Properties

Label 3675.2.a.bg.1.1
Level $3675$
Weight $2$
Character 3675.1
Self dual yes
Analytic conductor $29.345$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3675.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(29.3450227428\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3675.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.732051 q^{2} +1.00000 q^{3} -1.46410 q^{4} -0.732051 q^{6} +2.53590 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.732051 q^{2} +1.00000 q^{3} -1.46410 q^{4} -0.732051 q^{6} +2.53590 q^{8} +1.00000 q^{9} -2.73205 q^{11} -1.46410 q^{12} -5.73205 q^{13} +1.07180 q^{16} -6.73205 q^{17} -0.732051 q^{18} -2.46410 q^{19} +2.00000 q^{22} +1.26795 q^{23} +2.53590 q^{24} +4.19615 q^{26} +1.00000 q^{27} +6.19615 q^{29} +6.46410 q^{31} -5.85641 q^{32} -2.73205 q^{33} +4.92820 q^{34} -1.46410 q^{36} -7.19615 q^{37} +1.80385 q^{38} -5.73205 q^{39} +2.73205 q^{41} +7.19615 q^{43} +4.00000 q^{44} -0.928203 q^{46} -2.00000 q^{47} +1.07180 q^{48} -6.73205 q^{51} +8.39230 q^{52} +8.39230 q^{53} -0.732051 q^{54} -2.46410 q^{57} -4.53590 q^{58} +10.1962 q^{59} +4.00000 q^{61} -4.73205 q^{62} +2.14359 q^{64} +2.00000 q^{66} -2.66025 q^{67} +9.85641 q^{68} +1.26795 q^{69} -4.19615 q^{71} +2.53590 q^{72} +4.66025 q^{73} +5.26795 q^{74} +3.60770 q^{76} +4.19615 q^{78} +13.3923 q^{79} +1.00000 q^{81} -2.00000 q^{82} +9.12436 q^{83} -5.26795 q^{86} +6.19615 q^{87} -6.92820 q^{88} -9.12436 q^{89} -1.85641 q^{92} +6.46410 q^{93} +1.46410 q^{94} -5.85641 q^{96} -1.07180 q^{97} -2.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 4q^{4} + 2q^{6} + 12q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 4q^{4} + 2q^{6} + 12q^{8} + 2q^{9} - 2q^{11} + 4q^{12} - 8q^{13} + 16q^{16} - 10q^{17} + 2q^{18} + 2q^{19} + 4q^{22} + 6q^{23} + 12q^{24} - 2q^{26} + 2q^{27} + 2q^{29} + 6q^{31} + 16q^{32} - 2q^{33} - 4q^{34} + 4q^{36} - 4q^{37} + 14q^{38} - 8q^{39} + 2q^{41} + 4q^{43} + 8q^{44} + 12q^{46} - 4q^{47} + 16q^{48} - 10q^{51} - 4q^{52} - 4q^{53} + 2q^{54} + 2q^{57} - 16q^{58} + 10q^{59} + 8q^{61} - 6q^{62} + 32q^{64} + 4q^{66} + 12q^{67} - 8q^{68} + 6q^{69} + 2q^{71} + 12q^{72} - 8q^{73} + 14q^{74} + 28q^{76} - 2q^{78} + 6q^{79} + 2q^{81} - 4q^{82} - 6q^{83} - 14q^{86} + 2q^{87} + 6q^{89} + 24q^{92} + 6q^{93} - 4q^{94} + 16q^{96} - 16q^{97} - 2q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.732051 −0.517638 −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.46410 −0.732051
\(5\) 0 0
\(6\) −0.732051 −0.298858
\(7\) 0 0
\(8\) 2.53590 0.896575
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.73205 −0.823744 −0.411872 0.911242i \(-0.635125\pi\)
−0.411872 + 0.911242i \(0.635125\pi\)
\(12\) −1.46410 −0.422650
\(13\) −5.73205 −1.58978 −0.794892 0.606750i \(-0.792473\pi\)
−0.794892 + 0.606750i \(0.792473\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.07180 0.267949
\(17\) −6.73205 −1.63276 −0.816381 0.577514i \(-0.804023\pi\)
−0.816381 + 0.577514i \(0.804023\pi\)
\(18\) −0.732051 −0.172546
\(19\) −2.46410 −0.565304 −0.282652 0.959223i \(-0.591214\pi\)
−0.282652 + 0.959223i \(0.591214\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 1.26795 0.264386 0.132193 0.991224i \(-0.457798\pi\)
0.132193 + 0.991224i \(0.457798\pi\)
\(24\) 2.53590 0.517638
\(25\) 0 0
\(26\) 4.19615 0.822933
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.19615 1.15060 0.575298 0.817944i \(-0.304886\pi\)
0.575298 + 0.817944i \(0.304886\pi\)
\(30\) 0 0
\(31\) 6.46410 1.16099 0.580493 0.814265i \(-0.302860\pi\)
0.580493 + 0.814265i \(0.302860\pi\)
\(32\) −5.85641 −1.03528
\(33\) −2.73205 −0.475589
\(34\) 4.92820 0.845180
\(35\) 0 0
\(36\) −1.46410 −0.244017
\(37\) −7.19615 −1.18304 −0.591520 0.806290i \(-0.701472\pi\)
−0.591520 + 0.806290i \(0.701472\pi\)
\(38\) 1.80385 0.292623
\(39\) −5.73205 −0.917863
\(40\) 0 0
\(41\) 2.73205 0.426675 0.213337 0.976979i \(-0.431567\pi\)
0.213337 + 0.976979i \(0.431567\pi\)
\(42\) 0 0
\(43\) 7.19615 1.09740 0.548701 0.836018i \(-0.315122\pi\)
0.548701 + 0.836018i \(0.315122\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −0.928203 −0.136856
\(47\) −2.00000 −0.291730 −0.145865 0.989305i \(-0.546597\pi\)
−0.145865 + 0.989305i \(0.546597\pi\)
\(48\) 1.07180 0.154701
\(49\) 0 0
\(50\) 0 0
\(51\) −6.73205 −0.942676
\(52\) 8.39230 1.16380
\(53\) 8.39230 1.15277 0.576386 0.817178i \(-0.304463\pi\)
0.576386 + 0.817178i \(0.304463\pi\)
\(54\) −0.732051 −0.0996195
\(55\) 0 0
\(56\) 0 0
\(57\) −2.46410 −0.326378
\(58\) −4.53590 −0.595593
\(59\) 10.1962 1.32743 0.663713 0.747987i \(-0.268980\pi\)
0.663713 + 0.747987i \(0.268980\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −4.73205 −0.600971
\(63\) 0 0
\(64\) 2.14359 0.267949
