Properties

Label 2151.2.a.j.1.11
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(0.0888996\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.0888996 q^{2} -1.99210 q^{4} -0.674348 q^{5} +0.687902 q^{7} +0.354896 q^{8} +O(q^{10})\) \(q-0.0888996 q^{2} -1.99210 q^{4} -0.674348 q^{5} +0.687902 q^{7} +0.354896 q^{8} +0.0599493 q^{10} -0.536956 q^{11} -0.628799 q^{13} -0.0611542 q^{14} +3.95264 q^{16} +5.05676 q^{17} -0.724853 q^{19} +1.34337 q^{20} +0.0477352 q^{22} -7.36157 q^{23} -4.54525 q^{25} +0.0559000 q^{26} -1.37037 q^{28} -3.61524 q^{29} +5.74731 q^{31} -1.06118 q^{32} -0.449544 q^{34} -0.463885 q^{35} -0.110957 q^{37} +0.0644391 q^{38} -0.239323 q^{40} +8.05251 q^{41} +12.2197 q^{43} +1.06967 q^{44} +0.654440 q^{46} +1.48719 q^{47} -6.52679 q^{49} +0.404071 q^{50} +1.25263 q^{52} -6.92286 q^{53} +0.362095 q^{55} +0.244133 q^{56} +0.321393 q^{58} -9.81893 q^{59} +6.30376 q^{61} -0.510934 q^{62} -7.81095 q^{64} +0.424030 q^{65} -7.02929 q^{67} -10.0736 q^{68} +0.0412392 q^{70} -9.21843 q^{71} -12.1931 q^{73} +0.00986406 q^{74} +1.44398 q^{76} -0.369373 q^{77} -6.71164 q^{79} -2.66546 q^{80} -0.715865 q^{82} -8.10511 q^{83} -3.41002 q^{85} -1.08633 q^{86} -0.190563 q^{88} -13.1414 q^{89} -0.432552 q^{91} +14.6650 q^{92} -0.132211 q^{94} +0.488803 q^{95} +8.31742 q^{97} +0.580229 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.0888996 −0.0628615 −0.0314308 0.999506i \(-0.510006\pi\)
−0.0314308 + 0.999506i \(0.510006\pi\)
\(3\) 0 0
\(4\) −1.99210 −0.996048
\(5\) −0.674348 −0.301578 −0.150789 0.988566i \(-0.548181\pi\)
−0.150789 + 0.988566i \(0.548181\pi\)
\(6\) 0 0
\(7\) 0.687902 0.260002 0.130001 0.991514i \(-0.458502\pi\)
0.130001 + 0.991514i \(0.458502\pi\)
\(8\) 0.354896 0.125475
\(9\) 0 0
\(10\) 0.0599493 0.0189576
\(11\) −0.536956 −0.161898 −0.0809492 0.996718i \(-0.525795\pi\)
−0.0809492 + 0.996718i \(0.525795\pi\)
\(12\) 0 0
\(13\) −0.628799 −0.174397 −0.0871987 0.996191i \(-0.527792\pi\)
−0.0871987 + 0.996191i \(0.527792\pi\)
\(14\) −0.0611542 −0.0163441
\(15\) 0 0
\(16\) 3.95264 0.988161
\(17\) 5.05676 1.22644 0.613222 0.789911i \(-0.289873\pi\)
0.613222 + 0.789911i \(0.289873\pi\)
\(18\) 0 0
\(19\) −0.724853 −0.166293 −0.0831463 0.996537i \(-0.526497\pi\)
−0.0831463 + 0.996537i \(0.526497\pi\)
\(20\) 1.34337 0.300386
\(21\) 0 0
\(22\) 0.0477352 0.0101772
\(23\) −7.36157 −1.53499 −0.767496 0.641053i \(-0.778498\pi\)
−0.767496 + 0.641053i \(0.778498\pi\)
\(24\) 0 0
\(25\) −4.54525 −0.909051
\(26\) 0.0559000 0.0109629
\(27\) 0 0
\(28\) −1.37037 −0.258975
\(29\) −3.61524 −0.671332 −0.335666 0.941981i \(-0.608961\pi\)
−0.335666 + 0.941981i \(0.608961\pi\)
\(30\) 0 0
\(31\) 5.74731 1.03225 0.516124 0.856514i \(-0.327374\pi\)
0.516124 + 0.856514i \(0.327374\pi\)
\(32\) −1.06118 −0.187592
\(33\) 0 0
\(34\) −0.449544 −0.0770961
\(35\) −0.463885 −0.0784109
\(36\) 0 0
\(37\) −0.110957 −0.0182413 −0.00912063 0.999958i \(-0.502903\pi\)
−0.00912063 + 0.999958i \(0.502903\pi\)
\(38\) 0.0644391 0.0104534
\(39\) 0 0
\(40\) −0.239323 −0.0378403
\(41\) 8.05251 1.25759 0.628796 0.777571i \(-0.283548\pi\)
0.628796 + 0.777571i \(0.283548\pi\)
\(42\) 0 0
\(43\) 12.2197 1.86349 0.931746 0.363111i \(-0.118285\pi\)
0.931746 + 0.363111i \(0.118285\pi\)
\(44\) 1.06967 0.161259
\(45\) 0 0
\(46\) 0.654440 0.0964919
\(47\) 1.48719 0.216929 0.108465 0.994100i \(-0.465407\pi\)
0.108465 + 0.994100i \(0.465407\pi\)
\(48\) 0 0
\(49\) −6.52679 −0.932399
\(50\) 0.404071 0.0571443
\(51\) 0 0
\(52\) 1.25263 0.173708
\(53\) −6.92286 −0.950927 −0.475464 0.879735i \(-0.657720\pi\)
−0.475464 + 0.879735i \(0.657720\pi\)
\(54\) 0 0
\(55\) 0.362095 0.0488249
\(56\) 0.244133 0.0326237
\(57\) 0 0
\(58\) 0.321393 0.0422010
\(59\) −9.81893 −1.27832 −0.639158 0.769076i \(-0.720717\pi\)
−0.639158 + 0.769076i \(0.720717\pi\)
\(60\) 0 0
\(61\) 6.30376 0.807113 0.403557 0.914955i \(-0.367774\pi\)
0.403557 + 0.914955i \(0.367774\pi\)
\(62\) −0.510934 −0.0648886
\(63\) 0 0
\(64\) −7.81095 −0.976369
\(65\) 0.424030 0.0525944
\(66\) 0 0
\(67\) −7.02929 −0.858764 −0.429382 0.903123i \(-0.641269\pi\)
−0.429382 + 0.903123i \(0.641269\pi\)
\(68\) −10.0736 −1.22160
\(69\) 0 0
\(70\) 0.0412392 0.00492903
\(71\) −9.21843 −1.09403 −0.547013 0.837124i \(-0.684235\pi\)
−0.547013 + 0.837124i \(0.684235\pi\)
\(72\) 0 0
\(73\) −12.1931 −1.42710 −0.713548 0.700606i \(-0.752913\pi\)
−0.713548 + 0.700606i \(0.752913\pi\)
\(74\) 0.00986406 0.00114667
\(75\) 0 0
\(76\) 1.44398 0.165636
\(77\) −0.369373 −0.0420940
