Properties

Label 7623.2.a.x.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051 q^{2} -1.46410 q^{4} +0.267949 q^{5} -1.00000 q^{7} -2.53590 q^{8} +O(q^{10})\) \(q+0.732051 q^{2} -1.46410 q^{4} +0.267949 q^{5} -1.00000 q^{7} -2.53590 q^{8} +0.196152 q^{10} +2.73205 q^{13} -0.732051 q^{14} +1.07180 q^{16} +3.73205 q^{17} -2.19615 q^{19} -0.392305 q^{20} -3.26795 q^{23} -4.92820 q^{25} +2.00000 q^{26} +1.46410 q^{28} -1.26795 q^{29} +7.46410 q^{31} +5.85641 q^{32} +2.73205 q^{34} -0.267949 q^{35} -9.46410 q^{37} -1.60770 q^{38} -0.679492 q^{40} -4.00000 q^{41} +0.464102 q^{43} -2.39230 q^{46} +9.19615 q^{47} +1.00000 q^{49} -3.60770 q^{50} -4.00000 q^{52} -2.53590 q^{53} +2.53590 q^{56} -0.928203 q^{58} +10.1244 q^{59} +8.19615 q^{61} +5.46410 q^{62} +2.14359 q^{64} +0.732051 q^{65} -7.00000 q^{67} -5.46410 q^{68} -0.196152 q^{70} -9.12436 q^{71} -6.73205 q^{73} -6.92820 q^{74} +3.21539 q^{76} -0.535898 q^{79} +0.287187 q^{80} -2.92820 q^{82} -4.26795 q^{83} +1.00000 q^{85} +0.339746 q^{86} +14.6603 q^{89} -2.73205 q^{91} +4.78461 q^{92} +6.73205 q^{94} -0.588457 q^{95} +3.26795 q^{97} +0.732051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 4q^{4} + 4q^{5} - 2q^{7} - 12q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 4q^{4} + 4q^{5} - 2q^{7} - 12q^{8} - 10q^{10} + 2q^{13} + 2q^{14} + 16q^{16} + 4q^{17} + 6q^{19} + 20q^{20} - 10q^{23} + 4q^{25} + 4q^{26} - 4q^{28} - 6q^{29} + 8q^{31} - 16q^{32} + 2q^{34} - 4q^{35} - 12q^{37} - 24q^{38} - 36q^{40} - 8q^{41} - 6q^{43} + 16q^{46} + 8q^{47} + 2q^{49} - 28q^{50} - 8q^{52} - 12q^{53} + 12q^{56} + 12q^{58} - 4q^{59} + 6q^{61} + 4q^{62} + 32q^{64} - 2q^{65} - 14q^{67} - 4q^{68} + 10q^{70} + 6q^{71} - 10q^{73} + 48q^{76} - 8q^{79} + 56q^{80} + 8q^{82} - 12q^{83} + 2q^{85} + 18q^{86} + 12q^{89} - 2q^{91} - 32q^{92} + 10q^{94} + 30q^{95} + 10q^{97} - 2q^{98} + O(q^{100}) \)

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 \(-0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(3\) 0 0
\(4\) −1.46410 −0.732051
\(5\) 0.267949 0.119831 0.0599153 0.998203i \(-0.480917\pi\)
0.0599153 + 0.998203i \(0.480917\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.53590 −0.896575
\(9\) 0 0
\(10\) 0.196152 0.0620288
\(11\) 0 0
\(12\) 0 0
\(13\) 2.73205 0.757735 0.378867 0.925451i \(-0.376314\pi\)
0.378867 + 0.925451i \(0.376314\pi\)
\(14\) −0.732051 −0.195649
\(15\) 0 0
\(16\) 1.07180 0.267949
\(17\) 3.73205 0.905155 0.452578 0.891725i \(-0.350505\pi\)
0.452578 + 0.891725i \(0.350505\pi\)
\(18\) 0 0
\(19\) −2.19615 −0.503832 −0.251916 0.967749i \(-0.581061\pi\)
−0.251916 + 0.967749i \(0.581061\pi\)
\(20\) −0.392305 −0.0877220
\(21\) 0 0
\(22\) 0 0
\(23\) −3.26795 −0.681415 −0.340707 0.940169i \(-0.610666\pi\)
−0.340707 + 0.940169i \(0.610666\pi\)
\(24\) 0 0
\(25\) −4.92820 −0.985641
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 1.46410 0.276689
\(29\) −1.26795 −0.235452 −0.117726 0.993046i \(-0.537560\pi\)
−0.117726 + 0.993046i \(0.537560\pi\)
\(30\) 0 0
\(31\) 7.46410 1.34059 0.670296 0.742094i \(-0.266167\pi\)
0.670296 + 0.742094i \(0.266167\pi\)
\(32\) 5.85641 1.03528
\(33\) 0 0
\(34\) 2.73205 0.468543
\(35\) −0.267949 −0.0452917
\(36\) 0 0
\(37\) −9.46410 −1.55589 −0.777944 0.628333i \(-0.783737\pi\)
−0.777944 + 0.628333i \(0.783737\pi\)
\(38\) −1.60770 −0.260803
\(39\) 0 0
\(40\) −0.679492 −0.107437
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 0.464102 0.0707748 0.0353874 0.999374i \(-0.488733\pi\)
0.0353874 + 0.999374i \(0.488733\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.39230 −0.352726
\(47\) 9.19615 1.34140 0.670698 0.741730i \(-0.265995\pi\)
0.670698 + 0.741730i \(0.265995\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −3.60770 −0.510205
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −2.53590 −0.348332 −0.174166 0.984716i \(-0.555723\pi\)
−0.174166 + 0.984716i \(0.555723\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.53590 0.338874
\(57\) 0 0
\(58\) −0.928203 −0.121879
\(59\) 10.1244 1.31808 0.659039 0.752108i \(-0.270963\pi\)
0.659039 + 0.752108i \(0.270963\pi\)
\(60\) 0 0
\(61\) 8.19615 1.04941 0.524705 0.851284i \(-0.324176\pi\)
0.524705 + 0.851284i \(0.324176\pi\)
\(62\) 5.46410 0.693942
\(63\) 0 0
\(64\) 2.14359 0.267949
\(65\) 0.732051 0.0907997
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) −5.46410 −0.662620
\(69\) 0 0
\(70\) −0.196152 −0.0234447
