723468
\(835\) 3.92048 0.135674
\(836\) −41.8892 −1.44877
\(837\) 4.85911 0.167955
\(838\) 6.85972 0.236965
\(839\) 24.0282 0.829547 0.414773 0.909925i \(-0.363861\pi\)
0.414773 + 0.909925i \(0.363861\pi\)
\(840\) −0.815576 −0.0281400
\(841\) −16.5148 −0.569476
\(842\) −0.471790 −0.0162590
\(843\) −7.17004 −0.246949
\(844\) −48.6825 −1.67572
\(845\) 0 0
\(846\) −1.78692 −0.0614355
\(847\) 5.12953 0.176253
\(848\) 11.3351 0.389248
\(849\) −9.74389 −0.334409
\(850\) 2.05353 0.0704353
\(851\) −13.8692 −0.475431
\(852\) 6.38778 0.218842
\(853\) 42.2325 1.44601 0.723006 0.690841i \(-0.242760\pi\)
0.723006 + 0.690841i \(0.242760\pi\)
\(854\) −0.856575 −0.0293114
\(855\) 5.98637 0.204730
\(856\) 12.2065 0.417209
\(857\) 16.7604 0.572524 0.286262 0.958151i \(-0.407587\pi\)
0.286262 + 0.958151i \(0.407587\pi\)
\(858\) 0 0
\(859\) −37.9949 −1.29637 −0.648185 0.761483i \(-0.724472\pi\)
−0.648185 + 0.761483i \(0.724472\pi\)
\(860\) −5.19879 −0.177277
\(861\) −4.76127 −0.162264
\(862\) 7.27693 0.247853
\(863\) −31.3316 −1.06654 −0.533269 0.845945i \(-0.679037\pi\)
−0.533269 + 0.845945i \(0.679037\pi\)
\(864\) −2.13743 −0.0727168
\(865\) −22.2884 −0.757827
\(866\) −1.10916 −0.0376908
\(867\) −7.84145 −0.266310
\(868\) −9.55703 −0.324387
\(869\) −70.2758 −2.38394
\(870\) 0.726472 0.0246297
\(871\) 0 0
\(872\) −4.06130 −0.137533
\(873\) −1.76727 −0.0598130
\(874\) −2.09954 −0.0710180
\(875\) 9.85006 0.332993
\(876\) −15.3567 −0.518854
\(877\) −21.5909 −0.729075 −0.364537 0.931189i \(-0.618773\pi\)
−0.364537 + 0.931189i \(0.618773\pi\)
\(878\) 3.76638 0.127109
\(879\) −0.348391 −0.0117509
\(880\) 17.2374 0.581071
\(881\) −8.51629 −0.286921 −0.143461 0.989656i \(-0.545823\pi\)
−0.143461 + 0.989656i \(0.545823\pi\)
\(882\) −0.182130 −0.00613264
\(883\) 7.62183 0.256495 0.128248 0.991742i \(-0.459065\pi\)
0.128248 + 0.991742i \(0.459065\pi\)
\(884\) 0 0
\(885\) 12.9509 0.435341
\(886\) 1.68748 0.0566919
\(887\) −5.49147 −0.184385 −0.0921927 0.995741i \(-0.529388\pi\)
−0.0921927 + 0.995741i \(0.529388\pi\)
\(888\) 4.60955 0.154686
\(889\) 11.6456 0.390581
\(890\) −2.68445 −0.0899828
\(891\) −4.01616 −0.134546
\(892\) −47.4787 −1.58970
\(893\) −52.0293 −1.74109
\(894\) −0.545003 −0.0182276
\(895\) −21.5107 −0.719023
\(896\) 5.58890 0.186712
\(897\) 0 0
\(898\) 6.59688 0.220141
\(899\) 17.1694 0.572630
\(900\) 7.32777 0.244259
\(901\) 9.02230 0.300576
\(902\) −3.48270 −0.115961
\(903\) −2.34151 −0.0779205
\(904\) −1.96336 −0.0653003
\(905\) 3.90265 0.129729
\(906\) 0.156755 0.00520782
\(907\) 52.3003 1.73660 0.868301 0.496038i \(-0.165212\pi\)
0.868301 + 0.496038i \(0.165212\pi\)
\(908\) −40.6644 −1.34950
\(909\) 15.3230 0.508231
\(910\) 0 0
\(911\) −53.1804 −1.76195 −0.880973 0.473167i \(-0.843111\pi\)
−0.880973 + 0.473167i \(0.843111\pi\)
\(912\) −20.1625 −0.667648
\(913\) −2.83384 −0.0937865
\(914\) 3.04246 0.100636
\(915\) −5.30913 −0.175514
\(916\) −14.1135 −0.466324
\(917\) 14.7907 0.488431
\(918\) −0.551182 −0.0181917
\(919\) 9.23684 0.304695 0.152348 0.988327i \(-0.451317\pi\)
0.152348 + 0.988327i \(0.451317\pi\)