\(78\) 0 0
\(79\) −6.71164 −0.755118 −0.377559 0.925985i \(-0.623236\pi\)
−0.377559 + 0.925985i \(0.623236\pi\)
\(80\) −2.66546 −0.298007
\(81\) 0 0
\(82\) −0.715865 −0.0790541
\(83\) −8.10511 −0.889651 −0.444825 0.895617i \(-0.646734\pi\)
−0.444825 + 0.895617i \(0.646734\pi\)
\(84\) 0 0
\(85\) −3.41002 −0.369868
\(86\) −1.08633 −0.117142
\(87\) 0 0
\(88\) −0.190563 −0.0203141
\(89\) −13.1414 −1.39298 −0.696492 0.717564i \(-0.745257\pi\)
−0.696492 + 0.717564i \(0.745257\pi\)
\(90\) 0 0
\(91\) −0.432552 −0.0453438
\(92\) 14.6650 1.52893
\(93\) 0 0
\(94\) −0.132211 −0.0136365
\(95\) 0.488803 0.0501502
\(96\) 0 0
\(97\) 8.31742 0.844506 0.422253 0.906478i \(-0.361239\pi\)
0.422253 + 0.906478i \(0.361239\pi\)
\(98\) 0.580229 0.0586120
\(99\) 0 0
\(100\) 9.05459 0.905459
\(101\) −13.3756 −1.33092 −0.665462 0.746432i \(-0.731765\pi\)
−0.665462 + 0.746432i \(0.731765\pi\)
\(102\) 0 0
\(103\) 3.09332 0.304794 0.152397 0.988319i \(-0.451301\pi\)
0.152397 + 0.988319i \(0.451301\pi\)
\(104\) −0.223158 −0.0218825
\(105\) 0 0
\(106\) 0.615439 0.0597767
\(107\) −11.5001 −1.11175 −0.555877 0.831265i \(-0.687617\pi\)
−0.555877 + 0.831265i \(0.687617\pi\)
\(108\) 0 0
\(109\) 10.0986 0.967275 0.483637 0.875268i \(-0.339315\pi\)
0.483637 + 0.875268i \(0.339315\pi\)
\(110\) −0.0321901 −0.00306921
\(111\) 0 0
\(112\) 2.71903 0.256924
\(113\) −20.8956 −1.96570 −0.982848 0.184415i \(-0.940961\pi\)
−0.982848 + 0.184415i \(0.940961\pi\)
\(114\) 0 0
\(115\) 4.96426 0.462920
\(116\) 7.20190 0.668679
\(117\) 0 0
\(118\) 0.872898 0.0803568
\(119\) 3.47855 0.318878
\(120\) 0 0
\(121\) −10.7117 −0.973789
\(122\) −0.560401 −0.0507364
\(123\) 0 0
\(124\) −11.4492 −1.02817
\(125\) 6.43683 0.575727
\(126\) 0 0
\(127\) 5.25364 0.466185 0.233092 0.972455i \(-0.425116\pi\)
0.233092 + 0.972455i \(0.425116\pi\)
\(128\) 2.81675 0.248968
\(129\) 0 0
\(130\) −0.0376961 −0.00330616
\(131\) −7.68069 −0.671065 −0.335532 0.942029i \(-0.608916\pi\)
−0.335532 + 0.942029i \(0.608916\pi\)
\(132\) 0 0
\(133\) −0.498628 −0.0432365
\(134\) 0.624901 0.0539832
\(135\) 0 0
\(136\) 1.79462 0.153888
\(137\) 11.5008 0.982578 0.491289 0.870996i \(-0.336526\pi\)
0.491289 + 0.870996i \(0.336526\pi\)
\(138\) 0 0
\(139\) 17.1077 1.45106 0.725529 0.688192i \(-0.241595\pi\)
0.725529 + 0.688192i \(0.241595\pi\)
\(140\) 0.924105 0.0781011
\(141\) 0 0
\(142\) 0.819514 0.0687721
\(143\) 0.337638 0.0282347
\(144\) 0 0
\(145\) 2.43793 0.202459
\(146\) 1.08396 0.0897094
\(147\) 0 0
\(148\) 0.221038 0.0181692
\(149\) −23.5946 −1.93294 −0.966470 0.256778i \(-0.917339\pi\)
−0.966470 + 0.256778i \(0.917339\pi\)
\(150\) 0 0
\(151\) −11.3698 −0.925263 −0.462631 0.886551i \(-0.653095\pi\)
−0.462631 + 0.886551i \(0.653095\pi\)
\(152\) −0.257247 −0.0208655
\(153\) 0 0
\(154\) 0.0328371 0.00264609
\(155\) −3.87569 −0.311303
\(156\) 0 0
\(157\) −0.914383 −0.0729757 −0.0364878 0.999334i \(-0.511617\pi\)
−0.0364878 + 0.999334i \(0.511617\pi\)
\(158\) 0.596662 0.0474679
\(159\) 0 0
\(160\) 0.715605 0.0565735
\(161\) −5.06403 −0.399102
\(162\) 0 0
\(163\) −9.90233 −0.775610 −0.387805 0.921741i \(-0.626767\pi\)
−0.387805 + 0.921741i \(0.626767\pi\)
\(164\) −16.0414 −1.25262
\(165\) 0 0
\(166\) 0.720541 0.0559248
\(167\) −14.3435 −1.10993 −0.554965 0.831873i \(-0.687269\pi\)
−0.554965 + 0.831873i \(0.687269\pi\)
\(168\) 0 0
\(169\) −12.6046 −0.969586
\(170\) 0.303149 0.0232505
\(171\) 0 0
\(172\) −24.3429 −1.85613
\(173\) 7.17674 0.545637 0.272819 0.962065i \(-0.412044\pi\)
0.272819 + 0.962065i \(0.412044\pi\)
\(174\) 0 0
\(175\) −3.12669 −0.236355
\(176\) −2.12240 −0.159982
\(177\) 0 0
\(178\) 1.16826 0.0875651
\(179\) 18.9923 1.41955 0.709774 0.704429i \(-0.248797\pi\)
0.709774 + 0.704429i \(0.248797\pi\)
\(180\) 0 0
\(181\) 20.1558 1.49817 0.749086 0.662473i \(-0.230493\pi\)
0.749086 + 0.662473i \(0.230493\pi\)
\(182\) 0.0384537 0.00285038
\(183\) 0 0
\(184\) −2.61259 −0.192603
\(185\) 0.0748239 0.00550116
\(186\) 0 0
\(187\) −2.71526 −0.198559
\(188\) −2.96263 −0.216072
\(189\) 0 0
\(190\) −0.0434544 −0.00315251
\(191\) 7.35271 0.532023 0.266012 0.963970i \(-0.414294\pi\)
0.266012 + 0.963970i \(0.414294\pi\)
\(192\) 0 0
\(193\) 5.59970 0.403076 0.201538 0.979481i \(-0.435406\pi\)
0.201538 + 0.979481i \(0.435406\pi\)
\(194\) −0.739415 −0.0530869
\(195\) 0 0
\(196\) 13.0020 0.928714
\(197\) −23.4497 −1.67072 −0.835362 0.549700i \(-0.814742\pi\)
−0.835362 + 0.549700i \(0.814742\pi\)
\(198\) 0 0
\(199\) 8.09735 0.574006 0.287003 0.957930i \(-0.407341\pi\)