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −2.66025 −0.325002 −0.162501 0.986708i \(-0.551956\pi\)
−0.162501 + 0.986708i \(0.551956\pi\)
\(68\) 9.85641 1.19526
\(69\) 1.26795 0.152643
\(70\) 0 0
\(71\) −4.19615 −0.497992 −0.248996 0.968505i \(-0.580101\pi\)
−0.248996 + 0.968505i \(0.580101\pi\)
\(72\) 2.53590 0.298858
\(73\) 4.66025 0.545441 0.272721 0.962093i \(-0.412076\pi\)
0.272721 + 0.962093i \(0.412076\pi\)
\(74\) 5.26795 0.612387
\(75\) 0 0
\(76\) 3.60770 0.413831
\(77\) 0 0
\(78\) 4.19615 0.475121
\(79\) 13.3923 1.50675 0.753376 0.657590i \(-0.228424\pi\)
0.753376 + 0.657590i \(0.228424\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 9.12436 1.00153 0.500764 0.865584i \(-0.333052\pi\)
0.500764 + 0.865584i \(0.333052\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −5.26795 −0.568058
\(87\) 6.19615 0.664297
\(88\) −6.92820 −0.738549
\(89\) −9.12436 −0.967180 −0.483590 0.875295i \(-0.660667\pi\)
−0.483590 + 0.875295i \(0.660667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.85641 −0.193544
\(93\) 6.46410 0.670296
\(94\) 1.46410 0.151011
\(95\) 0 0
\(96\) −5.85641 −0.597717
\(97\) −1.07180 −0.108824 −0.0544122 0.998519i \(-0.517329\pi\)
−0.0544122 + 0.998519i \(0.517329\pi\)
\(98\) 0 0
\(99\) −2.73205 −0.274581
\(100\) 0 0
\(101\) −10.7321 −1.06788 −0.533939 0.845523i \(-0.679289\pi\)
−0.533939 + 0.845523i \(0.679289\pi\)
\(102\) 4.92820 0.487965
\(103\) −1.19615 −0.117860 −0.0589302 0.998262i \(-0.518769\pi\)
−0.0589302 + 0.998262i \(0.518769\pi\)
\(104\) −14.5359 −1.42536
\(105\) 0 0
\(106\) −6.14359 −0.596719
\(107\) 8.19615 0.792352 0.396176 0.918175i \(-0.370337\pi\)
0.396176 + 0.918175i \(0.370337\pi\)
\(108\) −1.46410 −0.140883
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) −7.19615 −0.683029
\(112\) 0 0
\(113\) 4.92820 0.463606 0.231803 0.972763i \(-0.425537\pi\)
0.231803 + 0.972763i \(0.425537\pi\)
\(114\) 1.80385 0.168946
\(115\) 0 0
\(116\) −9.07180 −0.842295
\(117\) −5.73205 −0.529928
\(118\) −7.46410 −0.687126
\(119\) 0 0
\(120\) 0 0
\(121\) −3.53590 −0.321445
\(122\) −2.92820 −0.265107
\(123\) 2.73205 0.246341
\(124\) −9.46410 −0.849901
\(125\) 0 0
\(126\) 0 0
\(127\) −15.1962 −1.34844 −0.674220 0.738530i \(-0.735520\pi\)
−0.674220 + 0.738530i \(0.735520\pi\)
\(128\) 10.1436 0.896575
\(129\) 7.19615 0.633586
\(130\) 0 0
\(131\) 8.53590 0.745785 0.372892 0.927875i \(-0.378366\pi\)
0.372892 + 0.927875i \(0.378366\pi\)
\(132\) 4.00000 0.348155
\(133\) 0 0
\(134\) 1.94744 0.168233
\(135\) 0 0
\(136\) −17.0718 −1.46389
\(137\) 8.19615 0.700245 0.350122 0.936704i \(-0.386140\pi\)
0.350122 + 0.936704i \(0.386140\pi\)
\(138\) −0.928203 −0.0790139
\(139\) −7.92820 −0.672461 −0.336231 0.941780i \(-0.609152\pi\)
−0.336231 + 0.941780i \(0.609152\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 3.07180 0.257779
\(143\) 15.6603 1.30958
\(144\) 1.07180 0.0893164
\(145\) 0 0
\(146\) −3.41154 −0.282341
\(147\) 0 0
\(148\) 10.5359 0.866046
\(149\) −21.8564 −1.79055 −0.895273 0.445517i \(-0.853020\pi\)
−0.895273 + 0.445517i \(0.853020\pi\)
\(150\) 0 0
\(151\) 4.92820 0.401051 0.200526 0.979688i \(-0.435735\pi\)
0.200526 + 0.979688i \(0.435735\pi\)
\(152\) −6.24871 −0.506837
\(153\) −6.73205 −0.544254
\(154\) 0 0
\(155\) 0 0
\(156\) 8.39230 0.671922
\(157\) 14.3923 1.14863 0.574315 0.818634i \(-0.305268\pi\)
0.574315 + 0.818634i \(0.305268\pi\)
\(158\) −9.80385 −0.779952
\(159\) 8.39230 0.665553
\(160\) 0 0
\(161\) 0 0
\(162\) −0.732051 −0.0575153
\(163\) 5.85641 0.458709 0.229355 0.973343i \(-0.426338\pi\)
0.229355 + 0.973343i \(0.426338\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −6.67949 −0.518429
\(167\) 0.339746 0.0262903 0.0131452 0.999914i \(-0.495816\pi\)
0.0131452 + 0.999914i \(0.495816\pi\)
\(168\) 0 0
\(169\) 19.8564 1.52742
\(170\) 0 0
\(171\) −2.46410 −0.188435
\(172\) −10.5359 −0.803355
\(173\) −21.4641 −1.63189 −0.815943 0.578133i \(-0.803782\pi\)
−0.815943 + 0.578133i \(0.803782\pi\)
\(174\) −4.53590 −0.343866
\(175\) 0 0
\(176\) −2.92820 −0.220722
\(177\) 10.1962 0.766390
\(178\) 6.67949 0.500649
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) 10.3205 0.767117 0.383559 0.923517i \(-0.374698\pi\)
0.383559 + 0.923517i \(0.374698\pi\)
\(182\) 0 0
\(183\) 4.00000 0.295689
\(184\) 3.21539 0.237042
\(185\) 0 0
\(186\) −4.73205 −0.346971
\(187\) 18.3923 1.34498
\(188\) 2.92820 0.213561
\(189\) 0 0
\(190\) 0 0
\(191\) 4.92820 0.356592 0.178296 0.983977i \(-0.442941\pi\)
0.178296 + 0.983977i \(0.442941\pi\)
\(192\) 2.14359 0.154701
\(193\) 9.19615 0.661954 0.330977 0.943639i \(-0.392622\pi\)
0.330977 + 0.943639i \(0.392622\pi\)
\(194\) 0.784610 0.0563317
\(195\) 0 0
\(196\) 0 0
\(197\) 17.6603 1.25824 0.629121 0.777308i \(-0.283415\pi\)