\(71\) −9.12436 −1.08286 −0.541431 0.840745i \(-0.682117\pi\)
−0.541431 + 0.840745i \(0.682117\pi\)
\(72\) 0 0
\(73\) −6.73205 −0.787927 −0.393963 0.919126i \(-0.628896\pi\)
−0.393963 + 0.919126i \(0.628896\pi\)
\(74\) −6.92820 −0.805387
\(75\) 0 0
\(76\) 3.21539 0.368831
\(77\) 0 0
\(78\) 0 0
\(79\) −0.535898 −0.0602933 −0.0301466 0.999545i \(-0.509597\pi\)
−0.0301466 + 0.999545i \(0.509597\pi\)
\(80\) 0.287187 0.0321085
\(81\) 0 0
\(82\) −2.92820 −0.323366
\(83\) −4.26795 −0.468468 −0.234234 0.972180i \(-0.575258\pi\)
−0.234234 + 0.972180i \(0.575258\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 0.339746 0.0366357
\(87\) 0 0
\(88\) 0 0
\(89\) 14.6603 1.55398 0.776992 0.629511i \(-0.216745\pi\)
0.776992 + 0.629511i \(0.216745\pi\)
\(90\) 0 0
\(91\) −2.73205 −0.286397
\(92\) 4.78461 0.498830
\(93\) 0 0
\(94\) 6.73205 0.694358
\(95\) −0.588457 −0.0603744
\(96\) 0 0
\(97\) 3.26795 0.331810 0.165905 0.986142i \(-0.446945\pi\)
0.165905 + 0.986142i \(0.446945\pi\)
\(98\) 0.732051 0.0739483
\(99\) 0 0
\(100\) 7.21539 0.721539
\(101\) −9.73205 −0.968375 −0.484188 0.874964i \(-0.660885\pi\)
−0.484188 + 0.874964i \(0.660885\pi\)
\(102\) 0 0
\(103\) 4.19615 0.413459 0.206730 0.978398i \(-0.433718\pi\)
0.206730 + 0.978398i \(0.433718\pi\)
\(104\) −6.92820 −0.679366
\(105\) 0 0
\(106\) −1.85641 −0.180310
\(107\) −18.5885 −1.79701 −0.898507 0.438959i \(-0.855347\pi\)
−0.898507 + 0.438959i \(0.855347\pi\)
\(108\) 0 0
\(109\) 17.3923 1.66588 0.832940 0.553363i \(-0.186656\pi\)
0.832940 + 0.553363i \(0.186656\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.07180 −0.101275
\(113\) −8.53590 −0.802990 −0.401495 0.915861i \(-0.631509\pi\)
−0.401495 + 0.915861i \(0.631509\pi\)
\(114\) 0 0
\(115\) −0.875644 −0.0816543
\(116\) 1.85641 0.172363
\(117\) 0 0
\(118\) 7.41154 0.682288
\(119\) −3.73205 −0.342117
\(120\) 0 0
\(121\) 0 0
\(122\) 6.00000 0.543214
\(123\) 0 0
\(124\) −10.9282 −0.981382
\(125\) −2.66025 −0.237940
\(126\) 0 0
\(127\) −3.53590 −0.313760 −0.156880 0.987618i \(-0.550144\pi\)
−0.156880 + 0.987618i \(0.550144\pi\)
\(128\) −10.1436 −0.896575
\(129\) 0 0
\(130\) 0.535898 0.0470014
\(131\) −5.19615 −0.453990 −0.226995 0.973896i \(-0.572890\pi\)
−0.226995 + 0.973896i \(0.572890\pi\)
\(132\) 0 0
\(133\) 2.19615 0.190431
\(134\) −5.12436 −0.442677
\(135\) 0 0
\(136\) −9.46410 −0.811540
\(137\) −13.6603 −1.16707 −0.583537 0.812086i \(-0.698332\pi\)
−0.583537 + 0.812086i \(0.698332\pi\)
\(138\) 0 0
\(139\) 7.80385 0.661914 0.330957 0.943646i \(-0.392629\pi\)
0.330957 + 0.943646i \(0.392629\pi\)
\(140\) 0.392305 0.0331558
\(141\) 0 0
\(142\) −6.67949 −0.560531
\(143\) 0 0
\(144\) 0 0
\(145\) −0.339746 −0.0282144
\(146\) −4.92820 −0.407861
\(147\) 0 0
\(148\) 13.8564 1.13899
\(149\) 2.73205 0.223818 0.111909 0.993718i \(-0.464303\pi\)
0.111909 + 0.993718i \(0.464303\pi\)
\(150\) 0 0
\(151\) 9.39230 0.764335 0.382167 0.924093i \(-0.375178\pi\)
0.382167 + 0.924093i \(0.375178\pi\)
\(152\) 5.56922 0.451723
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −6.53590 −0.521621 −0.260811 0.965390i \(-0.583990\pi\)
−0.260811 + 0.965390i \(0.583990\pi\)
\(158\) −0.392305 −0.0312101
\(159\) 0 0
\(160\) 1.56922 0.124058
\(161\) 3.26795 0.257550
\(162\) 0 0
\(163\) 7.46410 0.584634 0.292317 0.956322i \(-0.405574\pi\)
0.292317 + 0.956322i \(0.405574\pi\)
\(164\) 5.85641 0.457309
\(165\) 0 0
\(166\) −3.12436 −0.242497
\(167\) −25.0526 −1.93863 −0.969313 0.245831i \(-0.920939\pi\)
−0.969313 + 0.245831i \(0.920939\pi\)
\(168\) 0 0
\(169\) −5.53590 −0.425838
\(170\) 0.732051 0.0561457
\(171\) 0 0
\(172\) −0.679492 −0.0518108
\(173\) −5.19615 −0.395056 −0.197528 0.980297i \(-0.563291\pi\)
−0.197528 + 0.980297i \(0.563291\pi\)
\(174\) 0 0
\(175\) 4.92820 0.372537
\(176\) 0 0
\(177\) 0 0
\(178\) 10.7321 0.804401
\(179\) 4.39230 0.328296 0.164148 0.986436i \(-0.447512\pi\)
0.164148 + 0.986436i \(0.447512\pi\)
\(180\) 0 0
\(181\) −8.00000 −0.594635 −0.297318 0.954779i \(-0.596092\pi\)
−0.297318 + 0.954779i \(0.596092\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) 8.28719 0.610940
\(185\) −2.53590 −0.186443
\(186\) 0 0
\(187\) 0 0
\(188\) −13.4641 −0.981971
\(189\) 0 0
\(190\) −0.430781 −0.0312521
\(191\) −16.1962 −1.17191 −0.585956 0.810343i \(-0.699281\pi\)
−0.585956 + 0.810343i \(0.699281\pi\)
\(192\) 0 0
\(193\) −5.00000 −0.359908 −0.179954 0.983675i \(-0.557595\pi\)
−0.179954 + 0.983675i \(0.557595\pi\)