0.287003 + 0.957930i \(0.407341\pi\)
\(200\) −1.61309 −0.114063
\(201\) 0 0
\(202\) 1.18909 0.0836639
\(203\) −2.48693 −0.174548
\(204\) 0 0
\(205\) −5.43020 −0.379261
\(206\) −0.274995 −0.0191598
\(207\) 0 0
\(208\) −2.48542 −0.172333
\(209\) 0.389214 0.0269225
\(210\) 0 0
\(211\) −26.2444 −1.80674 −0.903371 0.428860i \(-0.858915\pi\)
−0.903371 + 0.428860i \(0.858915\pi\)
\(212\) 13.7910 0.947170
\(213\) 0 0
\(214\) 1.02235 0.0698865
\(215\) −8.24036 −0.561988
\(216\) 0 0
\(217\) 3.95359 0.268387
\(218\) −0.897765 −0.0608044
\(219\) 0 0
\(220\) −0.721329 −0.0486320
\(221\) −3.17968 −0.213889
\(222\) 0 0
\(223\) 5.41208 0.362420 0.181210 0.983444i \(-0.441999\pi\)
0.181210 + 0.983444i \(0.441999\pi\)
\(224\) −0.729987 −0.0487743
\(225\) 0 0
\(226\) 1.85761 0.123567
\(227\) 23.2617 1.54394 0.771968 0.635661i \(-0.219272\pi\)
0.771968 + 0.635661i \(0.219272\pi\)
\(228\) 0 0
\(229\) 21.9483 1.45039 0.725193 0.688546i \(-0.241751\pi\)
0.725193 + 0.688546i \(0.241751\pi\)
\(230\) −0.441321 −0.0290998
\(231\) 0 0
\(232\) −1.28303 −0.0842352
\(233\) 4.41858 0.289471 0.144736 0.989470i \(-0.453767\pi\)
0.144736 + 0.989470i \(0.453767\pi\)
\(234\) 0 0
\(235\) −1.00289 −0.0654210
\(236\) 19.5603 1.27326
\(237\) 0 0
\(238\) −0.309242 −0.0200452
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) −11.2098 −0.722087 −0.361044 0.932549i \(-0.617579\pi\)
−0.361044 + 0.932549i \(0.617579\pi\)
\(242\) 0.952264 0.0612138
\(243\) 0 0
\(244\) −12.5577 −0.803924
\(245\) 4.40133 0.281191
\(246\) 0 0
\(247\) 0.455787 0.0290010
\(248\) 2.03970 0.129521
\(249\) 0 0
\(250\) −0.572231 −0.0361911
\(251\) 24.1478 1.52419 0.762096 0.647464i \(-0.224170\pi\)
0.762096 + 0.647464i \(0.224170\pi\)
\(252\) 0 0
\(253\) 3.95284 0.248513
\(254\) −0.467046 −0.0293051
\(255\) 0 0
\(256\) 15.3715 0.960718
\(257\) −23.7037 −1.47859 −0.739297 0.673379i \(-0.764842\pi\)
−0.739297 + 0.673379i \(0.764842\pi\)
\(258\) 0 0
\(259\) −0.0763277 −0.00474277
\(260\) −0.844708 −0.0523866
\(261\) 0 0
\(262\) 0.682810 0.0421841
\(263\) 19.8888 1.22640 0.613198 0.789929i \(-0.289883\pi\)
0.613198 + 0.789929i \(0.289883\pi\)
\(264\) 0 0
\(265\) 4.66842 0.286779
\(266\) 0.0443278 0.00271791
\(267\) 0 0
\(268\) 14.0030 0.855371
\(269\) −16.7074 −1.01867 −0.509335 0.860568i \(-0.670109\pi\)
−0.509335 + 0.860568i \(0.670109\pi\)
\(270\) 0 0
\(271\) 18.0253 1.09496 0.547481 0.836818i \(-0.315587\pi\)
0.547481 + 0.836818i \(0.315587\pi\)
\(272\) 19.9876 1.21192
\(273\) 0 0
\(274\) −1.02242 −0.0617664
\(275\) 2.44060 0.147174
\(276\) 0 0
\(277\) −23.7188 −1.42513 −0.712563 0.701608i \(-0.752466\pi\)
−0.712563 + 0.701608i \(0.752466\pi\)
\(278\) −1.52087 −0.0912157
\(279\) 0 0
\(280\) −0.164631 −0.00983858
\(281\) 8.96828 0.535003 0.267501 0.963557i \(-0.413802\pi\)
0.267501 + 0.963557i \(0.413802\pi\)
\(282\) 0 0
\(283\) 2.74613 0.163240 0.0816201 0.996664i \(-0.473991\pi\)
0.0816201 + 0.996664i \(0.473991\pi\)
\(284\) 18.3640 1.08970
\(285\) 0 0
\(286\) −0.0300158 −0.00177487
\(287\) 5.53934 0.326977
\(288\) 0 0
\(289\) 8.57079 0.504164
\(290\) −0.216731 −0.0127269
\(291\) 0 0
\(292\) 24.2899 1.42146
\(293\) 26.5055 1.54847 0.774235 0.632898i \(-0.218135\pi\)
0.774235 + 0.632898i \(0.218135\pi\)
\(294\) 0 0
\(295\) 6.62138 0.385511
\(296\) −0.0393783 −0.00228882
\(297\) 0 0
\(298\) 2.09755 0.121508
\(299\) 4.62895 0.267699
\(300\) 0 0
\(301\) 8.40598 0.484512
\(302\) 1.01077 0.0581634
\(303\) 0 0
\(304\) −2.86509 −0.164324
\(305\) −4.25093 −0.243407
\(306\) 0 0
\(307\) −27.6982 −1.58082 −0.790410 0.612578i \(-0.790133\pi\)
−0.790410 + 0.612578i \(0.790133\pi\)
\(308\) 0.735827 0.0419276
\(309\) 0 0
\(310\) 0.344547 0.0195690
\(311\) 9.87576 0.560003 0.280001 0.960000i \(-0.409665\pi\)
0.280001 + 0.960000i \(0.409665\pi\)
\(312\) 0 0
\(313\) −14.8397 −0.838792 −0.419396 0.907803i \(-0.637758\pi\)
−0.419396 + 0.907803i \(0.637758\pi\)
\(314\) 0.0812883 0.00458736
\(315\) 0 0
\(316\) 13.3702 0.752134
\(317\) 19.8717 1.11610 0.558052 0.829806i \(-0.311549\pi\)
0.558052 + 0.829806i \(0.311549\pi\)
\(318\) 0 0
\(319\) 1.94122 0.108688
\(320\) 5.26730 0.294451
\(321\) 0 0
\(322\) 0.450190 0.0250881
\(323\) −3.66541 −0.203949
\(324\) 0 0
\(325\) 2.85805 0.158536
\(326\) 0.880313 0.0487560
\(327\) 0 0
\(328\) 2.85780 0.157796
\(329\) 1.02304 0.0564021
\(330\) 0 0
\(331\) −15.2217 −0.836658 −0.418329 0.908296i \(-0.637384\pi\)
−0.418329 + 0.908296i \(0.637384\pi\)
\(332\) 16.1462 0.886135
\(333\) 0 0