0.629121 + 0.777308i \(0.283415\pi\)
\(198\) 2.00000 0.142134
\(199\) 22.0000 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(200\) 0 0
\(201\) −2.66025 −0.187640
\(202\) 7.85641 0.552775
\(203\) 0 0
\(204\) 9.85641 0.690086
\(205\) 0 0
\(206\) 0.875644 0.0610090
\(207\) 1.26795 0.0881286
\(208\) −6.14359 −0.425982
\(209\) 6.73205 0.465666
\(210\) 0 0
\(211\) 20.9282 1.44076 0.720378 0.693581i \(-0.243968\pi\)
0.720378 + 0.693581i \(0.243968\pi\)
\(212\) −12.2872 −0.843887
\(213\) −4.19615 −0.287516
\(214\) −6.00000 −0.410152
\(215\) 0 0
\(216\) 2.53590 0.172546
\(217\) 0 0
\(218\) −8.05256 −0.545388
\(219\) 4.66025 0.314911
\(220\) 0 0
\(221\) 38.5885 2.59574
\(222\) 5.26795 0.353562
\(223\) −0.392305 −0.0262707 −0.0131353 0.999914i \(-0.504181\pi\)
−0.0131353 + 0.999914i \(0.504181\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −3.60770 −0.239980
\(227\) −15.6603 −1.03941 −0.519704 0.854347i \(-0.673958\pi\)
−0.519704 + 0.854347i \(0.673958\pi\)
\(228\) 3.60770 0.238925
\(229\) −3.00000 −0.198246 −0.0991228 0.995075i \(-0.531604\pi\)
−0.0991228 + 0.995075i \(0.531604\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 15.7128 1.03160
\(233\) 17.3205 1.13470 0.567352 0.823475i \(-0.307968\pi\)
0.567352 + 0.823475i \(0.307968\pi\)
\(234\) 4.19615 0.274311
\(235\) 0 0
\(236\) −14.9282 −0.971743
\(237\) 13.3923 0.869924
\(238\) 0 0
\(239\) 20.9282 1.35373 0.676866 0.736106i \(-0.263337\pi\)
0.676866 + 0.736106i \(0.263337\pi\)
\(240\) 0 0
\(241\) 6.53590 0.421014 0.210507 0.977592i \(-0.432488\pi\)
0.210507 + 0.977592i \(0.432488\pi\)
\(242\) 2.58846 0.166392
\(243\) 1.00000 0.0641500
\(244\) −5.85641 −0.374918
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 14.1244 0.898711
\(248\) 16.3923 1.04091
\(249\) 9.12436 0.578233
\(250\) 0 0
\(251\) 6.58846 0.415860 0.207930 0.978144i \(-0.433327\pi\)
0.207930 + 0.978144i \(0.433327\pi\)
\(252\) 0 0
\(253\) −3.46410 −0.217786
\(254\) 11.1244 0.698004
\(255\) 0 0
\(256\) −11.7128 −0.732051
\(257\) −11.6603 −0.727347 −0.363673 0.931527i \(-0.618478\pi\)
−0.363673 + 0.931527i \(0.618478\pi\)
\(258\) −5.26795 −0.327968
\(259\) 0 0
\(260\) 0 0
\(261\) 6.19615 0.383532
\(262\) −6.24871 −0.386047
\(263\) 12.3923 0.764142 0.382071 0.924133i \(-0.375211\pi\)
0.382071 + 0.924133i \(0.375211\pi\)
\(264\) −6.92820 −0.426401
\(265\) 0 0
\(266\) 0 0
\(267\) −9.12436 −0.558401
\(268\) 3.89488 0.237918
\(269\) −19.4641 −1.18675 −0.593374 0.804927i \(-0.702204\pi\)
−0.593374 + 0.804927i \(0.702204\pi\)
\(270\) 0 0
\(271\) 16.9282 1.02832 0.514158 0.857696i \(-0.328105\pi\)
0.514158 + 0.857696i \(0.328105\pi\)
\(272\) −7.21539 −0.437497
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −1.85641 −0.111743
\(277\) −2.66025 −0.159839 −0.0799196 0.996801i \(-0.525466\pi\)
−0.0799196 + 0.996801i \(0.525466\pi\)
\(278\) 5.80385 0.348092
\(279\) 6.46410 0.386996
\(280\) 0 0
\(281\) −13.8564 −0.826604 −0.413302 0.910594i \(-0.635625\pi\)
−0.413302 + 0.910594i \(0.635625\pi\)
\(282\) 1.46410 0.0871860
\(283\) −0.124356 −0.00739218 −0.00369609 0.999993i \(-0.501177\pi\)
−0.00369609 + 0.999993i \(0.501177\pi\)
\(284\) 6.14359 0.364555
\(285\) 0 0
\(286\) −11.4641 −0.677887
\(287\) 0 0
\(288\) −5.85641 −0.345092
\(289\) 28.3205 1.66591
\(290\) 0 0
\(291\) −1.07180 −0.0628298
\(292\) −6.82309 −0.399291
\(293\) −5.07180 −0.296298 −0.148149 0.988965i \(-0.547331\pi\)
−0.148149 + 0.988965i \(0.547331\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −18.2487 −1.06068
\(297\) −2.73205 −0.158530
\(298\) 16.0000 0.926855
\(299\) −7.26795 −0.420316
\(300\) 0 0
\(301\) 0 0
\(302\) −3.60770 −0.207600
\(303\) −10.7321 −0.616540
\(304\) −2.64102 −0.151473
\(305\) 0 0
\(306\) 4.92820 0.281727
\(307\) 7.87564 0.449487 0.224743 0.974418i \(-0.427846\pi\)
0.224743 + 0.974418i \(0.427846\pi\)
\(308\) 0 0
\(309\) −1.19615 −0.0680467
\(310\) 0 0
\(311\) −15.1244 −0.857624 −0.428812 0.903394i \(-0.641068\pi\)
−0.428812 + 0.903394i \(0.641068\pi\)
\(312\) −14.5359 −0.822933
\(313\) −4.66025 −0.263413 −0.131707 0.991289i \(-0.542046\pi\)
−0.131707 + 0.991289i \(0.542046\pi\)
\(314\) −10.5359 −0.594575
\(315\) 0 0
\(316\) −19.6077 −1.10302
\(317\) −30.4449 −1.70995 −0.854977 0.518666i \(-0.826429\pi\)
−0.854977 + 0.518666i \(0.826429\pi\)
\(318\) −6.14359 −0.344516
\(319\) −16.9282 −0.947797
\(320\) 0 0
\(321\) 8.19615 0.457465
\(322\) 0 0
\(323\) 16.5885 0.923006
\(324\) −1.46410 −0.0813390
\(325\) 0 0
\(326\) −4.28719 −0.237445
\(327\) 11.0000 0.608301
\(328\) 6.92820 0.382546
\(329\) 0 0
\(330\) 0 0
\(331\) 21.9282 1.20528 0.602642 0.798012i \(-0.294115\pi\)
0.602642 + 0.798012i \(0.294115\pi\)
\(332\) −13.3590 −0.733169
\(333\) −7.19615 −0.394347