\(194\) 2.39230 0.171757
\(195\) 0 0
\(196\) −1.46410 −0.104579
\(197\) −12.5359 −0.893146 −0.446573 0.894747i \(-0.647356\pi\)
−0.446573 + 0.894747i \(0.647356\pi\)
\(198\) 0 0
\(199\) −20.1962 −1.43167 −0.715834 0.698271i \(-0.753953\pi\)
−0.715834 + 0.698271i \(0.753953\pi\)
\(200\) 12.4974 0.883701
\(201\) 0 0
\(202\) −7.12436 −0.501268
\(203\) 1.26795 0.0889926
\(204\) 0 0
\(205\) −1.07180 −0.0748575
\(206\) 3.07180 0.214022
\(207\) 0 0
\(208\) 2.92820 0.203034
\(209\) 0 0
\(210\) 0 0
\(211\) −17.7846 −1.22434 −0.612172 0.790725i \(-0.709704\pi\)
−0.612172 + 0.790725i \(0.709704\pi\)
\(212\) 3.71281 0.254997
\(213\) 0 0
\(214\) −13.6077 −0.930203
\(215\) 0.124356 0.00848099
\(216\) 0 0
\(217\) −7.46410 −0.506696
\(218\) 12.7321 0.862323
\(219\) 0 0
\(220\) 0 0
\(221\) 10.1962 0.685867
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) −5.85641 −0.391298
\(225\) 0 0
\(226\) −6.24871 −0.415658
\(227\) −14.6603 −0.973035 −0.486518 0.873671i \(-0.661733\pi\)
−0.486518 + 0.873671i \(0.661733\pi\)
\(228\) 0 0
\(229\) 6.92820 0.457829 0.228914 0.973447i \(-0.426482\pi\)
0.228914 + 0.973447i \(0.426482\pi\)
\(230\) −0.641016 −0.0422674
\(231\) 0 0
\(232\) 3.21539 0.211101
\(233\) −20.1962 −1.32309 −0.661547 0.749904i \(-0.730100\pi\)
−0.661547 + 0.749904i \(0.730100\pi\)
\(234\) 0 0
\(235\) 2.46410 0.160740
\(236\) −14.8231 −0.964901
\(237\) 0 0
\(238\) −2.73205 −0.177093
\(239\) −24.9282 −1.61247 −0.806236 0.591594i \(-0.798499\pi\)
−0.806236 + 0.591594i \(0.798499\pi\)
\(240\) 0 0
\(241\) 15.6603 1.00877 0.504383 0.863480i \(-0.331720\pi\)
0.504383 + 0.863480i \(0.331720\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −12.0000 −0.768221
\(245\) 0.267949 0.0171186
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) −18.9282 −1.20194
\(249\) 0 0
\(250\) −1.94744 −0.123167
\(251\) 1.07180 0.0676512 0.0338256 0.999428i \(-0.489231\pi\)
0.0338256 + 0.999428i \(0.489231\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.58846 −0.162414
\(255\) 0 0
\(256\) −11.7128 −0.732051
\(257\) −4.12436 −0.257270 −0.128635 0.991692i \(-0.541060\pi\)
−0.128635 + 0.991692i \(0.541060\pi\)
\(258\) 0 0
\(259\) 9.46410 0.588071
\(260\) −1.07180 −0.0664700
\(261\) 0 0
\(262\) −3.80385 −0.235002
\(263\) −5.80385 −0.357881 −0.178940 0.983860i \(-0.557267\pi\)
−0.178940 + 0.983860i \(0.557267\pi\)
\(264\) 0 0
\(265\) −0.679492 −0.0417409
\(266\) 1.60770 0.0985741
\(267\) 0 0
\(268\) 10.2487 0.626040
\(269\) −23.8564 −1.45455 −0.727275 0.686346i \(-0.759214\pi\)
−0.727275 + 0.686346i \(0.759214\pi\)
\(270\) 0 0
\(271\) 3.60770 0.219152 0.109576 0.993978i \(-0.465051\pi\)
0.109576 + 0.993978i \(0.465051\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) 0 0
\(276\) 0 0
\(277\) −25.7846 −1.54925 −0.774624 0.632423i \(-0.782061\pi\)
−0.774624 + 0.632423i \(0.782061\pi\)
\(278\) 5.71281 0.342632
\(279\) 0 0
\(280\) 0.679492 0.0406074
\(281\) 27.7128 1.65321 0.826604 0.562784i \(-0.190270\pi\)
0.826604 + 0.562784i \(0.190270\pi\)
\(282\) 0 0
\(283\) −26.2487 −1.56032 −0.780162 0.625578i \(-0.784863\pi\)
−0.780162 + 0.625578i \(0.784863\pi\)
\(284\) 13.3590 0.792710
\(285\) 0 0
\(286\) 0 0
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) −3.07180 −0.180694
\(290\) −0.248711 −0.0146048
\(291\) 0 0
\(292\) 9.85641 0.576803
\(293\) 16.2679 0.950384 0.475192 0.879882i \(-0.342379\pi\)
0.475192 + 0.879882i \(0.342379\pi\)
\(294\) 0 0
\(295\) 2.71281 0.157946
\(296\) 24.0000 1.39497
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) −8.92820 −0.516331
\(300\) 0 0
\(301\) −0.464102 −0.0267504
\(302\) 6.87564 0.395649
\(303\) 0 0
\(304\) −2.35383 −0.135001
\(305\) 2.19615 0.125751
\(306\) 0 0
\(307\) 26.5885 1.51748 0.758742 0.651392i \(-0.225814\pi\)
0.758742 + 0.651392i \(0.225814\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.46410 0.0831554
\(311\) −19.0526 −1.08037 −0.540186 0.841546i \(-0.681646\pi\)
−0.540186 + 0.841546i \(0.681646\pi\)
\(312\) 0 0
\(313\) 0.392305 0.0221744 0.0110872 0.999939i \(-0.496471\pi\)
0.0110872 + 0.999939i \(0.496471\pi\)
\(314\) −4.78461 −0.270011
\(315\) 0 0
\(316\) 0.784610 0.0441377
\(317\) −29.5167 −1.65782 −0.828910 0.559381i \(-0.811039\pi\)
−0.828910 + 0.559381i \(0.811039\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.574374 0.0321085
\(321\) 0 0
\(322\) 2.39230 0.133318
\(323\) −8.19615 −0.456046
\(324\) 0 0
\(325\) −13.4641 −0.746854
\(326\) 5.46410 0.302629