\(334\) 1.27513 0.0697719
\(335\) 4.74019 0.258984
\(336\) 0 0
\(337\) −31.2836 −1.70413 −0.852063 0.523440i \(-0.824648\pi\)
−0.852063 + 0.523440i \(0.824648\pi\)
\(338\) 1.12054 0.0609496
\(339\) 0 0
\(340\) 6.79308 0.368407
\(341\) −3.08605 −0.167119
\(342\) 0 0
\(343\) −9.30510 −0.502428
\(344\) 4.33673 0.233821
\(345\) 0 0
\(346\) −0.638009 −0.0342996
\(347\) −2.32374 −0.124745 −0.0623726 0.998053i \(-0.519867\pi\)
−0.0623726 + 0.998053i \(0.519867\pi\)
\(348\) 0 0
\(349\) 17.1644 0.918788 0.459394 0.888233i \(-0.348067\pi\)
0.459394 + 0.888233i \(0.348067\pi\)
\(350\) 0.277961 0.0148577
\(351\) 0 0
\(352\) 0.569807 0.0303708
\(353\) −21.0255 −1.11907 −0.559537 0.828805i \(-0.689021\pi\)
−0.559537 + 0.828805i \(0.689021\pi\)
\(354\) 0 0
\(355\) 6.21643 0.329934
\(356\) 26.1789 1.38748
\(357\) 0 0
\(358\) −1.68840 −0.0892350
\(359\) −31.0325 −1.63783 −0.818917 0.573912i \(-0.805425\pi\)
−0.818917 + 0.573912i \(0.805425\pi\)
\(360\) 0 0
\(361\) −18.4746 −0.972347
\(362\) −1.79185 −0.0941773
\(363\) 0 0
\(364\) 0.861685 0.0451646
\(365\) 8.22240 0.430380
\(366\) 0 0
\(367\) −0.316706 −0.0165319 −0.00826595 0.999966i \(-0.502631\pi\)
−0.00826595 + 0.999966i \(0.502631\pi\)
\(368\) −29.0976 −1.51682
\(369\) 0 0
\(370\) −0.00665181 −0.000345811 0
\(371\) −4.76224 −0.247243
\(372\) 0 0
\(373\) −2.67698 −0.138609 −0.0693045 0.997596i \(-0.522078\pi\)
−0.0693045 + 0.997596i \(0.522078\pi\)
\(374\) 0.241385 0.0124817
\(375\) 0 0
\(376\) 0.527798 0.0272191
\(377\) 2.27326 0.117079
\(378\) 0 0
\(379\) −17.6822 −0.908275 −0.454137 0.890932i \(-0.650052\pi\)
−0.454137 + 0.890932i \(0.650052\pi\)
\(380\) −0.973744 −0.0499520
\(381\) 0 0
\(382\) −0.653653 −0.0334438
\(383\) −10.1403 −0.518145 −0.259072 0.965858i \(-0.583417\pi\)
−0.259072 + 0.965858i \(0.583417\pi\)
\(384\) 0 0
\(385\) 0.249086 0.0126946
\(386\) −0.497811 −0.0253379
\(387\) 0 0
\(388\) −16.5691 −0.841169
\(389\) 9.21319 0.467127 0.233564 0.972341i \(-0.424961\pi\)
0.233564 + 0.972341i \(0.424961\pi\)
\(390\) 0 0
\(391\) −37.2256 −1.88258
\(392\) −2.31633 −0.116992
\(393\) 0 0
\(394\) 2.08467 0.105024
\(395\) 4.52598 0.227727
\(396\) 0 0
\(397\) 30.4727 1.52938 0.764689 0.644399i \(-0.222892\pi\)
0.764689 + 0.644399i \(0.222892\pi\)
\(398\) −0.719851 −0.0360829
\(399\) 0 0
\(400\) −17.9658 −0.898289
\(401\) 34.4093 1.71832 0.859160 0.511707i \(-0.170987\pi\)
0.859160 + 0.511707i \(0.170987\pi\)
\(402\) 0 0
\(403\) −3.61391 −0.180021
\(404\) 26.6455 1.32566
\(405\) 0 0
\(406\) 0.221087 0.0109724
\(407\) 0.0595792 0.00295323
\(408\) 0 0
\(409\) −19.0633 −0.942618 −0.471309 0.881968i \(-0.656218\pi\)
−0.471309 + 0.881968i \(0.656218\pi\)
\(410\) 0.482742 0.0238409
\(411\) 0 0
\(412\) −6.16219 −0.303589
\(413\) −6.75446 −0.332365
\(414\) 0 0
\(415\) 5.46566 0.268299
\(416\) 0.667269 0.0327156
\(417\) 0 0
\(418\) −0.0346010 −0.00169239
\(419\) −3.95215 −0.193075 −0.0965375 0.995329i \(-0.530777\pi\)
−0.0965375 + 0.995329i \(0.530777\pi\)
\(420\) 0 0
\(421\) 27.8805 1.35881 0.679406 0.733762i \(-0.262237\pi\)
0.679406 + 0.733762i \(0.262237\pi\)
\(422\) 2.33312 0.113574
\(423\) 0 0
\(424\) −2.45689 −0.119317
\(425\) −22.9842 −1.11490
\(426\) 0 0
\(427\) 4.33637 0.209851
\(428\) 22.9093 1.10736
\(429\) 0 0
\(430\) 0.732564 0.0353274
\(431\) −8.80087 −0.423923 −0.211961 0.977278i \(-0.567985\pi\)
−0.211961 + 0.977278i \(0.567985\pi\)
\(432\) 0 0
\(433\) 38.8304 1.86607 0.933034 0.359788i \(-0.117151\pi\)
0.933034 + 0.359788i \(0.117151\pi\)
\(434\) −0.351472 −0.0168712
\(435\) 0 0
\(436\) −20.1175 −0.963453
\(437\) 5.33605 0.255258
\(438\) 0 0
\(439\) 26.9487 1.28619 0.643095 0.765786i \(-0.277650\pi\)
0.643095 + 0.765786i \(0.277650\pi\)
\(440\) 0.128506 0.00612629
\(441\) 0 0
\(442\) 0.282673 0.0134454
\(443\) 14.5012 0.688972 0.344486 0.938792i \(-0.388053\pi\)
0.344486 + 0.938792i \(0.388053\pi\)
\(444\) 0 0
\(445\) 8.86187 0.420093
\(446\) −0.481132 −0.0227823
\(447\) 0 0
\(448\) −5.37316 −0.253858
\(449\) −8.58911 −0.405345 −0.202673 0.979247i \(-0.564963\pi\)
−0.202673 + 0.979247i \(0.564963\pi\)
\(450\) 0 0
\(451\) −4.32385 −0.203602
\(452\) 41.6261 1.95793
\(453\) 0 0
\(454\) −2.06796 −0.0970541
\(455\) 0.291691 0.0136747
\(456\) 0 0
\(457\) 13.4901 0.631040 0.315520 0.948919i \(-0.397821\pi\)
0.315520 + 0.948919i \(0.397821\pi\)
\(458\) −1.95120 −0.0911734
\(459\) 0 0
\(460\) −9.88929 −0.461090
\(461\) 20.1254 0.937332 0.468666 0.883375i \(-0.344735\pi\)
0.468666 + 0.883375i \(0.344735\pi\)
\(462\) 0 0