\(334\) −0.248711 −0.0136089
\(335\) 0 0
\(336\) 0 0
\(337\) 33.9808 1.85105 0.925525 0.378686i \(-0.123624\pi\)
0.925525 + 0.378686i \(0.123624\pi\)
\(338\) −14.5359 −0.790649
\(339\) 4.92820 0.267663
\(340\) 0 0
\(341\) −17.6603 −0.956356
\(342\) 1.80385 0.0975409
\(343\) 0 0
\(344\) 18.2487 0.983905
\(345\) 0 0
\(346\) 15.7128 0.844726
\(347\) 34.9282 1.87504 0.937522 0.347926i \(-0.113114\pi\)
0.937522 + 0.347926i \(0.113114\pi\)
\(348\) −9.07180 −0.486299
\(349\) 22.0000 1.17763 0.588817 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) −5.73205 −0.305954
\(352\) 16.0000 0.852803
\(353\) −21.1244 −1.12434 −0.562168 0.827023i \(-0.690033\pi\)
−0.562168 + 0.827023i \(0.690033\pi\)
\(354\) −7.46410 −0.396713
\(355\) 0 0
\(356\) 13.3590 0.708025
\(357\) 0 0
\(358\) 7.32051 0.386901
\(359\) −4.73205 −0.249748 −0.124874 0.992173i \(-0.539853\pi\)
−0.124874 + 0.992173i \(0.539853\pi\)
\(360\) 0 0
\(361\) −12.9282 −0.680432
\(362\) −7.55514 −0.397089
\(363\) −3.53590 −0.185587
\(364\) 0 0
\(365\) 0 0
\(366\) −2.92820 −0.153060
\(367\) −0.803848 −0.0419605 −0.0209803 0.999780i \(-0.506679\pi\)
−0.0209803 + 0.999780i \(0.506679\pi\)
\(368\) 1.35898 0.0708419
\(369\) 2.73205 0.142225
\(370\) 0 0
\(371\) 0 0
\(372\) −9.46410 −0.490691
\(373\) 18.5167 0.958756 0.479378 0.877608i \(-0.340862\pi\)
0.479378 + 0.877608i \(0.340862\pi\)
\(374\) −13.4641 −0.696212
\(375\) 0 0
\(376\) −5.07180 −0.261558
\(377\) −35.5167 −1.82920
\(378\) 0 0
\(379\) −28.3205 −1.45473 −0.727363 0.686253i \(-0.759255\pi\)
−0.727363 + 0.686253i \(0.759255\pi\)
\(380\) 0 0
\(381\) −15.1962 −0.778522
\(382\) −3.60770 −0.184586
\(383\) −11.3205 −0.578451 −0.289225 0.957261i \(-0.593398\pi\)
−0.289225 + 0.957261i \(0.593398\pi\)
\(384\) 10.1436 0.517638
\(385\) 0 0
\(386\) −6.73205 −0.342652
\(387\) 7.19615 0.365801
\(388\) 1.56922 0.0796650
\(389\) 36.5885 1.85511 0.927554 0.373689i \(-0.121907\pi\)
0.927554 + 0.373689i \(0.121907\pi\)
\(390\) 0 0
\(391\) −8.53590 −0.431679
\(392\) 0 0
\(393\) 8.53590 0.430579
\(394\) −12.9282 −0.651313
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 20.8038 1.04412 0.522058 0.852910i \(-0.325165\pi\)
0.522058 + 0.852910i \(0.325165\pi\)
\(398\) −16.1051 −0.807277
\(399\) 0 0
\(400\) 0 0
\(401\) 4.39230 0.219341 0.109671 0.993968i \(-0.465020\pi\)
0.109671 + 0.993968i \(0.465020\pi\)
\(402\) 1.94744 0.0971295
\(403\) −37.0526 −1.84572
\(404\) 15.7128 0.781742
\(405\) 0 0
\(406\) 0 0
\(407\) 19.6603 0.974523
\(408\) −17.0718 −0.845180
\(409\) 30.8564 1.52575 0.762876 0.646545i \(-0.223787\pi\)
0.762876 + 0.646545i \(0.223787\pi\)
\(410\) 0 0
\(411\) 8.19615 0.404286
\(412\) 1.75129 0.0862798
\(413\) 0 0
\(414\) −0.928203 −0.0456187
\(415\) 0 0
\(416\) 33.5692 1.64587
\(417\) −7.92820 −0.388246
\(418\) −4.92820 −0.241046
\(419\) −28.5359 −1.39407 −0.697035 0.717037i \(-0.745498\pi\)
−0.697035 + 0.717037i \(0.745498\pi\)
\(420\) 0 0
\(421\) 13.9282 0.678819 0.339410 0.940639i \(-0.389773\pi\)
0.339410 + 0.940639i \(0.389773\pi\)
\(422\) −15.3205 −0.745791
\(423\) −2.00000 −0.0972433
\(424\) 21.2820 1.03355
\(425\) 0 0
\(426\) 3.07180 0.148829
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 15.6603 0.756084
\(430\) 0 0
\(431\) −17.3205 −0.834300 −0.417150 0.908838i \(-0.636971\pi\)
−0.417150 + 0.908838i \(0.636971\pi\)
\(432\) 1.07180 0.0515668
\(433\) −4.80385 −0.230858 −0.115429 0.993316i \(-0.536824\pi\)
−0.115429 + 0.993316i \(0.536824\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −16.1051 −0.771295
\(437\) −3.12436 −0.149458
\(438\) −3.41154 −0.163010
\(439\) 7.46410 0.356242 0.178121 0.984009i \(-0.442998\pi\)
0.178121 + 0.984009i \(0.442998\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −28.2487 −1.34365
\(443\) −2.53590 −0.120484 −0.0602421 0.998184i \(-0.519187\pi\)
−0.0602421 + 0.998184i \(0.519187\pi\)
\(444\) 10.5359 0.500012
\(445\) 0 0
\(446\) 0.287187 0.0135987
\(447\) −21.8564 −1.03377
\(448\) 0 0
\(449\) −8.14359 −0.384320 −0.192160 0.981364i \(-0.561549\pi\)
−0.192160 + 0.981364i \(0.561549\pi\)
\(450\) 0 0
\(451\) −7.46410 −0.351471
\(452\) −7.21539 −0.339383
\(453\) 4.92820 0.231547
\(454\) 11.4641 0.538037
\(455\) 0 0
\(456\) −6.24871 −0.292623
\(457\) −0.660254 −0.0308854 −0.0154427 0.999881i \(-0.504916\pi\)
−0.0154427 + 0.999881i \(0.504916\pi\)
\(458\) 2.19615 0.102619
\(459\) −6.73205 −0.314225
\(460\) 0 0
\(461\) −34.9808 −1.62922 −0.814608 0.580012i \(-0.803048\pi\)
−0.814608 + 0.580012i \(0.803048\pi\)
\(462\) 0 0
\(463\) −22.2679 −1.03488 −0.517440 0.855720i \(-0.673115\pi\)
−0.517440 + 0.855720i \(0.673115\pi\)
\(464\) 6.64102 0.308301
\(465\) 0 0
\(466\) −12.6795 −0.587366
\(467\) 27.8564 1.28904 0.644520 0.764587i \(-0.277057\pi\)