\(327\) 0 0
\(328\) 10.1436 0.560086
\(329\) −9.19615 −0.507000
\(330\) 0 0
\(331\) 26.1769 1.43881 0.719407 0.694589i \(-0.244414\pi\)
0.719407 + 0.694589i \(0.244414\pi\)
\(332\) 6.24871 0.342943
\(333\) 0 0
\(334\) −18.3397 −1.00351
\(335\) −1.87564 −0.102477
\(336\) 0 0
\(337\) −28.0000 −1.52526 −0.762629 0.646837i \(-0.776092\pi\)
−0.762629 + 0.646837i \(0.776092\pi\)
\(338\) −4.05256 −0.220430
\(339\) 0 0
\(340\) −1.46410 −0.0794021
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −1.17691 −0.0634550
\(345\) 0 0
\(346\) −3.80385 −0.204496
\(347\) −26.0526 −1.39857 −0.699287 0.714841i \(-0.746499\pi\)
−0.699287 + 0.714841i \(0.746499\pi\)
\(348\) 0 0
\(349\) 1.60770 0.0860579 0.0430290 0.999074i \(-0.486299\pi\)
0.0430290 + 0.999074i \(0.486299\pi\)
\(350\) 3.60770 0.192839
\(351\) 0 0
\(352\) 0 0
\(353\) 11.0718 0.589292 0.294646 0.955606i \(-0.404798\pi\)
0.294646 + 0.955606i \(0.404798\pi\)
\(354\) 0 0
\(355\) −2.44486 −0.129760
\(356\) −21.4641 −1.13760
\(357\) 0 0
\(358\) 3.21539 0.169939
\(359\) −25.1244 −1.32601 −0.663006 0.748614i \(-0.730720\pi\)
−0.663006 + 0.748614i \(0.730720\pi\)
\(360\) 0 0
\(361\) −14.1769 −0.746153
\(362\) −5.85641 −0.307806
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −1.80385 −0.0944177
\(366\) 0 0
\(367\) −12.7321 −0.664608 −0.332304 0.943172i \(-0.607826\pi\)
−0.332304 + 0.943172i \(0.607826\pi\)
\(368\) −3.50258 −0.182584
\(369\) 0 0
\(370\) −1.85641 −0.0965100
\(371\) 2.53590 0.131657
\(372\) 0 0
\(373\) −23.9282 −1.23896 −0.619478 0.785014i \(-0.712656\pi\)
−0.619478 + 0.785014i \(0.712656\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −23.3205 −1.20266
\(377\) −3.46410 −0.178410
\(378\) 0 0
\(379\) 3.53590 0.181627 0.0908135 0.995868i \(-0.471053\pi\)
0.0908135 + 0.995868i \(0.471053\pi\)
\(380\) 0.861561 0.0441972
\(381\) 0 0
\(382\) −11.8564 −0.606627
\(383\) 34.1244 1.74367 0.871837 0.489797i \(-0.162929\pi\)
0.871837 + 0.489797i \(0.162929\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.66025 −0.186302
\(387\) 0 0
\(388\) −4.78461 −0.242902
\(389\) −16.3923 −0.831123 −0.415561 0.909565i \(-0.636415\pi\)
−0.415561 + 0.909565i \(0.636415\pi\)
\(390\) 0 0
\(391\) −12.1962 −0.616786
\(392\) −2.53590 −0.128082
\(393\) 0 0
\(394\) −9.17691 −0.462326
\(395\) −0.143594 −0.00722498
\(396\) 0 0
\(397\) −13.3205 −0.668537 −0.334269 0.942478i \(-0.608489\pi\)
−0.334269 + 0.942478i \(0.608489\pi\)
\(398\) −14.7846 −0.741086
\(399\) 0 0
\(400\) −5.28203 −0.264102
\(401\) 16.0000 0.799002 0.399501 0.916733i \(-0.369183\pi\)
0.399501 + 0.916733i \(0.369183\pi\)
\(402\) 0 0
\(403\) 20.3923 1.01581
\(404\) 14.2487 0.708900
\(405\) 0 0
\(406\) 0.928203 0.0460660
\(407\) 0 0
\(408\) 0 0
\(409\) −23.6603 −1.16992 −0.584962 0.811061i \(-0.698891\pi\)
−0.584962 + 0.811061i \(0.698891\pi\)
\(410\) −0.784610 −0.0387491
\(411\) 0 0
\(412\) −6.14359 −0.302673
\(413\) −10.1244 −0.498187
\(414\) 0 0
\(415\) −1.14359 −0.0561368
\(416\) 16.0000 0.784465
\(417\) 0 0
\(418\) 0 0
\(419\) −9.73205 −0.475442 −0.237721 0.971334i \(-0.576400\pi\)
−0.237721 + 0.971334i \(0.576400\pi\)
\(420\) 0 0
\(421\) 38.8564 1.89375 0.946873 0.321609i \(-0.104224\pi\)
0.946873 + 0.321609i \(0.104224\pi\)
\(422\) −13.0192 −0.633767
\(423\) 0 0
\(424\) 6.43078 0.312306
\(425\) −18.3923 −0.892158
\(426\) 0 0
\(427\) −8.19615 −0.396640
\(428\) 27.2154 1.31551
\(429\) 0 0
\(430\) 0.0910347 0.00439008
\(431\) −23.3205 −1.12331 −0.561655 0.827372i \(-0.689835\pi\)
−0.561655 + 0.827372i \(0.689835\pi\)
\(432\) 0 0
\(433\) 33.9090 1.62956 0.814780 0.579770i \(-0.196857\pi\)
0.814780 + 0.579770i \(0.196857\pi\)
\(434\) −5.46410 −0.262285
\(435\) 0 0
\(436\) −25.4641 −1.21951
\(437\) 7.17691 0.343318
\(438\) 0 0
\(439\) −0.143594 −0.00685335 −0.00342667 0.999994i \(-0.501091\pi\)
−0.00342667 + 0.999994i \(0.501091\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 7.46410 0.355031
\(443\) 5.80385 0.275749 0.137875 0.990450i \(-0.455973\pi\)
0.137875 + 0.990450i \(0.455973\pi\)
\(444\) 0 0
\(445\) 3.92820 0.186215
\(446\) 11.7128 0.554618
\(447\) 0 0
\(448\) −2.14359 −0.101275
\(449\) 16.2487 0.766824 0.383412 0.923577i \(-0.374749\pi\)
0.383412 + 0.923577i \(0.374749\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 12.4974 0.587829
\(453\) 0 0
\(454\) −10.7321 −0.503680
\(455\) −0.732051 −0.0343191
\(456\) 0 0
\(457\) −33.9282 −1.58709 −0.793547 0.608509i \(-0.791768\pi\)