\(463\) 19.2303 0.893709 0.446855 0.894607i \(-0.352544\pi\)
0.446855 + 0.894607i \(0.352544\pi\)
\(464\) −14.2897 −0.663384
\(465\) 0 0
\(466\) −0.392810 −0.0181966
\(467\) 17.7598 0.821828 0.410914 0.911674i \(-0.365210\pi\)
0.410914 + 0.911674i \(0.365210\pi\)
\(468\) 0 0
\(469\) −4.83546 −0.223281
\(470\) 0.0891561 0.00411246
\(471\) 0 0
\(472\) −3.48470 −0.160396
\(473\) −6.56146 −0.301696
\(474\) 0 0
\(475\) 3.29464 0.151168
\(476\) −6.92961 −0.317618
\(477\) 0 0
\(478\) 0.0888996 0.00406617
\(479\) 28.8979 1.32038 0.660189 0.751100i \(-0.270476\pi\)
0.660189 + 0.751100i \(0.270476\pi\)
\(480\) 0 0
\(481\) 0.0697698 0.00318123
\(482\) 0.996548 0.0453915
\(483\) 0 0
\(484\) 21.3387 0.969941
\(485\) −5.60884 −0.254684
\(486\) 0 0
\(487\) −8.62899 −0.391017 −0.195508 0.980702i \(-0.562636\pi\)
−0.195508 + 0.980702i \(0.562636\pi\)
\(488\) 2.23718 0.101272
\(489\) 0 0
\(490\) −0.391276 −0.0176761
\(491\) −24.6724 −1.11345 −0.556725 0.830697i \(-0.687942\pi\)
−0.556725 + 0.830697i \(0.687942\pi\)
\(492\) 0 0
\(493\) −18.2814 −0.823351
\(494\) −0.0405193 −0.00182305
\(495\) 0 0
\(496\) 22.7171 1.02003
\(497\) −6.34137 −0.284449
\(498\) 0 0
\(499\) −0.336524 −0.0150649 −0.00753244 0.999972i \(-0.502398\pi\)
−0.00753244 + 0.999972i \(0.502398\pi\)
\(500\) −12.8228 −0.573452
\(501\) 0 0
\(502\) −2.14673 −0.0958131
\(503\) −33.4007 −1.48926 −0.744632 0.667476i \(-0.767375\pi\)
−0.744632 + 0.667476i \(0.767375\pi\)
\(504\) 0 0
\(505\) 9.01983 0.401377
\(506\) −0.351406 −0.0156219
\(507\) 0 0
\(508\) −10.4658 −0.464343
\(509\) −35.3725 −1.56786 −0.783929 0.620851i \(-0.786787\pi\)
−0.783929 + 0.620851i \(0.786787\pi\)
\(510\) 0 0
\(511\) −8.38766 −0.371048
\(512\) −7.00002 −0.309360
\(513\) 0 0
\(514\) 2.10725 0.0929466
\(515\) −2.08598 −0.0919191
\(516\) 0 0
\(517\) −0.798557 −0.0351205
\(518\) 0.00678550 0.000298138 0
\(519\) 0 0
\(520\) 0.150486 0.00659926
\(521\) 11.3345 0.496572 0.248286 0.968687i \(-0.420133\pi\)
0.248286 + 0.968687i \(0.420133\pi\)
\(522\) 0 0
\(523\) −40.7825 −1.78329 −0.891646 0.452733i \(-0.850449\pi\)
−0.891646 + 0.452733i \(0.850449\pi\)
\(524\) 15.3007 0.668413
\(525\) 0 0
\(526\) −1.76811 −0.0770931
\(527\) 29.0628 1.26599
\(528\) 0 0
\(529\) 31.1926 1.35620
\(530\) −0.415020 −0.0180273
\(531\) 0 0
\(532\) 0.993314 0.0430656
\(533\) −5.06341 −0.219321
\(534\) 0 0
\(535\) 7.75505 0.335280
\(536\) −2.49466 −0.107753
\(537\) 0 0
\(538\) 1.48528 0.0640352
\(539\) 3.50460 0.150954
\(540\) 0 0
\(541\) 15.1438 0.651082 0.325541 0.945528i \(-0.394454\pi\)
0.325541 + 0.945528i \(0.394454\pi\)
\(542\) −1.60245 −0.0688310
\(543\) 0 0
\(544\) −5.36613 −0.230071
\(545\) −6.81000 −0.291709
\(546\) 0 0
\(547\) 28.0014 1.19725 0.598627 0.801028i \(-0.295713\pi\)
0.598627 + 0.801028i \(0.295713\pi\)
\(548\) −22.9107 −0.978696
\(549\) 0 0
\(550\) −0.216969 −0.00925157
\(551\) 2.62051 0.111638
\(552\) 0 0
\(553\) −4.61695 −0.196333
\(554\) 2.10859 0.0895855
\(555\) 0 0
\(556\) −34.0802 −1.44532
\(557\) 39.4275 1.67060 0.835298 0.549798i \(-0.185295\pi\)
0.835298 + 0.549798i \(0.185295\pi\)
\(558\) 0 0
\(559\) −7.68376 −0.324988
\(560\) −1.83357 −0.0774826
\(561\) 0 0
\(562\) −0.797276 −0.0336311
\(563\) −11.7452 −0.495003 −0.247502 0.968887i \(-0.579610\pi\)
−0.247502 + 0.968887i \(0.579610\pi\)
\(564\) 0 0
\(565\) 14.0909 0.592810
\(566\) −0.244129 −0.0102615
\(567\) 0 0
\(568\) −3.27158 −0.137272
\(569\) 16.2860 0.682743 0.341371 0.939928i \(-0.389109\pi\)
0.341371 + 0.939928i \(0.389109\pi\)
\(570\) 0 0
\(571\) 3.21609 0.134589 0.0672946 0.997733i \(-0.478563\pi\)
0.0672946 + 0.997733i \(0.478563\pi\)
\(572\) −0.672607 −0.0281231
\(573\) 0 0
\(574\) −0.492445 −0.0205542
\(575\) 33.4602 1.39539
\(576\) 0 0
\(577\) 22.9258 0.954412 0.477206 0.878792i \(-0.341650\pi\)
0.477206 + 0.878792i \(0.341650\pi\)
\(578\) −0.761940 −0.0316925
\(579\) 0 0
\(580\) −4.85659 −0.201659
\(581\) −5.57552 −0.231311
\(582\) 0 0
\(583\) 3.71727 0.153954
\(584\) −4.32728 −0.179064
\(585\) 0 0
\(586\) −2.35633 −0.0973392
\(587\) 7.86672 0.324694 0.162347 0.986734i \(-0.448094\pi\)
0.162347 + 0.986734i \(0.448094\pi\)
\(588\) 0 0
\(589\) −4.16596 −0.171655
\(590\) −0.588638 −0.0242338
\(591\) 0 0
\(592\) −0.438575 −0.0180253
\(593\) −6.59271 −0.270730 −0.135365 0.990796i \(-0.543221\pi\)
−0.135365 + 0.990796i \(0.543221\pi\)
\(594\) 0 0
\(595\) −2.34576 −0.0961666
\(596\) 47.0026 1.92530
\(597\) 0 0
\(598\) −0.411511 −0.0168280
\(599\) 32.3184 1.32050 0.660248 0.751048i \(-0.270451\pi\)