0.644520 + 0.764587i \(0.277057\pi\)
\(468\) 8.39230 0.387934
\(469\) 0 0
\(470\) 0 0
\(471\) 14.3923 0.663162
\(472\) 25.8564 1.19014
\(473\) −19.6603 −0.903979
\(474\) −9.80385 −0.450306
\(475\) 0 0
\(476\) 0 0
\(477\) 8.39230 0.384257
\(478\) −15.3205 −0.700744
\(479\) −32.7846 −1.49797 −0.748984 0.662589i \(-0.769458\pi\)
−0.748984 + 0.662589i \(0.769458\pi\)
\(480\) 0 0
\(481\) 41.2487 1.88078
\(482\) −4.78461 −0.217933
\(483\) 0 0
\(484\) 5.17691 0.235314
\(485\) 0 0
\(486\) −0.732051 −0.0332065
\(487\) 31.5885 1.43141 0.715705 0.698403i \(-0.246106\pi\)
0.715705 + 0.698403i \(0.246106\pi\)
\(488\) 10.1436 0.459179
\(489\) 5.85641 0.264836
\(490\) 0 0
\(491\) 10.2487 0.462518 0.231259 0.972892i \(-0.425716\pi\)
0.231259 + 0.972892i \(0.425716\pi\)
\(492\) −4.00000 −0.180334
\(493\) −41.7128 −1.87865
\(494\) −10.3397 −0.465207
\(495\) 0 0
\(496\) 6.92820 0.311086
\(497\) 0 0
\(498\) −6.67949 −0.299315
\(499\) −20.4641 −0.916099 −0.458050 0.888927i \(-0.651452\pi\)
−0.458050 + 0.888927i \(0.651452\pi\)
\(500\) 0 0
\(501\) 0.339746 0.0151787
\(502\) −4.82309 −0.215265
\(503\) 6.39230 0.285019 0.142509 0.989793i \(-0.454483\pi\)
0.142509 + 0.989793i \(0.454483\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 2.53590 0.112734
\(507\) 19.8564 0.881854
\(508\) 22.2487 0.987127
\(509\) 11.4641 0.508137 0.254069 0.967186i \(-0.418231\pi\)
0.254069 + 0.967186i \(0.418231\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −11.7128 −0.517638
\(513\) −2.46410 −0.108793
\(514\) 8.53590 0.376502
\(515\) 0 0
\(516\) −10.5359 −0.463817
\(517\) 5.46410 0.240311
\(518\) 0 0
\(519\) −21.4641 −0.942169
\(520\) 0 0
\(521\) 1.46410 0.0641435 0.0320717 0.999486i \(-0.489790\pi\)
0.0320717 + 0.999486i \(0.489790\pi\)
\(522\) −4.53590 −0.198531
\(523\) 24.2679 1.06116 0.530582 0.847634i \(-0.321974\pi\)
0.530582 + 0.847634i \(0.321974\pi\)
\(524\) −12.4974 −0.545952
\(525\) 0 0
\(526\) −9.07180 −0.395549
\(527\) −43.5167 −1.89562
\(528\) −2.92820 −0.127434
\(529\) −21.3923 −0.930100
\(530\) 0 0
\(531\) 10.1962 0.442475
\(532\) 0 0
\(533\) −15.6603 −0.678321
\(534\) 6.67949 0.289050
\(535\) 0 0
\(536\) −6.74613 −0.291389
\(537\) −10.0000 −0.431532
\(538\) 14.2487 0.614306
\(539\) 0 0
\(540\) 0 0
\(541\) −35.7846 −1.53850 −0.769250 0.638948i \(-0.779370\pi\)
−0.769250 + 0.638948i \(0.779370\pi\)
\(542\) −12.3923 −0.532295
\(543\) 10.3205 0.442895
\(544\) 39.4256 1.69036
\(545\) 0 0
\(546\) 0 0
\(547\) −22.2487 −0.951286 −0.475643 0.879638i \(-0.657785\pi\)
−0.475643 + 0.879638i \(0.657785\pi\)
\(548\) −12.0000 −0.512615
\(549\) 4.00000 0.170716
\(550\) 0 0
\(551\) −15.2679 −0.650437
\(552\) 3.21539 0.136856
\(553\) 0 0
\(554\) 1.94744 0.0827388
\(555\) 0 0
\(556\) 11.6077 0.492276
\(557\) 26.7846 1.13490 0.567450 0.823408i \(-0.307930\pi\)
0.567450 + 0.823408i \(0.307930\pi\)
\(558\) −4.73205 −0.200324
\(559\) −41.2487 −1.74463
\(560\) 0 0
\(561\) 18.3923 0.776524
\(562\) 10.1436 0.427882
\(563\) 18.0000 0.758610 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(564\) 2.92820 0.123300
\(565\) 0 0
\(566\) 0.0910347 0.00382647
\(567\) 0 0
\(568\) −10.6410 −0.446487
\(569\) −26.4449 −1.10863 −0.554313 0.832308i \(-0.687019\pi\)
−0.554313 + 0.832308i \(0.687019\pi\)
\(570\) 0 0
\(571\) 39.3923 1.64852 0.824258 0.566214i \(-0.191592\pi\)
0.824258 + 0.566214i \(0.191592\pi\)
\(572\) −22.9282 −0.958676
\(573\) 4.92820 0.205879
\(574\) 0 0
\(575\) 0 0
\(576\) 2.14359 0.0893164
\(577\) −11.3397 −0.472080 −0.236040 0.971743i \(-0.575850\pi\)
−0.236040 + 0.971743i \(0.575850\pi\)
\(578\) −20.7321 −0.862340
\(579\) 9.19615 0.382179
\(580\) 0 0
\(581\) 0 0
\(582\) 0.784610 0.0325231
\(583\) −22.9282 −0.949589
\(584\) 11.8179 0.489029
\(585\) 0 0
\(586\) 3.71281 0.153375
\(587\) −37.2679 −1.53821 −0.769106 0.639121i \(-0.779298\pi\)
−0.769106 + 0.639121i \(0.779298\pi\)
\(588\) 0 0
\(589\) −15.9282 −0.656310
\(590\) 0 0
\(591\) 17.6603 0.726446
\(592\) −7.71281 −0.316995
\(593\) 37.9090 1.55673 0.778367 0.627809i \(-0.216048\pi\)
0.778367 + 0.627809i \(0.216048\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) 32.0000 1.31077
\(597\) 22.0000 0.900400
\(598\) 5.32051 0.217572
\(599\) 10.2487 0.418751 0.209375 0.977835i \(-0.432857\pi\)
0.209375 + 0.977835i \(0.432857\pi\)
\(600\) 0 0
\(601\) −13.9282 −0.568143 −0.284072 0.958803i \(-0.591685\pi\)
−0.284072 + 0.958803i \(0.591685\pi\)
\(602\) 0 0
\(603\) −2.66025 −0.108334
\(604\) −7.21539 −0.293590
\(605\) 0 0
\(606\) 7.85641 0.319145
\(607\) 7.19615 0.292083 0.146041 0.989278i \(-0.453347\pi\)
0.146041 + 0.989278i \(0.453347\pi\)
\(608\) 14.4308 0.585245
\(609\) 0 0
\(610\) 0 0
\(611\) 11.4641 0.463788
\(612\) 9.85641 0.398422