−0.793547 + 0.608509i \(0.791768\pi\)
\(458\) 5.07180 0.236989
\(459\) 0 0
\(460\) 1.28203 0.0597751
\(461\) 27.5885 1.28492 0.642461 0.766318i \(-0.277913\pi\)
0.642461 + 0.766318i \(0.277913\pi\)
\(462\) 0 0
\(463\) 31.4641 1.46226 0.731130 0.682238i \(-0.238993\pi\)
0.731130 + 0.682238i \(0.238993\pi\)
\(464\) −1.35898 −0.0630892
\(465\) 0 0
\(466\) −14.7846 −0.684884
\(467\) 34.3923 1.59149 0.795743 0.605634i \(-0.207081\pi\)
0.795743 + 0.605634i \(0.207081\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 1.80385 0.0832053
\(471\) 0 0
\(472\) −25.6743 −1.18176
\(473\) 0 0
\(474\) 0 0
\(475\) 10.8231 0.496597
\(476\) 5.46410 0.250447
\(477\) 0 0
\(478\) −18.2487 −0.834677
\(479\) −32.3731 −1.47916 −0.739582 0.673067i \(-0.764977\pi\)
−0.739582 + 0.673067i \(0.764977\pi\)
\(480\) 0 0
\(481\) −25.8564 −1.17895
\(482\) 11.4641 0.522176
\(483\) 0 0
\(484\) 0 0
\(485\) 0.875644 0.0397610
\(486\) 0 0
\(487\) 42.1769 1.91122 0.955609 0.294637i \(-0.0951988\pi\)
0.955609 + 0.294637i \(0.0951988\pi\)
\(488\) −20.7846 −0.940875
\(489\) 0 0
\(490\) 0.196152 0.00886126
\(491\) 0.0525589 0.00237195 0.00118597 0.999999i \(-0.499622\pi\)
0.00118597 + 0.999999i \(0.499622\pi\)
\(492\) 0 0
\(493\) −4.73205 −0.213121
\(494\) −4.39230 −0.197619
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 9.12436 0.409283
\(498\) 0 0
\(499\) 40.3205 1.80499 0.902497 0.430696i \(-0.141732\pi\)
0.902497 + 0.430696i \(0.141732\pi\)
\(500\) 3.89488 0.174184
\(501\) 0 0
\(502\) 0.784610 0.0350188
\(503\) 18.5167 0.825617 0.412809 0.910818i \(-0.364548\pi\)
0.412809 + 0.910818i \(0.364548\pi\)
\(504\) 0 0
\(505\) −2.60770 −0.116041
\(506\) 0 0
\(507\) 0 0
\(508\) 5.17691 0.229688
\(509\) 17.1962 0.762206 0.381103 0.924533i \(-0.375544\pi\)
0.381103 + 0.924533i \(0.375544\pi\)
\(510\) 0 0
\(511\) 6.73205 0.297808
\(512\) 11.7128 0.517638
\(513\) 0 0
\(514\) −3.01924 −0.133173
\(515\) 1.12436 0.0495450
\(516\) 0 0
\(517\) 0 0
\(518\) 6.92820 0.304408
\(519\) 0 0
\(520\) −1.85641 −0.0814088
\(521\) 19.9808 0.875373 0.437687 0.899128i \(-0.355798\pi\)
0.437687 + 0.899128i \(0.355798\pi\)
\(522\) 0 0
\(523\) 26.5885 1.16263 0.581316 0.813678i \(-0.302538\pi\)
0.581316 + 0.813678i \(0.302538\pi\)
\(524\) 7.60770 0.332344
\(525\) 0 0
\(526\) −4.24871 −0.185253
\(527\) 27.8564 1.21344
\(528\) 0 0
\(529\) −12.3205 −0.535674
\(530\) −0.497423 −0.0216067
\(531\) 0 0
\(532\) −3.21539 −0.139405
\(533\) −10.9282 −0.473353
\(534\) 0 0
\(535\) −4.98076 −0.215337
\(536\) 17.7513 0.766739
\(537\) 0 0
\(538\) −17.4641 −0.752931
\(539\) 0 0
\(540\) 0 0
\(541\) 41.3923 1.77959 0.889797 0.456356i \(-0.150846\pi\)
0.889797 + 0.456356i \(0.150846\pi\)
\(542\) 2.64102 0.113441
\(543\) 0 0
\(544\) 21.8564 0.937086
\(545\) 4.66025 0.199623
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 20.0000 0.854358
\(549\) 0 0
\(550\) 0 0
\(551\) 2.78461 0.118628
\(552\) 0 0
\(553\) 0.535898 0.0227887
\(554\) −18.8756 −0.801949
\(555\) 0 0
\(556\) −11.4256 −0.484554
\(557\) −16.9282 −0.717271 −0.358635 0.933478i \(-0.616758\pi\)
−0.358635 + 0.933478i \(0.616758\pi\)
\(558\) 0 0
\(559\) 1.26795 0.0536285
\(560\) −0.287187 −0.0121359
\(561\) 0 0
\(562\) 20.2872 0.855763
\(563\) 9.73205 0.410157 0.205079 0.978746i \(-0.434255\pi\)
0.205079 + 0.978746i \(0.434255\pi\)
\(564\) 0 0
\(565\) −2.28719 −0.0962227
\(566\) −19.2154 −0.807683
\(567\) 0 0
\(568\) 23.1384 0.970867
\(569\) −25.2679 −1.05929 −0.529644 0.848220i \(-0.677674\pi\)
−0.529644 + 0.848220i \(0.677674\pi\)
\(570\) 0 0
\(571\) 1.07180 0.0448533 0.0224266 0.999748i \(-0.492861\pi\)
0.0224266 + 0.999748i \(0.492861\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 2.92820 0.122221
\(575\) 16.1051 0.671630
\(576\) 0 0
\(577\) −14.4449 −0.601348 −0.300674 0.953727i \(-0.597212\pi\)
−0.300674 + 0.953727i \(0.597212\pi\)
\(578\) −2.24871 −0.0935341
\(579\) 0 0
\(580\) 0.497423 0.0206543
\(581\) 4.26795 0.177064
\(582\) 0 0
\(583\) 0 0
\(584\) 17.0718 0.706436
\(585\) 0 0
\(586\) 11.9090 0.491955
\(587\) 8.12436 0.335328 0.167664 0.985844i \(-0.446378\pi\)
0.167664 + 0.985844i \(0.446378\pi\)
\(588\) 0 0
\(589\) −16.3923 −0.675433
\(590\) 1.98592 0.0817589
\(591\) 0 0
\(592\) −10.1436 −0.416899
\(593\) −35.5885 −1.46144 −0.730721 0.682676i \(-0.760816\pi\)
−0.730721 + 0.682676i \(0.760816\pi\)
\(594\) 0 0
\(595\) −1.00000 −0.0409960
\(596\) −4.00000 −0.163846
\(597\) 0 0
\(598\) −6.53590 −0.267273