0.660248 + 0.751048i \(0.270451\pi\)
\(600\) 0 0
\(601\) −34.1450 −1.39280 −0.696402 0.717652i \(-0.745217\pi\)
−0.696402 + 0.717652i \(0.745217\pi\)
\(602\) −0.747288 −0.0304572
\(603\) 0 0
\(604\) 22.6498 0.921606
\(605\) 7.22340 0.293673
\(606\) 0 0
\(607\) −19.5667 −0.794186 −0.397093 0.917778i \(-0.629981\pi\)
−0.397093 + 0.917778i \(0.629981\pi\)
\(608\) 0.769199 0.0311952
\(609\) 0 0
\(610\) 0.377906 0.0153010
\(611\) −0.935145 −0.0378319
\(612\) 0 0
\(613\) −46.3571 −1.87235 −0.936174 0.351538i \(-0.885659\pi\)
−0.936174 + 0.351538i \(0.885659\pi\)
\(614\) 2.46236 0.0993728
\(615\) 0 0
\(616\) −0.131089 −0.00528172
\(617\) −33.5775 −1.35178 −0.675889 0.737004i \(-0.736240\pi\)
−0.675889 + 0.737004i \(0.736240\pi\)
\(618\) 0 0
\(619\) −9.59081 −0.385487 −0.192744 0.981249i \(-0.561739\pi\)
−0.192744 + 0.981249i \(0.561739\pi\)
\(620\) 7.72075 0.310073
\(621\) 0 0
\(622\) −0.877951 −0.0352026
\(623\) −9.03998 −0.362179
\(624\) 0 0
\(625\) 18.3856 0.735424
\(626\) 1.31925 0.0527277
\(627\) 0 0
\(628\) 1.82154 0.0726873
\(629\) −0.561084 −0.0223719
\(630\) 0 0
\(631\) −30.4228 −1.21111 −0.605556 0.795803i \(-0.707049\pi\)
−0.605556 + 0.795803i \(0.707049\pi\)
\(632\) −2.38193 −0.0947482
\(633\) 0 0
\(634\) −1.76658 −0.0701600
\(635\) −3.54278 −0.140591
\(636\) 0 0
\(637\) 4.10404 0.162608
\(638\) −0.172574 −0.00683227
\(639\) 0 0
\(640\) −1.89947 −0.0750832
\(641\) −19.5297 −0.771376 −0.385688 0.922629i \(-0.626036\pi\)
−0.385688 + 0.922629i \(0.626036\pi\)
\(642\) 0 0
\(643\) 41.7587 1.64680 0.823401 0.567460i \(-0.192074\pi\)
0.823401 + 0.567460i \(0.192074\pi\)
\(644\) 10.0880 0.397525
\(645\) 0 0
\(646\) 0.325853 0.0128205
\(647\) −25.7021 −1.01046 −0.505228 0.862986i \(-0.668591\pi\)
−0.505228 + 0.862986i \(0.668591\pi\)
\(648\) 0 0
\(649\) 5.27233 0.206957
\(650\) −0.254080 −0.00996582
\(651\) 0 0
\(652\) 19.7264 0.772545
\(653\) −5.47922 −0.214418 −0.107209 0.994236i \(-0.534191\pi\)
−0.107209 + 0.994236i \(0.534191\pi\)
\(654\) 0 0
\(655\) 5.17946 0.202378
\(656\) 31.8287 1.24270
\(657\) 0 0
\(658\) −0.0909480 −0.00354552
\(659\) 41.0759 1.60009 0.800044 0.599941i \(-0.204809\pi\)
0.800044 + 0.599941i \(0.204809\pi\)
\(660\) 0 0
\(661\) 0.166915 0.00649225 0.00324612 0.999995i \(-0.498967\pi\)
0.00324612 + 0.999995i \(0.498967\pi\)
\(662\) 1.35320 0.0525936
\(663\) 0 0
\(664\) −2.87647 −0.111629
\(665\) 0.336249 0.0130392
\(666\) 0 0
\(667\) 26.6138 1.03049
\(668\) 28.5736 1.10554
\(669\) 0 0
\(670\) −0.421401 −0.0162801
\(671\) −3.38484 −0.130670
\(672\) 0 0
\(673\) −31.5591 −1.21651 −0.608257 0.793741i \(-0.708131\pi\)
−0.608257 + 0.793741i \(0.708131\pi\)
\(674\) 2.78110 0.107124
\(675\) 0 0
\(676\) 25.1096 0.965754
\(677\) −28.9919 −1.11425 −0.557124 0.830429i \(-0.688095\pi\)
−0.557124 + 0.830429i \(0.688095\pi\)
\(678\) 0 0
\(679\) 5.72157 0.219574
\(680\) −1.21020 −0.0464091
\(681\) 0 0
\(682\) 0.274349 0.0105054
\(683\) −29.7746 −1.13929 −0.569646 0.821890i \(-0.692920\pi\)
−0.569646 + 0.821890i \(0.692920\pi\)
\(684\) 0 0
\(685\) −7.75554 −0.296324
\(686\) 0.827220 0.0315834
\(687\) 0 0
\(688\) 48.3003 1.84143
\(689\) 4.35309 0.165839
\(690\) 0 0
\(691\) −21.5776 −0.820852 −0.410426 0.911894i \(-0.634620\pi\)
−0.410426 + 0.911894i \(0.634620\pi\)
\(692\) −14.2968 −0.543481
\(693\) 0 0
\(694\) 0.206580 0.00784167
\(695\) −11.5366 −0.437607
\(696\) 0 0
\(697\) 40.7196 1.54236
\(698\) −1.52591 −0.0577564
\(699\) 0 0
\(700\) 6.22867 0.235421
\(701\) −2.79934 −0.105730 −0.0528649 0.998602i \(-0.516835\pi\)
−0.0528649 + 0.998602i \(0.516835\pi\)
\(702\) 0 0
\(703\) 0.0804277 0.00303339
\(704\) 4.19414 0.158072
\(705\) 0 0
\(706\) 1.86916 0.0703467
\(707\) −9.20111 −0.346043
\(708\) 0 0
\(709\) −46.7987 −1.75756 −0.878781 0.477225i \(-0.841643\pi\)
−0.878781 + 0.477225i \(0.841643\pi\)
\(710\) −0.552638 −0.0207401
\(711\) 0 0
\(712\) −4.66382 −0.174784
\(713\) −42.3092 −1.58449
\(714\) 0 0
\(715\) −0.227685 −0.00851495
\(716\) −37.8344 −1.41394
\(717\) 0 0
\(718\) 2.75878 0.102957
\(719\) −6.28186 −0.234274 −0.117137 0.993116i \(-0.537372\pi\)
−0.117137 + 0.993116i \(0.537372\pi\)
\(720\) 0 0
\(721\) 2.12790 0.0792471
\(722\) 1.64238 0.0611232
\(723\) 0 0
\(724\) −40.1524 −1.49225
\(725\) 16.4322 0.610275
\(726\) 0 0
\(727\) 5.34229 0.198135 0.0990673 0.995081i \(-0.468414\pi\)
0.0990673 + 0.995081i \(0.468414\pi\)
\(728\) −0.153511 −0.00568949
\(729\) 0 0
\(730\) −0.730968 −0.0270544
\(731\) 61.7922 2.28547
\(732\) 0 0