\(613\) 13.0718 0.527965 0.263982 0.964527i \(-0.414964\pi\)
0.263982 + 0.964527i \(0.414964\pi\)
\(614\) −5.76537 −0.232671
\(615\) 0 0
\(616\) 0 0
\(617\) −12.2487 −0.493115 −0.246557 0.969128i \(-0.579299\pi\)
−0.246557 + 0.969128i \(0.579299\pi\)
\(618\) 0.875644 0.0352236
\(619\) −43.9282 −1.76562 −0.882812 0.469727i \(-0.844352\pi\)
−0.882812 + 0.469727i \(0.844352\pi\)
\(620\) 0 0
\(621\) 1.26795 0.0508810
\(622\) 11.0718 0.443939
\(623\) 0 0
\(624\) −6.14359 −0.245941
\(625\) 0 0
\(626\) 3.41154 0.136353
\(627\) 6.73205 0.268852
\(628\) −21.0718 −0.840856
\(629\) 48.4449 1.93162
\(630\) 0 0
\(631\) 7.21539 0.287240 0.143620 0.989633i \(-0.454126\pi\)
0.143620 + 0.989633i \(0.454126\pi\)
\(632\) 33.9615 1.35092
\(633\) 20.9282 0.831821
\(634\) 22.2872 0.885137
\(635\) 0 0
\(636\) −12.2872 −0.487219
\(637\) 0 0
\(638\) 12.3923 0.490616
\(639\) −4.19615 −0.165997
\(640\) 0 0
\(641\) 14.1962 0.560714 0.280357 0.959896i \(-0.409547\pi\)
0.280357 + 0.959896i \(0.409547\pi\)
\(642\) −6.00000 −0.236801
\(643\) 40.5167 1.59782 0.798911 0.601450i \(-0.205410\pi\)
0.798911 + 0.601450i \(0.205410\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −12.1436 −0.477783
\(647\) −37.9090 −1.49036 −0.745178 0.666866i \(-0.767635\pi\)
−0.745178 + 0.666866i \(0.767635\pi\)
\(648\) 2.53590 0.0996195
\(649\) −27.8564 −1.09346
\(650\) 0 0
\(651\) 0 0
\(652\) −8.57437 −0.335798
\(653\) 13.4115 0.524834 0.262417 0.964955i \(-0.415480\pi\)
0.262417 + 0.964955i \(0.415480\pi\)
\(654\) −8.05256 −0.314880
\(655\) 0 0
\(656\) 2.92820 0.114327
\(657\) 4.66025 0.181814
\(658\) 0 0
\(659\) −10.9282 −0.425702 −0.212851 0.977085i \(-0.568275\pi\)
−0.212851 + 0.977085i \(0.568275\pi\)
\(660\) 0 0
\(661\) 3.53590 0.137531 0.0687653 0.997633i \(-0.478094\pi\)
0.0687653 + 0.997633i \(0.478094\pi\)
\(662\) −16.0526 −0.623900
\(663\) 38.5885 1.49865
\(664\) 23.1384 0.897946
\(665\) 0 0
\(666\) 5.26795 0.204129
\(667\) 7.85641 0.304201
\(668\) −0.497423 −0.0192459
\(669\) −0.392305 −0.0151674
\(670\) 0 0
\(671\) −10.9282 −0.421879
\(672\) 0 0
\(673\) 44.6603 1.72153 0.860763 0.509006i \(-0.169987\pi\)
0.860763 + 0.509006i \(0.169987\pi\)
\(674\) −24.8756 −0.958174
\(675\) 0 0
\(676\) −29.0718 −1.11815
\(677\) 8.87564 0.341119 0.170559 0.985347i \(-0.445443\pi\)
0.170559 + 0.985347i \(0.445443\pi\)
\(678\) −3.60770 −0.138553
\(679\) 0 0
\(680\) 0 0
\(681\) −15.6603 −0.600102
\(682\) 12.9282 0.495046
\(683\) −10.0526 −0.384650 −0.192325 0.981331i \(-0.561603\pi\)
−0.192325 + 0.981331i \(0.561603\pi\)
\(684\) 3.60770 0.137944
\(685\) 0 0
\(686\) 0 0
\(687\) −3.00000 −0.114457
\(688\) 7.71281 0.294048
\(689\) −48.1051 −1.83266
\(690\) 0 0
\(691\) −18.8564 −0.717332 −0.358666 0.933466i \(-0.616768\pi\)
−0.358666 + 0.933466i \(0.616768\pi\)
\(692\) 31.4256 1.19462
\(693\) 0 0
\(694\) −25.5692 −0.970594
\(695\) 0 0
\(696\) 15.7128 0.595593
\(697\) −18.3923 −0.696658
\(698\) −16.1051 −0.609588
\(699\) 17.3205 0.655122
\(700\) 0 0
\(701\) 22.5885 0.853154 0.426577 0.904451i \(-0.359719\pi\)
0.426577 + 0.904451i \(0.359719\pi\)
\(702\) 4.19615 0.158374
\(703\) 17.7321 0.668777
\(704\) −5.85641 −0.220722
\(705\) 0 0
\(706\) 15.4641 0.581999
\(707\) 0 0
\(708\) −14.9282 −0.561036
\(709\) −14.9282 −0.560640 −0.280320 0.959907i \(-0.590441\pi\)
−0.280320 + 0.959907i \(0.590441\pi\)
\(710\) 0 0
\(711\) 13.3923 0.502251
\(712\) −23.1384 −0.867150
\(713\) 8.19615 0.306948
\(714\) 0 0
\(715\) 0 0
\(716\) 14.6410 0.547160
\(717\) 20.9282 0.781578
\(718\) 3.46410 0.129279
\(719\) −27.4641 −1.02424 −0.512119 0.858914i \(-0.671139\pi\)
−0.512119 + 0.858914i \(0.671139\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 9.46410 0.352217
\(723\) 6.53590 0.243073
\(724\) −15.1103 −0.561569
\(725\) 0 0
\(726\) 2.58846 0.0960667
\(727\) 30.6603 1.13713 0.568563 0.822640i \(-0.307500\pi\)
0.568563 + 0.822640i \(0.307500\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −48.4449 −1.79180
\(732\) −5.85641 −0.216459
\(733\) 18.6603 0.689232 0.344616 0.938744i \(-0.388009\pi\)
0.344616 + 0.938744i \(0.388009\pi\)
\(734\) 0.588457 0.0217204
\(735\) 0 0
\(736\) −7.42563 −0.273712
\(737\) 7.26795 0.267718
\(738\) −2.00000 −0.0736210
\(739\) −13.7846 −0.507075 −0.253538 0.967326i \(-0.581594\pi\)
−0.253538 + 0.967326i \(0.581594\pi\)
\(740\) 0 0
\(741\) 14.1244 0.518871
\(742\) 0 0
\(743\) −49.9090 −1.83098 −0.915491 0.402338i \(-0.868198\pi\)
−0.915491 + 0.402338i \(0.868198\pi\)
\(744\) 16.3923 0.600971
\(745\) 0 0
\(746\) −13.5551 −0.496289
\(747\) 9.12436 0.333843
\(748\) −26.9282 −0.984593
\(749\) 0 0
\(750\) 0 0
\(751\) 31.9282 1.16508 0.582538 0.812803i \(-0.302060\pi\)
0.582538 + 0.812803i \(0.302060\pi\)
\(752\) −2.14359 −0.0781688