\(599\) −23.1769 −0.946983 −0.473492 0.880798i \(-0.657007\pi\)
−0.473492 + 0.880798i \(0.657007\pi\)
\(600\) 0 0
\(601\) 31.9090 1.30159 0.650797 0.759252i \(-0.274435\pi\)
0.650797 + 0.759252i \(0.274435\pi\)
\(602\) −0.339746 −0.0138470
\(603\) 0 0
\(604\) −13.7513 −0.559532
\(605\) 0 0
\(606\) 0 0
\(607\) −4.58846 −0.186240 −0.0931199 0.995655i \(-0.529684\pi\)
−0.0931199 + 0.995655i \(0.529684\pi\)
\(608\) −12.8616 −0.521605
\(609\) 0 0
\(610\) 1.60770 0.0650937
\(611\) 25.1244 1.01642
\(612\) 0 0
\(613\) −14.6077 −0.589999 −0.295000 0.955497i \(-0.595320\pi\)
−0.295000 + 0.955497i \(0.595320\pi\)
\(614\) 19.4641 0.785507
\(615\) 0 0
\(616\) 0 0
\(617\) 39.4641 1.58876 0.794382 0.607418i \(-0.207795\pi\)
0.794382 + 0.607418i \(0.207795\pi\)
\(618\) 0 0
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) −2.92820 −0.117599
\(621\) 0 0
\(622\) −13.9474 −0.559241
\(623\) −14.6603 −0.587351
\(624\) 0 0
\(625\) 23.9282 0.957128
\(626\) 0.287187 0.0114783
\(627\) 0 0
\(628\) 9.56922 0.381853
\(629\) −35.3205 −1.40832
\(630\) 0 0
\(631\) −0.856406 −0.0340930 −0.0170465 0.999855i \(-0.505426\pi\)
−0.0170465 + 0.999855i \(0.505426\pi\)
\(632\) 1.35898 0.0540575
\(633\) 0 0
\(634\) −21.6077 −0.858151
\(635\) −0.947441 −0.0375981
\(636\) 0 0
\(637\) 2.73205 0.108248
\(638\) 0 0
\(639\) 0 0
\(640\) −2.71797 −0.107437
\(641\) −35.9090 −1.41832 −0.709159 0.705048i \(-0.750925\pi\)
−0.709159 + 0.705048i \(0.750925\pi\)
\(642\) 0 0
\(643\) −41.8564 −1.65066 −0.825328 0.564654i \(-0.809010\pi\)
−0.825328 + 0.564654i \(0.809010\pi\)
\(644\) −4.78461 −0.188540
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) 27.1962 1.06919 0.534596 0.845108i \(-0.320464\pi\)
0.534596 + 0.845108i \(0.320464\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −9.85641 −0.386600
\(651\) 0 0
\(652\) −10.9282 −0.427981
\(653\) −20.9808 −0.821041 −0.410520 0.911851i \(-0.634653\pi\)
−0.410520 + 0.911851i \(0.634653\pi\)
\(654\) 0 0
\(655\) −1.39230 −0.0544019
\(656\) −4.28719 −0.167387
\(657\) 0 0
\(658\) −6.73205 −0.262443
\(659\) 3.51666 0.136990 0.0684948 0.997651i \(-0.478180\pi\)
0.0684948 + 0.997651i \(0.478180\pi\)
\(660\) 0 0
\(661\) −45.8564 −1.78361 −0.891804 0.452422i \(-0.850560\pi\)
−0.891804 + 0.452422i \(0.850560\pi\)
\(662\) 19.1628 0.744785
\(663\) 0 0
\(664\) 10.8231 0.420017
\(665\) 0.588457 0.0228194
\(666\) 0 0
\(667\) 4.14359 0.160441
\(668\) 36.6795 1.41917
\(669\) 0 0
\(670\) −1.37307 −0.0530462
\(671\) 0 0
\(672\) 0 0
\(673\) −8.60770 −0.331802 −0.165901 0.986142i \(-0.553053\pi\)
−0.165901 + 0.986142i \(0.553053\pi\)
\(674\) −20.4974 −0.789531
\(675\) 0 0
\(676\) 8.10512 0.311735
\(677\) −27.5885 −1.06031 −0.530155 0.847901i \(-0.677866\pi\)
−0.530155 + 0.847901i \(0.677866\pi\)
\(678\) 0 0
\(679\) −3.26795 −0.125412
\(680\) −2.53590 −0.0972473
\(681\) 0 0
\(682\) 0 0
\(683\) 28.2487 1.08091 0.540453 0.841374i \(-0.318253\pi\)
0.540453 + 0.841374i \(0.318253\pi\)
\(684\) 0 0
\(685\) −3.66025 −0.139851
\(686\) −0.732051 −0.0279498
\(687\) 0 0
\(688\) 0.497423 0.0189641
\(689\) −6.92820 −0.263944
\(690\) 0 0
\(691\) −8.39230 −0.319258 −0.159629 0.987177i \(-0.551030\pi\)
−0.159629 + 0.987177i \(0.551030\pi\)
\(692\) 7.60770 0.289201
\(693\) 0 0
\(694\) −19.0718 −0.723956
\(695\) 2.09103 0.0793175
\(696\) 0 0
\(697\) −14.9282 −0.565446
\(698\) 1.17691 0.0445469
\(699\) 0 0
\(700\) −7.21539 −0.272716
\(701\) −13.6603 −0.515941 −0.257970 0.966153i \(-0.583054\pi\)
−0.257970 + 0.966153i \(0.583054\pi\)
\(702\) 0 0
\(703\) 20.7846 0.783906
\(704\) 0 0
\(705\) 0 0
\(706\) 8.10512 0.305040
\(707\) 9.73205 0.366011
\(708\) 0 0
\(709\) −8.60770 −0.323269 −0.161634 0.986851i \(-0.551677\pi\)
−0.161634 + 0.986851i \(0.551677\pi\)
\(710\) −1.78976 −0.0671687
\(711\) 0 0
\(712\) −37.1769 −1.39326
\(713\) −24.3923 −0.913499
\(714\) 0 0
\(715\) 0 0
\(716\) −6.43078 −0.240330
\(717\) 0 0
\(718\) −18.3923 −0.686395
\(719\) 44.1051 1.64484 0.822422 0.568878i \(-0.192622\pi\)
0.822422 + 0.568878i \(0.192622\pi\)
\(720\) 0 0
\(721\) −4.19615 −0.156273
\(722\) −10.3782 −0.386237
\(723\) 0 0
\(724\) 11.7128 0.435303
\(725\) 6.24871 0.232071
\(726\) 0 0
\(727\) 37.5167 1.39142 0.695708 0.718325i \(-0.255091\pi\)
0.695708 + 0.718325i \(0.255091\pi\)
\(728\) 6.92820 0.256776
\(729\) 0 0
\(730\) −1.32051 −0.0488742
\(731\) 1.73205 0.0640622
\(732\) 0 0
\(733\) −14.4449 −0.533533 −0.266767 0.963761i \(-0.585955\pi\)