\(920\) 1.77289 0.0584506
\(921\) −5.97146 −0.196766
\(922\) 4.46983 0.147206
\(923\) 0 0
\(924\) 7.89910 0.259861
\(925\) −23.7705 −0.781570
\(926\) 3.38263 0.111160
\(927\) 8.19962 0.269311
\(928\) −7.55248 −0.247922
\(929\) 25.2833 0.829517 0.414759 0.909931i \(-0.363866\pi\)
0.414759 + 0.909931i \(0.363866\pi\)
\(930\) 0.999028 0.0327594
\(931\) −5.30303 −0.173800
\(932\) 31.4206 1.02922
\(933\) −6.16012 −0.201673
\(934\) −2.16519 −0.0708473
\(935\) 13.7203 0.448702
\(936\) 0 0
\(937\) −42.6293 −1.39264 −0.696320 0.717732i \(-0.745180\pi\)
−0.696320 + 0.717732i \(0.745180\pi\)
\(938\) −0.322530 −0.0105310
\(939\) 22.5210 0.734947
\(940\) 21.7836 0.710502
\(941\) 4.78825 0.156093 0.0780463 0.996950i \(-0.475132\pi\)
0.0780463 + 0.996950i \(0.475132\pi\)
\(942\) 0.842228 0.0274413
\(943\) 10.3500 0.337043
\(944\) −43.6196 −1.41970
\(945\) −1.12886 −0.0367218
\(946\) −1.71273 −0.0556856
\(947\) −9.40557 −0.305640 −0.152820 0.988254i \(-0.548835\pi\)
−0.152820 + 0.988254i \(0.548835\pi\)
\(948\) −34.4161 −1.11778
\(949\) 0 0
\(950\) −3.59841 −0.116748
\(951\) 30.4700 0.988057
\(952\) 2.18644 0.0708630
\(953\) −11.7968 −0.382135 −0.191068 0.981577i \(-0.561195\pi\)
−0.191068 + 0.981577i \(0.561195\pi\)
\(954\) −0.542982 −0.0175797
\(955\) 23.7274 0.767800
\(956\) −32.0191 −1.03557
\(957\) −14.1909 −0.458725
\(958\) 2.15433 0.0696033
\(959\) −15.4499 −0.498904
\(960\) 8.14455 0.262864
\(961\) −7.38907 −0.238357
\(962\) 0 0
\(963\) 16.8953 0.544443
\(964\) 45.7353 1.47304
\(965\) −21.3818 −0.688304
\(966\) 0.395913 0.0127383
\(967\) −30.0746 −0.967135 −0.483568 0.875307i \(-0.660659\pi\)
−0.483568 + 0.875307i \(0.660659\pi\)
\(968\) 3.70597 0.119115
\(969\) −16.0486 −0.515556
\(970\) −0.363349 −0.0116664
\(971\) −19.2259 −0.616987 −0.308494 0.951226i \(-0.599825\pi\)
−0.308494 + 0.951226i \(0.599825\pi\)
\(972\) −1.96683 −0.0630861
\(973\) −11.4715 −0.367760
\(974\) −5.51903 −0.176841
\(975\) 0 0
\(976\) 17.8815 0.572373
\(977\) −19.0860 −0.610615 −0.305308 0.952254i \(-0.598759\pi\)
−0.305308 + 0.952254i \(0.598759\pi\)
\(978\) −4.19711 −0.134209
\(979\) 52.4378 1.67592
\(980\) 2.22027 0.0709240
\(981\) −5.62135 −0.179476
\(982\) 0.962686 0.0307206
\(983\) 42.0109 1.33994 0.669970 0.742389i \(-0.266307\pi\)
0.669970 + 0.742389i \(0.266307\pi\)
\(984\) −3.43992 −0.109661
\(985\) −9.52188 −0.303392
\(986\) −1.94757 −0.0620232
\(987\) 9.81123 0.312295
\(988\) 0 0
\(989\) 5.08996 0.161851
\(990\) −0.825718 −0.0262430
\(991\) −20.0564 −0.637112 −0.318556 0.947904i \(-0.603198\pi\)
−0.318556 + 0.947904i \(0.603198\pi\)
\(992\) −10.3860 −0.329756
\(993\) 1.62675 0.0516233
\(994\) 0.591514 0.0187617
\(995\) −23.0043 −0.729284
\(996\) −1.38781 −0.0439746
\(997\) 31.9271 1.01114 0.505570 0.862785i \(-0.331282\pi\)
0.505570 + 0.862785i \(0.331282\pi\)
\(998\) −6.32413 −0.200187
\(999\) 6.38019 0.201860
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.bg.1.8 15
13.12 even 2 3549.2.a.bh.1.8 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.bg.1.8 15 1.1 even 1 trivial
3549.2.a.bh.1.8 yes 15 13.12 even 2