Properties

Label 2151.2.a.k.1.10
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
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: \(0\)
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 1.10
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 \(-0.489994\pi\)
0.0314308 + 0.999506i \(0.489994\pi\)
\(3\) 0 0
\(4\) −1.99210 −0.996048
\(5\) 0.674348 0.301578 0.150789 0.988566i \(-0.451819\pi\)
0.150789 + 0.988566i \(0.451819\pi\)
\(6\) 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 \(-0.474205\pi\)
0.0809492 + 0.996718i \(0.474205\pi\)
\(12\) 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 \(-0.710127\pi\)
−0.613222 + 0.789911i \(0.710127\pi\)
\(18\) 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 \(-0.221502\pi\)
0.767496 + 0.641053i \(0.221502\pi\)
\(24\) 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 \(-0.391039\pi\)
0.335666 + 0.941981i \(0.391039\pi\)
\(30\) 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 \(-0.716452\pi\)
−0.628796 + 0.777571i \(0.716452\pi\)
\(42\) 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 \(-0.534593\pi\)
−0.108465 + 0.994100i \(0.534593\pi\)
\(48\) 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 \(-0.342280\pi\)
0.475464 + 0.879735i \(0.342280\pi\)
\(54\) 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 \(-0.279283\pi\)
0.639158 + 0.769076i \(0.279283\pi\)
\(60\) 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 \(-0.315765\pi\)
0.547013 + 0.837124i \(0.315765\pi\)
\(72\) 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 \(-0.353266\pi\)
0.444825 + 0.895617i \(0.353266\pi\)
\(84\) 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 \(-0.254743\pi\)
0.696492 + 0.717564i \(0.254743\pi\)
\(90\) 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 \(-0.268235\pi\)
0.665462 + 0.746432i \(0.268235\pi\)
\(102\) 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 \(-0.312383\pi\)
0.555877 + 0.831265i \(0.312383\pi\)
\(108\) 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 \(-0.0590392\pi\)
0.982848 + 0.184415i \(0.0590392\pi\)
\(114\) 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 \(-0.391084\pi\)
0.335532 + 0.942029i \(0.391084\pi\)
\(132\) 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 \(-0.663474\pi\)
−0.491289 + 0.870996i \(0.663474\pi\)
\(138\) 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 \(-0.0826609\pi\)
0.966470 + 0.256778i \(0.0826609\pi\)
\(150\) 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 \(-0.312731\pi\)
0.554965 + 0.831873i \(0.312731\pi\)
\(168\) 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 \(-0.587956\pi\)
−0.272819 + 0.962065i \(0.587956\pi\)
\(174\) 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 \(-0.751203\pi\)
−0.709774 + 0.704429i \(0.751203\pi\)
\(180\) 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 \(-0.585706\pi\)
−0.266012 + 0.963970i \(0.585706\pi\)
\(192\) 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 \(-0.185258\pi\)
0.835362 + 0.549700i \(0.185258\pi\)
\(198\) 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 \(-0.780728\pi\)
−0.771968 + 0.635661i \(0.780728\pi\)
\(228\) 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 \(-0.546233\pi\)
−0.144736 + 0.989470i \(0.546233\pi\)
\(234\) 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 \(-0.775830\pi\)
−0.762096 + 0.647464i \(0.775830\pi\)
\(252\) 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 \(-0.235158\pi\)
0.739297 + 0.673379i \(0.235158\pi\)
\(258\) 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 \(-0.710117\pi\)
−0.613198 + 0.789929i \(0.710117\pi\)
\(264\) 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 \(-0.329891\pi\)
0.509335 + 0.860568i \(0.329891\pi\)
\(270\) 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 \(-0.586198\pi\)
−0.267501 + 0.963557i \(0.586198\pi\)
\(282\) 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 \(-0.781865\pi\)
−0.774235 + 0.632898i \(0.781865\pi\)
\(294\) 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 \(-0.590335\pi\)
−0.280001 + 0.960000i \(0.590335\pi\)
\(312\) 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 \(-0.688451\pi\)
−0.558052 + 0.829806i \(0.688451\pi\)
\(318\) 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 \(-0.480133\pi\)
0.0623726 + 0.998053i \(0.480133\pi\)
\(348\) 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 \(-0.310979\pi\)
0.559537 + 0.828805i \(0.310979\pi\)
\(354\) 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 \(-0.194575\pi\)
0.818917 + 0.573912i \(0.194575\pi\)
\(360\) 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 \(-0.416583\pi\)
0.259072 + 0.965858i \(0.416583\pi\)
\(384\) 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 \(-0.575039\pi\)
−0.233564 + 0.972341i \(0.575039\pi\)
\(390\) 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 \(-0.829013\pi\)
−0.859160 + 0.511707i \(0.829013\pi\)
\(402\) 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 \(-0.469223\pi\)
0.0965375 + 0.995329i \(0.469223\pi\)
\(420\) 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 \(-0.432015\pi\)
0.211961 + 0.977278i \(0.432015\pi\)
\(432\) 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 \(-0.611947\pi\)
−0.344486 + 0.938792i \(0.611947\pi\)