−0.266767 + 0.963761i \(0.585955\pi\)
\(734\) −9.32051 −0.344026
\(735\) 0 0
\(736\) −19.1384 −0.705452
\(737\) 0 0
\(738\) 0 0
\(739\) 1.32051 0.0485757 0.0242878 0.999705i \(-0.492268\pi\)
0.0242878 + 0.999705i \(0.492268\pi\)
\(740\) 3.71281 0.136486
\(741\) 0 0
\(742\) 1.85641 0.0681508
\(743\) 17.1244 0.628232 0.314116 0.949385i \(-0.398292\pi\)
0.314116 + 0.949385i \(0.398292\pi\)
\(744\) 0 0
\(745\) 0.732051 0.0268203
\(746\) −17.5167 −0.641331
\(747\) 0 0
\(748\) 0 0
\(749\) 18.5885 0.679207
\(750\) 0 0
\(751\) 11.3923 0.415711 0.207856 0.978160i \(-0.433352\pi\)
0.207856 + 0.978160i \(0.433352\pi\)
\(752\) 9.85641 0.359426
\(753\) 0 0
\(754\) −2.53590 −0.0923520
\(755\) 2.51666 0.0915907
\(756\) 0 0
\(757\) −7.92820 −0.288155 −0.144078 0.989566i \(-0.546022\pi\)
−0.144078 + 0.989566i \(0.546022\pi\)
\(758\) 2.58846 0.0940170
\(759\) 0 0
\(760\) 1.49227 0.0541302
\(761\) −20.9090 −0.757949 −0.378975 0.925407i \(-0.623723\pi\)
−0.378975 + 0.925407i \(0.623723\pi\)
\(762\) 0 0
\(763\) −17.3923 −0.629644
\(764\) 23.7128 0.857899
\(765\) 0 0
\(766\) 24.9808 0.902592
\(767\) 27.6603 0.998754
\(768\) 0 0
\(769\) −38.6410 −1.39343 −0.696715 0.717348i \(-0.745356\pi\)
−0.696715 + 0.717348i \(0.745356\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.32051 0.263471
\(773\) 1.73205 0.0622975 0.0311488 0.999515i \(-0.490083\pi\)
0.0311488 + 0.999515i \(0.490083\pi\)
\(774\) 0 0
\(775\) −36.7846 −1.32134
\(776\) −8.28719 −0.297493
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 8.78461 0.314741
\(780\) 0 0
\(781\) 0 0
\(782\) −8.92820 −0.319272
\(783\) 0 0
\(784\) 1.07180 0.0382785
\(785\) −1.75129 −0.0625062
\(786\) 0 0
\(787\) −15.1769 −0.540999 −0.270499 0.962720i \(-0.587189\pi\)
−0.270499 + 0.962720i \(0.587189\pi\)
\(788\) 18.3538 0.653828
\(789\) 0 0
\(790\) −0.105118 −0.00373992
\(791\) 8.53590 0.303502
\(792\) 0 0
\(793\) 22.3923 0.795174
\(794\) −9.75129 −0.346060
\(795\) 0 0
\(796\) 29.5692 1.04805
\(797\) 18.9090 0.669790 0.334895 0.942255i \(-0.391299\pi\)
0.334895 + 0.942255i \(0.391299\pi\)
\(798\) 0 0
\(799\) 34.3205 1.21417
\(800\) −28.8616 −1.02041
\(801\) 0 0
\(802\) 11.7128 0.413594
\(803\) 0 0
\(804\) 0 0
\(805\) 0.875644 0.0308624
\(806\) 14.9282 0.525824
\(807\) 0 0
\(808\) 24.6795 0.868221
\(809\) −20.4449 −0.718803 −0.359402 0.933183i \(-0.617019\pi\)
−0.359402 + 0.933183i \(0.617019\pi\)
\(810\) 0 0
\(811\) 45.9090 1.61208 0.806041 0.591860i \(-0.201606\pi\)
0.806041 + 0.591860i \(0.201606\pi\)
\(812\) −1.85641 −0.0651471
\(813\) 0 0
\(814\) 0 0
\(815\) 2.00000 0.0700569
\(816\) 0 0
\(817\) −1.01924 −0.0356586
\(818\) −17.3205 −0.605597
\(819\) 0 0
\(820\) 1.56922 0.0547995
\(821\) −23.8564 −0.832594 −0.416297 0.909229i \(-0.636672\pi\)
−0.416297 + 0.909229i \(0.636672\pi\)
\(822\) 0 0
\(823\) −29.3205 −1.02205 −0.511024 0.859566i \(-0.670734\pi\)
−0.511024 + 0.859566i \(0.670734\pi\)
\(824\) −10.6410 −0.370697
\(825\) 0 0
\(826\) −7.41154 −0.257881
\(827\) 29.5167 1.02639 0.513197 0.858271i \(-0.328461\pi\)
0.513197 + 0.858271i \(0.328461\pi\)
\(828\) 0 0
\(829\) 31.8038 1.10459 0.552297 0.833648i \(-0.313752\pi\)
0.552297 + 0.833648i \(0.313752\pi\)
\(830\) −0.837169 −0.0290585
\(831\) 0 0
\(832\) 5.85641 0.203034
\(833\) 3.73205 0.129308
\(834\) 0 0
\(835\) −6.71281 −0.232306
\(836\) 0 0
\(837\) 0 0
\(838\) −7.12436 −0.246107
\(839\) −48.2295 −1.66507 −0.832533 0.553975i \(-0.813110\pi\)
−0.832533 + 0.553975i \(0.813110\pi\)
\(840\) 0 0
\(841\) −27.3923 −0.944562
\(842\) 28.4449 0.980275
\(843\) 0 0
\(844\) 26.0385 0.896281
\(845\) −1.48334 −0.0510284
\(846\) 0 0
\(847\) 0 0
\(848\) −2.71797 −0.0933354
\(849\) 0 0
\(850\) −13.4641 −0.461815
\(851\) 30.9282 1.06021
\(852\) 0 0
\(853\) −4.00000 −0.136957 −0.0684787 0.997653i \(-0.521815\pi\)
−0.0684787 + 0.997653i \(0.521815\pi\)
\(854\) −6.00000 −0.205316
\(855\) 0 0
\(856\) 47.1384 1.61116
\(857\) −26.1244 −0.892391 −0.446195 0.894936i \(-0.647221\pi\)
−0.446195 + 0.894936i \(0.647221\pi\)
\(858\) 0 0
\(859\) −30.0000 −1.02359 −0.511793 0.859109i \(-0.671019\pi\)
−0.511793 + 0.859109i \(0.671019\pi\)
\(860\) −0.182069 −0.00620851
\(861\) 0 0
\(862\) −17.0718 −0.581468
\(863\) 0.928203 0.0315964 0.0157982 0.999875i \(-0.494971\pi\)
0.0157982 + 0.999875i \(0.494971\pi\)
\(864\) 0 0
\(865\) −1.39230 −0.0473398
\(866\) 24.8231 0.843523
\(867\) 0 0
\(868\) 10.9282 0.370927
\(869\) 0 0
\(870\) 0 0