\(753\) 6.58846 0.240097
\(754\) 26.0000 0.946864
\(755\) 0 0
\(756\) 0 0
\(757\) 0.143594 0.00521900 0.00260950 0.999997i \(-0.499169\pi\)
0.00260950 + 0.999997i \(0.499169\pi\)
\(758\) 20.7321 0.753022
\(759\) −3.46410 −0.125739
\(760\) 0 0
\(761\) 43.2679 1.56846 0.784231 0.620469i \(-0.213058\pi\)
0.784231 + 0.620469i \(0.213058\pi\)
\(762\) 11.1244 0.402993
\(763\) 0 0
\(764\) −7.21539 −0.261044
\(765\) 0 0
\(766\) 8.28719 0.299428
\(767\) −58.4449 −2.11032
\(768\) −11.7128 −0.422650
\(769\) 17.6795 0.637539 0.318769 0.947832i \(-0.396730\pi\)
0.318769 + 0.947832i \(0.396730\pi\)
\(770\) 0 0
\(771\) −11.6603 −0.419934
\(772\) −13.4641 −0.484584
\(773\) 1.51666 0.0545505 0.0272752 0.999628i \(-0.491317\pi\)
0.0272752 + 0.999628i \(0.491317\pi\)
\(774\) −5.26795 −0.189353
\(775\) 0 0
\(776\) −2.71797 −0.0975694
\(777\) 0 0
\(778\) −26.7846 −0.960275
\(779\) −6.73205 −0.241201
\(780\) 0 0
\(781\) 11.4641 0.410218
\(782\) 6.24871 0.223453
\(783\) 6.19615 0.221432
\(784\) 0 0
\(785\) 0 0
\(786\) −6.24871 −0.222884
\(787\) −6.53590 −0.232980 −0.116490 0.993192i \(-0.537164\pi\)
−0.116490 + 0.993192i \(0.537164\pi\)
\(788\) −25.8564 −0.921096
\(789\) 12.3923 0.441178
\(790\) 0 0
\(791\) 0 0
\(792\) −6.92820 −0.246183
\(793\) −22.9282 −0.814204
\(794\) −15.2295 −0.540474
\(795\) 0 0
\(796\) −32.2102 −1.14166
\(797\) 42.0526 1.48958 0.744789 0.667300i \(-0.232550\pi\)
0.744789 + 0.667300i \(0.232550\pi\)
\(798\) 0 0
\(799\) 13.4641 0.476326
\(800\) 0 0
\(801\) −9.12436 −0.322393
\(802\) −3.21539 −0.113539
\(803\) −12.7321 −0.449304
\(804\) 3.89488 0.137362
\(805\) 0 0
\(806\) 27.1244 0.955415
\(807\) −19.4641 −0.685169
\(808\) −27.2154 −0.957434
\(809\) −29.7128 −1.04465 −0.522323 0.852747i \(-0.674935\pi\)
−0.522323 + 0.852747i \(0.674935\pi\)
\(810\) 0 0
\(811\) 3.46410 0.121641 0.0608205 0.998149i \(-0.480628\pi\)
0.0608205 + 0.998149i \(0.480628\pi\)
\(812\) 0 0
\(813\) 16.9282 0.593698
\(814\) −14.3923 −0.504450
\(815\) 0 0
\(816\) −7.21539 −0.252589
\(817\) −17.7321 −0.620366
\(818\) −22.5885 −0.789787
\(819\) 0 0
\(820\) 0 0
\(821\) 19.5167 0.681136 0.340568 0.940220i \(-0.389381\pi\)
0.340568 + 0.940220i \(0.389381\pi\)
\(822\) −6.00000 −0.209274
\(823\) 23.1769 0.807896 0.403948 0.914782i \(-0.367638\pi\)
0.403948 + 0.914782i \(0.367638\pi\)
\(824\) −3.03332 −0.105671
\(825\) 0 0
\(826\) 0 0
\(827\) 52.2487 1.81687 0.908433 0.418031i \(-0.137280\pi\)
0.908433 + 0.418031i \(0.137280\pi\)
\(828\) −1.85641 −0.0645146
\(829\) −25.3923 −0.881911 −0.440956 0.897529i \(-0.645360\pi\)
−0.440956 + 0.897529i \(0.645360\pi\)
\(830\) 0 0
\(831\) −2.66025 −0.0922832
\(832\) −12.2872 −0.425982
\(833\) 0 0
\(834\) 5.80385 0.200971
\(835\) 0 0
\(836\) −9.85641 −0.340891
\(837\) 6.46410 0.223432
\(838\) 20.8897 0.721624
\(839\) −40.4449 −1.39631 −0.698156 0.715946i \(-0.745996\pi\)
−0.698156 + 0.715946i \(0.745996\pi\)
\(840\) 0 0
\(841\) 9.39230 0.323873
\(842\) −10.1962 −0.351383
\(843\) −13.8564 −0.477240
\(844\) −30.6410 −1.05471
\(845\) 0 0
\(846\) 1.46410 0.0503369
\(847\) 0 0
\(848\) 8.99485 0.308884
\(849\) −0.124356 −0.00426787
\(850\) 0 0
\(851\) −9.12436 −0.312779
\(852\) 6.14359 0.210476
\(853\) −19.9808 −0.684128 −0.342064 0.939677i \(-0.611126\pi\)
−0.342064 + 0.939677i \(0.611126\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 20.7846 0.710403
\(857\) −4.87564 −0.166549 −0.0832744 0.996527i \(-0.526538\pi\)
−0.0832744 + 0.996527i \(0.526538\pi\)
\(858\) −11.4641 −0.391378
\(859\) 0.535898 0.0182846 0.00914231 0.999958i \(-0.497090\pi\)
0.00914231 + 0.999958i \(0.497090\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 12.6795 0.431865
\(863\) 6.39230 0.217597 0.108798 0.994064i \(-0.465300\pi\)
0.108798 + 0.994064i \(0.465300\pi\)
\(864\) −5.85641 −0.199239
\(865\) 0 0
\(866\) 3.51666 0.119501
\(867\) 28.3205 0.961815
\(868\) 0 0
\(869\) −36.5885 −1.24118
\(870\) 0 0
\(871\) 15.2487 0.516683
\(872\) 27.8949 0.944640
\(873\) −1.07180 −0.0362748
\(874\) 2.28719 0.0773653
\(875\) 0 0
\(876\) −6.82309 −0.230531
\(877\) 31.8564 1.07571 0.537857 0.843036i \(-0.319234\pi\)
0.537857 + 0.843036i \(0.319234\pi\)
\(878\) −5.46410 −0.184404
\(879\) −5.07180 −0.171067
\(880\) 0 0
\(881\) −17.8564 −0.601598 −0.300799 0.953688i \(-0.597253\pi\)
−0.300799 + 0.953688i \(0.597253\pi\)
\(882\) 0 0
\(883\) −22.4115 −0.754208 −0.377104 0.926171i \(-0.623080\pi\)
−0.377104 + 0.926171i \(0.623080\pi\)
\(884\) −56.4974 −1.90021
\(885\) 0 0
\(886\) 1.85641 0.0623672
\(887\) 28.7321 0.964728 0.482364 0.875971i \(-0.339778\pi\)
0.482364 + 0.875971i \(0.339778\pi\)
\(888\) −18.2487 −0.612387
\(889\) 0 0
\(890\) 0 0
\(891\) −2.73205 −0.0915271
\(892\) 0.574374 0.0192315
\(893\) 4.92820 0.164916