\(444\) 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 \(-0.435037\pi\)
0.202673 + 0.979247i \(0.435037\pi\)
\(450\) 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 \(-0.655265\pi\)
−0.468666 + 0.883375i \(0.655265\pi\)
\(462\) 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 \(-0.634790\pi\)
−0.410914 + 0.911674i \(0.634790\pi\)
\(468\) 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 \(-0.729524\pi\)
−0.660189 + 0.751100i \(0.729524\pi\)
\(480\) 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 \(-0.312058\pi\)
0.556725 + 0.830697i \(0.312058\pi\)
\(492\) 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 \(-0.232625\pi\)
0.744632 + 0.667476i \(0.232625\pi\)
\(504\) 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 \(-0.213213\pi\)
0.783929 + 0.620851i \(0.213213\pi\)
\(510\) 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 \(-0.579867\pi\)
−0.248286 + 0.968687i \(0.579867\pi\)
\(522\) 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 \(-0.814705\pi\)
−0.835298 + 0.549798i \(0.814705\pi\)
\(558\) 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 \(-0.420390\pi\)
0.247502 + 0.968887i \(0.420390\pi\)
\(564\) 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 \(-0.610891\pi\)
−0.341371 + 0.939928i \(0.610891\pi\)
\(570\) 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 \(-0.551906\pi\)
−0.162347 + 0.986734i \(0.551906\pi\)
\(588\) 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 \(-0.456779\pi\)
0.135365 + 0.990796i \(0.456779\pi\)
\(594\) 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 \(-0.729549\pi\)
−0.660248 + 0.751048i \(0.729549\pi\)
\(600\) 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 \(-0.263760\pi\)
0.675889 + 0.737004i \(0.263760\pi\)
\(618\) 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 \(-0.373964\pi\)
0.385688 + 0.922629i \(0.373964\pi\)
\(642\) 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 \(-0.331409\pi\)
0.505228 + 0.862986i \(0.331409\pi\)
\(648\) 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 \(-0.465809\pi\)
0.107209 + 0.994236i \(0.465809\pi\)
\(654\) 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 \(-0.795191\pi\)
−0.800044 + 0.599941i \(0.795191\pi\)
\(660\) 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 \(-0.311905\pi\)
0.557124 + 0.830429i \(0.311905\pi\)
\(678\) 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 \(-0.307080\pi\)
0.569646 + 0.821890i \(0.307080\pi\)
\(684\) 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 \(-0.483165\pi\)
0.0528649 + 0.998602i \(0.483165\pi\)
\(702\) 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 \(-0.462628\pi\)
0.117137 + 0.993116i \(0.462628\pi\)
\(720\) 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 \(-0.497168\pi\)
0.00889560 + 0.999960i \(0.497168\pi\)
\(744\) 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 \(-0.582857\pi\)
−0.257374 + 0.966312i \(0.582857\pi\)
\(762\) 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 \(-0.758720\pi\)
−0.726210 + 0.687473i \(0.758720\pi\)
\(774\) 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 \(-0.668950\pi\)
−0.506200 + 0.862416i \(0.668950\pi\)
\(798\) 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 \(-0.229937\pi\)
0.750241 + 0.661164i \(0.229937\pi\)
\(810\) 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 \(-0.350906\pi\)
0.451452 + 0.892295i \(0.350906\pi\)
\(822\) 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 \(-0.766764\pi\)
−0.743349 + 0.668903i \(0.766764\pi\)
\(828\) 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 \(-0.878983\pi\)
−0.928595 + 0.371095i \(0.878983\pi\)
\(840\) 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 \(-0.430482\pi\)
0.216665 + 0.976246i \(0.430482\pi\)
\(858\) 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 \(-0.345390\pi\)
0.466846 + 0.884339i \(0.345390\pi\)
\(864\) 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 \(-0.438246\pi\)
0.192792 + 0.981240i \(0.438246\pi\)
\(882\) 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 \(-0.762528\pi\)
−0.734382 + 0.678737i \(0.762528\pi\)
\(888\) 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 \(-0.189059\pi\)
0.828738 + 0.559637i \(0.189059\pi\)
\(912\) 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 \(-0.202526\pi\)
0.804327 + 0.594186i \(0.202526\pi\)
\(930\) 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 \(-0.564995\pi\)
−0.202771 + 0.979226i \(0.564995\pi\)
\(942\) 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 \(-0.710217\pi\)
−0.613445 + 0.789738i \(0.710217\pi\)
\(948\) 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 \(-0.532430\pi\)
−0.101704 + 0.994815i \(0.532430\pi\)
\(954\) 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 \(-0.620731\pi\)
−0.370258 + 0.928929i \(0.620731\pi\)
\(972\) 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 \(-0.578683\pi\)
−0.244682 + 0.969603i \(0.578683\pi\)
\(978\) 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 \(-0.255433\pi\)
0.694935 + 0.719073i \(0.255433\pi\)
\(984\) 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.k.1.10 yes 20
3.2 odd 2 2151.2.a.j.1.11 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.11 20 3.2 odd 2
2151.2.a.k.1.10 yes 20 1.1 even 1 trivial