\(871\) −19.1244 −0.648004
\(872\) −44.1051 −1.49359
\(873\) 0 0
\(874\) 5.25387 0.177715
\(875\) 2.66025 0.0899330
\(876\) 0 0
\(877\) 25.1769 0.850164 0.425082 0.905155i \(-0.360245\pi\)
0.425082 + 0.905155i \(0.360245\pi\)
\(878\) −0.105118 −0.00354755
\(879\) 0 0
\(880\) 0 0
\(881\) −39.8564 −1.34280 −0.671398 0.741097i \(-0.734306\pi\)
−0.671398 + 0.741097i \(0.734306\pi\)
\(882\) 0 0
\(883\) 27.7846 0.935027 0.467513 0.883986i \(-0.345150\pi\)
0.467513 + 0.883986i \(0.345150\pi\)
\(884\) −14.9282 −0.502090
\(885\) 0 0
\(886\) 4.24871 0.142738
\(887\) 10.1244 0.339943 0.169971 0.985449i \(-0.445632\pi\)
0.169971 + 0.985449i \(0.445632\pi\)
\(888\) 0 0
\(889\) 3.53590 0.118590
\(890\) 2.87564 0.0963918
\(891\) 0 0
\(892\) −23.4256 −0.784348
\(893\) −20.1962 −0.675838
\(894\) 0 0
\(895\) 1.17691 0.0393399
\(896\) 10.1436 0.338874
\(897\) 0 0
\(898\) 11.8949 0.396937
\(899\) −9.46410 −0.315645
\(900\) 0 0
\(901\) −9.46410 −0.315295
\(902\) 0 0
\(903\) 0 0
\(904\) 21.6462 0.719941
\(905\) −2.14359 −0.0712555
\(906\) 0 0
\(907\) 2.78461 0.0924614 0.0462307 0.998931i \(-0.485279\pi\)
0.0462307 + 0.998931i \(0.485279\pi\)
\(908\) 21.4641 0.712311
\(909\) 0 0
\(910\) −0.535898 −0.0177649
\(911\) −26.7846 −0.887414 −0.443707 0.896172i \(-0.646337\pi\)
−0.443707 + 0.896172i \(0.646337\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −24.8372 −0.821541
\(915\) 0 0
\(916\) −10.1436 −0.335154
\(917\) 5.19615 0.171592
\(918\) 0 0
\(919\) −6.39230 −0.210863 −0.105431 0.994427i \(-0.533622\pi\)
−0.105431 + 0.994427i \(0.533622\pi\)
\(920\) 2.22055 0.0732092
\(921\) 0 0
\(922\) 20.1962 0.665125
\(923\) −24.9282 −0.820522
\(924\) 0 0
\(925\) 46.6410 1.53355
\(926\) 23.0333 0.756922
\(927\) 0 0
\(928\) −7.42563 −0.243758
\(929\) −14.6603 −0.480987 −0.240494 0.970651i \(-0.577309\pi\)
−0.240494 + 0.970651i \(0.577309\pi\)
\(930\) 0 0
\(931\) −2.19615 −0.0719760
\(932\) 29.5692 0.968572
\(933\) 0 0
\(934\) 25.1769 0.823814
\(935\) 0 0
\(936\) 0 0
\(937\) −48.4449 −1.58262 −0.791312 0.611412i \(-0.790602\pi\)
−0.791312 + 0.611412i \(0.790602\pi\)
\(938\) 5.12436 0.167316
\(939\) 0 0
\(940\) −3.60770 −0.117670
\(941\) 13.6077 0.443598 0.221799 0.975092i \(-0.428807\pi\)
0.221799 + 0.975092i \(0.428807\pi\)
\(942\) 0 0
\(943\) 13.0718 0.425676
\(944\) 10.8513 0.353178
\(945\) 0 0
\(946\) 0 0
\(947\) −48.9282 −1.58995 −0.794976 0.606640i \(-0.792517\pi\)
−0.794976 + 0.606640i \(0.792517\pi\)
\(948\) 0 0
\(949\) −18.3923 −0.597039
\(950\) 7.92305 0.257058
\(951\) 0 0
\(952\) 9.46410 0.306733
\(953\) 49.3205 1.59765 0.798824 0.601565i \(-0.205456\pi\)
0.798824 + 0.601565i \(0.205456\pi\)
\(954\) 0 0
\(955\) −4.33975 −0.140431
\(956\) 36.4974 1.18041
\(957\) 0 0
\(958\) −23.6987 −0.765671
\(959\) 13.6603 0.441113
\(960\) 0 0
\(961\) 24.7128 0.797188
\(962\) −18.9282 −0.610270
\(963\) 0 0
\(964\) −22.9282 −0.738468
\(965\) −1.33975 −0.0431279
\(966\) 0 0
\(967\) 29.3923 0.945193 0.472596 0.881279i \(-0.343317\pi\)
0.472596 + 0.881279i \(0.343317\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0.641016 0.0205818
\(971\) 21.8756 0.702023 0.351011 0.936371i \(-0.385838\pi\)
0.351011 + 0.936371i \(0.385838\pi\)
\(972\) 0 0
\(973\) −7.80385 −0.250180
\(974\) 30.8756 0.989319
\(975\) 0 0
\(976\) 8.78461 0.281189
\(977\) −50.1962 −1.60592 −0.802959 0.596035i \(-0.796742\pi\)
−0.802959 + 0.596035i \(0.796742\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.392305 −0.0125317
\(981\) 0 0
\(982\) 0.0384758 0.00122781
\(983\) −8.78461 −0.280186 −0.140093 0.990138i \(-0.544740\pi\)
−0.140093 + 0.990138i \(0.544740\pi\)
\(984\) 0 0
\(985\) −3.35898 −0.107026
\(986\) −3.46410 −0.110319
\(987\) 0 0
\(988\) 8.78461 0.279476
\(989\) −1.51666 −0.0482270
\(990\) 0 0
\(991\) 17.6077 0.559327 0.279663 0.960098i \(-0.409777\pi\)
0.279663 + 0.960098i \(0.409777\pi\)
\(992\) 43.7128 1.38788
\(993\) 0 0
\(994\) 6.67949 0.211861
\(995\) −5.41154 −0.171557
\(996\) 0 0
\(997\) −61.0333 −1.93294 −0.966472 0.256771i \(-0.917341\pi\)
−0.966472 + 0.256771i \(0.917341\pi\)
\(998\) 29.5167 0.934334
\(999\) 0 0
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 7623.2.a.x.1.2 2
3.2 odd 2 2541.2.a.be.1.1 yes 2
11.10 odd 2 7623.2.a.bv.1.1 2
33.32 even 2 2541.2.a.o.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.o.1.2 2 33.32 even 2
2541.2.a.be.1.1 yes 2 3.2 odd 2
7623.2.a.x.1.2 2 1.1 even 1 trivial
7623.2.a.bv.1.1 2 11.10 odd 2