\(894\) 16.0000 0.535120
\(895\) 0 0
\(896\) 0 0
\(897\) −7.26795 −0.242670
\(898\) 5.96152 0.198939
\(899\) 40.0526 1.33583
\(900\) 0 0
\(901\) −56.4974 −1.88220
\(902\) 5.46410 0.181935
\(903\) 0 0
\(904\) 12.4974 0.415658
\(905\) 0 0
\(906\) −3.60770 −0.119858
\(907\) −2.41154 −0.0800740 −0.0400370 0.999198i \(-0.512748\pi\)
−0.0400370 + 0.999198i \(0.512748\pi\)
\(908\) 22.9282 0.760899
\(909\) −10.7321 −0.355960
\(910\) 0 0
\(911\) −11.2679 −0.373324 −0.186662 0.982424i \(-0.559767\pi\)
−0.186662 + 0.982424i \(0.559767\pi\)
\(912\) −2.64102 −0.0874528
\(913\) −24.9282 −0.825003
\(914\) 0.483340 0.0159874
\(915\) 0 0
\(916\) 4.39230 0.145126
\(917\) 0 0
\(918\) 4.92820 0.162655
\(919\) −3.14359 −0.103698 −0.0518488 0.998655i \(-0.516511\pi\)
−0.0518488 + 0.998655i \(0.516511\pi\)
\(920\) 0 0
\(921\) 7.87564 0.259511
\(922\) 25.6077 0.843345
\(923\) 24.0526 0.791700
\(924\) 0 0
\(925\) 0 0
\(926\) 16.3013 0.535693
\(927\) −1.19615 −0.0392868
\(928\) −36.2872 −1.19119
\(929\) 6.44486 0.211449 0.105725 0.994395i \(-0.466284\pi\)
0.105725 + 0.994395i \(0.466284\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −25.3590 −0.830661
\(933\) −15.1244 −0.495149
\(934\) −20.3923 −0.667257
\(935\) 0 0
\(936\) −14.5359 −0.475121
\(937\) −28.2679 −0.923474 −0.461737 0.887017i \(-0.652774\pi\)
−0.461737 + 0.887017i \(0.652774\pi\)
\(938\) 0 0
\(939\) −4.66025 −0.152082
\(940\) 0 0
\(941\) 8.05256 0.262506 0.131253 0.991349i \(-0.458100\pi\)
0.131253 + 0.991349i \(0.458100\pi\)
\(942\) −10.5359 −0.343278
\(943\) 3.46410 0.112807
\(944\) 10.9282 0.355683
\(945\) 0 0
\(946\) 14.3923 0.467934
\(947\) −11.6603 −0.378907 −0.189454 0.981890i \(-0.560672\pi\)
−0.189454 + 0.981890i \(0.560672\pi\)
\(948\) −19.6077 −0.636828
\(949\) −26.7128 −0.867135
\(950\) 0 0
\(951\) −30.4449 −0.987242
\(952\) 0 0
\(953\) −40.1051 −1.29913 −0.649566 0.760305i \(-0.725049\pi\)
−0.649566 + 0.760305i \(0.725049\pi\)
\(954\) −6.14359 −0.198906
\(955\) 0 0
\(956\) −30.6410 −0.991001
\(957\) −16.9282 −0.547211
\(958\) 24.0000 0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 10.7846 0.347891
\(962\) −30.1962 −0.973563
\(963\) 8.19615 0.264117
\(964\) −9.56922 −0.308204
\(965\) 0 0
\(966\) 0 0
\(967\) −14.1244 −0.454209 −0.227104 0.973870i \(-0.572926\pi\)
−0.227104 + 0.973870i \(0.572926\pi\)
\(968\) −8.96668 −0.288200
\(969\) 16.5885 0.532898
\(970\) 0 0
\(971\) −24.0000 −0.770197 −0.385098 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) −1.46410 −0.0469611
\(973\) 0 0
\(974\) −23.1244 −0.740952
\(975\) 0 0
\(976\) 4.28719 0.137230
\(977\) −14.5885 −0.466726 −0.233363 0.972390i \(-0.574973\pi\)
−0.233363 + 0.972390i \(0.574973\pi\)
\(978\) −4.28719 −0.137089
\(979\) 24.9282 0.796709
\(980\) 0 0
\(981\) 11.0000 0.351203
\(982\) −7.50258 −0.239417
\(983\) −20.1962 −0.644157 −0.322079 0.946713i \(-0.604382\pi\)
−0.322079 + 0.946713i \(0.604382\pi\)
\(984\) 6.92820 0.220863
\(985\) 0 0
\(986\) 30.5359 0.972461
\(987\) 0 0
\(988\) −20.6795 −0.657902
\(989\) 9.12436 0.290138
\(990\) 0 0
\(991\) 55.1051 1.75047 0.875236 0.483696i \(-0.160706\pi\)
0.875236 + 0.483696i \(0.160706\pi\)
\(992\) −37.8564 −1.20194
\(993\) 21.9282 0.695870
\(994\) 0 0
\(995\) 0 0
\(996\) −13.3590 −0.423296
\(997\) 4.01924 0.127291 0.0636453 0.997973i \(-0.479727\pi\)
0.0636453 + 0.997973i \(0.479727\pi\)
\(998\) 14.9808 0.474208
\(999\) −7.19615 −0.227676
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 3675.2.a.bg.1.1 2
5.4 even 2 735.2.a.g.1.2 2
7.2 even 3 525.2.i.f.151.2 4
7.4 even 3 525.2.i.f.226.2 4
7.6 odd 2 3675.2.a.be.1.1 2
15.14 odd 2 2205.2.a.z.1.1 2
35.2 odd 12 525.2.r.a.424.2 4
35.4 even 6 105.2.i.d.16.1 4
35.9 even 6 105.2.i.d.46.1 yes 4
35.18 odd 12 525.2.r.a.499.2 4
35.19 odd 6 735.2.i.l.361.1 4
35.23 odd 12 525.2.r.f.424.1 4
35.24 odd 6 735.2.i.l.226.1 4
35.32 odd 12 525.2.r.f.499.1 4
35.34 odd 2 735.2.a.h.1.2 2
105.44 odd 6 315.2.j.c.46.2 4
105.74 odd 6 315.2.j.c.226.2 4
105.104 even 2 2205.2.a.ba.1.1 2
140.39 odd 6 1680.2.bg.o.961.2 4
140.79 odd 6 1680.2.bg.o.1201.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.i.d.16.1 4 35.4 even 6
105.2.i.d.46.1 yes 4 35.9 even 6
315.2.j.c.46.2 4 105.44 odd 6
315.2.j.c.226.2 4 105.74 odd 6
525.2.i.f.151.2 4 7.2 even 3
525.2.i.f.226.2 4 7.4 even 3
525.2.r.a.424.2 4 35.2 odd 12
525.2.r.a.499.2 4 35.18 odd 12
525.2.r.f.424.1 4 35.23 odd 12
525.2.r.f.499.1 4 35.32 odd 12
735.2.a.g.1.2 2 5.4 even 2
735.2.a.h.1.2 2 35.34 odd 2
735.2.i.l.226.1 4 35.24 odd 6
735.2.i.l.361.1 4 35.19 odd 6
1680.2.bg.o.961.2 4 140.39 odd 6
1680.2.bg.o.1201.2 4 140.79 odd 6
2205.2.a.z.1.1 2 15.14 odd 2
2205.2.a.ba.1.1 2 105.104 even 2
3675.2.a.be.1.1 2 7.6 odd 2
3675.2.a.bg.1.1 2 1.1 even 1 trivial