\(733\) −3.43108 −0.126730 −0.0633650 0.997990i \(-0.520183\pi\)
−0.0633650 + 0.997990i \(0.520183\pi\)
\(734\) 0.0281550 0.00103922
\(735\) 0 0
\(736\) 7.81195 0.287952
\(737\) 3.77442 0.139033
\(738\) 0 0
\(739\) 45.0227 1.65619 0.828093 0.560591i \(-0.189426\pi\)
0.828093 + 0.560591i \(0.189426\pi\)
\(740\) −0.149056 −0.00547942
\(741\) 0 0
\(742\) 0.423362 0.0155421
\(743\) −0.484953 −0.0177912 −0.00889560 0.999960i \(-0.502832\pi\)
−0.00889560 + 0.999960i \(0.502832\pi\)
\(744\) 0 0
\(745\) 15.9109 0.582932
\(746\) 0.237983 0.00871317
\(747\) 0 0
\(748\) 5.40905 0.197775
\(749\) −7.91092 −0.289059
\(750\) 0 0
\(751\) 3.09423 0.112910 0.0564550 0.998405i \(-0.482020\pi\)
0.0564550 + 0.998405i \(0.482020\pi\)
\(752\) 5.87834 0.214361
\(753\) 0 0
\(754\) −0.202092 −0.00735974
\(755\) 7.66722 0.279039
\(756\) 0 0
\(757\) −21.9908 −0.799271 −0.399635 0.916674i \(-0.630863\pi\)
−0.399635 + 0.916674i \(0.630863\pi\)
\(758\) 1.57194 0.0570955
\(759\) 0 0
\(760\) 0.173474 0.00629257
\(761\) 14.2000 0.514748 0.257374 0.966312i \(-0.417143\pi\)
0.257374 + 0.966312i \(0.417143\pi\)
\(762\) 0 0
\(763\) 6.94688 0.251494
\(764\) −14.6473 −0.529921
\(765\) 0 0
\(766\) 0.901468 0.0325713
\(767\) 6.17413 0.222935
\(768\) 0 0
\(769\) −5.78312 −0.208545 −0.104272 0.994549i \(-0.533251\pi\)
−0.104272 + 0.994549i \(0.533251\pi\)
\(770\) −0.0221437 −0.000798002 0
\(771\) 0 0
\(772\) −11.1552 −0.401483
\(773\) 40.3814 1.45242 0.726210 0.687473i \(-0.241280\pi\)
0.726210 + 0.687473i \(0.241280\pi\)
\(774\) 0 0
\(775\) −26.1230 −0.938366
\(776\) 2.95182 0.105964
\(777\) 0 0
\(778\) −0.819049 −0.0293643
\(779\) −5.83689 −0.209128
\(780\) 0 0
\(781\) 4.94989 0.177121
\(782\) 3.30935 0.118342
\(783\) 0 0
\(784\) −25.7981 −0.921360
\(785\) 0.616612 0.0220078
\(786\) 0 0
\(787\) 30.2097 1.07686 0.538430 0.842670i \(-0.319017\pi\)
0.538430 + 0.842670i \(0.319017\pi\)
\(788\) 46.7141 1.66412
\(789\) 0 0
\(790\) −0.402358 −0.0143153
\(791\) −14.3741 −0.511086
\(792\) 0 0
\(793\) −3.96380 −0.140759
\(794\) −2.70901 −0.0961390
\(795\) 0 0
\(796\) −16.1307 −0.571738
\(797\) 28.5813 1.01240 0.506200 0.862416i \(-0.331050\pi\)
0.506200 + 0.862416i \(0.331050\pi\)
\(798\) 0 0
\(799\) 7.52037 0.266052
\(800\) 4.82333 0.170531
\(801\) 0 0
\(802\) −3.05898 −0.108016
\(803\) 6.54717 0.231044
\(804\) 0 0
\(805\) 3.41492 0.120360
\(806\) 0.321275 0.0113164
\(807\) 0 0
\(808\) −4.74695 −0.166997
\(809\) −42.6781 −1.50048 −0.750241 0.661164i \(-0.770063\pi\)
−0.750241 + 0.661164i \(0.770063\pi\)
\(810\) 0 0
\(811\) −14.9127 −0.523655 −0.261828 0.965115i \(-0.584325\pi\)
−0.261828 + 0.965115i \(0.584325\pi\)
\(812\) 4.95420 0.173858
\(813\) 0 0
\(814\) −0.00529657 −0.000185645 0
\(815\) 6.67762 0.233907
\(816\) 0 0
\(817\) −8.85751 −0.309885
\(818\) 1.69472 0.0592544
\(819\) 0 0
\(820\) 10.8175 0.377763
\(821\) −25.8710 −0.902905 −0.451452 0.892295i \(-0.649094\pi\)
−0.451452 + 0.892295i \(0.649094\pi\)
\(822\) 0 0
\(823\) 19.4859 0.679236 0.339618 0.940563i \(-0.389702\pi\)
0.339618 + 0.940563i \(0.389702\pi\)
\(824\) 1.09781 0.0382439
\(825\) 0 0
\(826\) 0.600468 0.0208930
\(827\) 42.7539 1.48670 0.743349 0.668903i \(-0.233236\pi\)
0.743349 + 0.668903i \(0.233236\pi\)
\(828\) 0 0
\(829\) −21.6962 −0.753542 −0.376771 0.926307i \(-0.622966\pi\)
−0.376771 + 0.926307i \(0.622966\pi\)
\(830\) −0.485895 −0.0168657
\(831\) 0 0
\(832\) 4.91152 0.170276
\(833\) −33.0044 −1.14353
\(834\) 0 0
\(835\) 9.67249 0.334730
\(836\) −0.775352 −0.0268161
\(837\) 0 0
\(838\) 0.351344 0.0121370
\(839\) 53.7944 1.85719 0.928595 0.371095i \(-0.121017\pi\)
0.928595 + 0.371095i \(0.121017\pi\)
\(840\) 0 0
\(841\) −15.9301 −0.549313
\(842\) −2.47857 −0.0854170
\(843\) 0 0
\(844\) 52.2815 1.79960
\(845\) 8.49990 0.292405
\(846\) 0 0
\(847\) −7.36858 −0.253187
\(848\) −27.3636 −0.939669
\(849\) 0 0
\(850\) 2.04329 0.0700843
\(851\) 0.816819 0.0280002
\(852\) 0 0
\(853\) −0.311946 −0.0106808 −0.00534041 0.999986i \(-0.501700\pi\)
−0.00534041 + 0.999986i \(0.501700\pi\)
\(854\) −0.385501 −0.0131916
\(855\) 0 0
\(856\) −4.08133 −0.139497
\(857\) −12.6856 −0.433330 −0.216665 0.976246i \(-0.569518\pi\)
−0.216665 + 0.976246i \(0.569518\pi\)
\(858\) 0 0
\(859\) −1.87231 −0.0638823 −0.0319411 0.999490i \(-0.510169\pi\)
−0.0319411 + 0.999490i \(0.510169\pi\)
\(860\) 16.4156 0.559767
\(861\) 0 0
\(862\) 0.782394 0.0266484
\(863\) −27.4289 −0.933692 −0.466846 0.884339i \(-0.654610\pi\)
−0.466846 + 0.884339i \(0.654610\pi\)
\(864\) 0 0
\(865\) −4.83962 −0.164552
\(866\) −3.45200 −0.117304
\(867\) 0 0
\(868\) −7.87593 −0.267326
\(869\) 3.60385 0.122252
\(870\) 0 0
\(871\) 4.42001 0.149766
\(872\) 3.58397 0.121368
\(873\) 0 0
\(874\) −0.474373 −0.0160459
\(875\) 4.42790 0.149690
\(876\) 0 0
\(877\) −20.8318 −0.703441 −0.351720 0.936105i \(-0.614403\pi\)
−0.351720 + 0.936105i \(0.614403\pi\)
\(878\) −2.39573 −0.0808519
\(879\) 0 0
\(880\) 1.43123 0.0482469
\(881\) −11.4448 −0.385584 −0.192792 0.981240i \(-0.561754\pi\)
−0.192792 + 0.981240i \(0.561754\pi\)
\(882\) 0 0
\(883\) 2.46065 0.0828076 0.0414038 0.999142i \(-0.486817\pi\)
0.0414038 + 0.999142i \(0.486817\pi\)
\(884\) 6.33424 0.213044
\(885\) 0 0
\(886\) −1.28915 −0.0433098
\(887\) 43.7435 1.46876 0.734382 0.678737i \(-0.237472\pi\)
0.734382 + 0.678737i \(0.237472\pi\)
\(888\) 0 0
\(889\) 3.61399 0.121209
\(890\) −0.787817 −0.0264077
\(891\) 0 0
\(892\) −10.7814 −0.360988
\(893\) −1.07800 −0.0360737
\(894\) 0 0
\(895\) −12.8074 −0.428104
\(896\) 1.93765 0.0647322
\(897\) 0 0
\(898\) 0.763568 0.0254806
\(899\) −20.7779 −0.692981
\(900\) 0 0
\(901\) −35.0072 −1.16626
\(902\) 0.384388 0.0127987
\(903\) 0 0
\(904\) −7.41578 −0.246645
\(905\) −13.5921 −0.451815
\(906\) 0 0
\(907\) −29.9761 −0.995339 −0.497669 0.867367i \(-0.665811\pi\)
−0.497669 + 0.867367i \(0.665811\pi\)
\(908\) −46.3396 −1.53784
\(909\) 0 0
\(910\) −0.0259312 −0.000859610 0
\(911\) −50.0272 −1.65748 −0.828738 0.559637i \(-0.810941\pi\)
−0.828738 + 0.559637i \(0.810941\pi\)
\(912\) 0 0
\(913\) 4.35209 0.144033
\(914\) −1.19926 −0.0396682
\(915\) 0 0
\(916\) −43.7232 −1.44465
\(917\) −5.28356 −0.174478
\(918\) 0 0
\(919\) 24.1329 0.796070 0.398035 0.917370i \(-0.369692\pi\)
0.398035 + 0.917370i \(0.369692\pi\)
\(920\) 1.76179 0.0580846
\(921\) 0 0
\(922\) −1.78914 −0.0589221
\(923\) 5.79654 0.190795
\(924\) 0 0
\(925\) 0.504329 0.0165822
\(926\) −1.70957 −0.0561799
\(927\) 0 0
\(928\) 3.83641 0.125936
\(929\) −49.0310 −1.60865 −0.804327 0.594186i \(-0.797474\pi\)
−0.804327 + 0.594186i \(0.797474\pi\)
\(930\) 0 0
\(931\) 4.73096 0.155051
\(932\) −8.80225 −0.288327
\(933\) 0 0
\(934\) −1.57884 −0.0516613
\(935\) 1.83103 0.0598810
\(936\) 0 0
\(937\) 0.755623 0.0246851 0.0123426 0.999924i \(-0.496071\pi\)
0.0123426 + 0.999924i \(0.496071\pi\)
\(938\) 0.429870 0.0140358
\(939\) 0 0
\(940\) 1.99784 0.0651625
\(941\) 12.4403 0.405542 0.202771 0.979226i \(-0.435005\pi\)
0.202771 + 0.979226i \(0.435005\pi\)
\(942\) 0 0
\(943\) −59.2791 −1.93039
\(944\) −38.8107 −1.26318
\(945\) 0 0
\(946\) 0.583311 0.0189651
\(947\) 37.7555 1.22689 0.613445 0.789738i \(-0.289783\pi\)
0.613445 + 0.789738i \(0.289783\pi\)
\(948\) 0 0
\(949\) 7.66702 0.248882
\(950\) −0.292892 −0.00950268
\(951\) 0 0
\(952\) 1.23452 0.0400111
\(953\) 6.27937 0.203409 0.101704 0.994815i \(-0.467570\pi\)
0.101704 + 0.994815i \(0.467570\pi\)
\(954\) 0 0
\(955\) −4.95829 −0.160446
\(956\) 1.99210 0.0644290
\(957\) 0 0
\(958\) −2.56901 −0.0830009
\(959\) 7.91141 0.255473
\(960\) 0 0
\(961\) 2.03160 0.0655355
\(962\) −0.00620251 −0.000199977 0
\(963\) 0 0
\(964\) 22.3310 0.719234
\(965\) −3.77615 −0.121559
\(966\) 0 0
\(967\) 41.8287 1.34512 0.672560 0.740042i \(-0.265195\pi\)
0.672560 + 0.740042i \(0.265195\pi\)
\(968\) −3.80153 −0.122186
\(969\) 0 0
\(970\) 0.498623 0.0160098
\(971\) 23.0751 0.740515 0.370258 0.928929i \(-0.379269\pi\)
0.370258 + 0.928929i \(0.379269\pi\)
\(972\) 0 0
\(973\) 11.7684 0.377278
\(974\) 0.767114 0.0245799
\(975\) 0 0
\(976\) 24.9165 0.797558
\(977\) 15.2960 0.489363 0.244682 0.969603i \(-0.421317\pi\)
0.244682 + 0.969603i \(0.421317\pi\)
\(978\) 0 0
\(979\) 7.05635 0.225522
\(980\) −8.76788 −0.280080
\(981\) 0 0
\(982\) 2.19337 0.0699931
\(983\) −43.5763 −1.38987 −0.694935 0.719073i \(-0.744567\pi\)
−0.694935 + 0.719073i \(0.744567\pi\)
\(984\) 0 0
\(985\) 15.8133 0.503853
\(986\) 1.62521 0.0517571
\(987\) 0 0
\(988\) −0.907972 −0.0288864
\(989\) −89.9564 −2.86045
\(990\) 0 0
\(991\) 33.3453 1.05925 0.529624 0.848232i \(-0.322333\pi\)
0.529624 + 0.848232i \(0.322333\pi\)
\(992\) −6.09893 −0.193641
\(993\) 0 0
\(994\) 0.563745 0.0178809
\(995\) −5.46044 −0.173107
\(996\) 0 0
\(997\) 42.7971 1.35540 0.677699 0.735340i \(-0.262977\pi\)
0.677699 + 0.735340i \(0.262977\pi\)
\(998\) 0.0299168 0.000947001 0
\(999\) 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 2151.2.a.j.1.11 20
3.2 odd 2 2151.2.a.k.1.10 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.11 20 1.1 even 1 trivial
2151.2.a.k.1.10 yes 20 3.2 odd 2