Properties

Label 171.2.a.c.1.1
Level $171$
Weight $2$
Character 171.1
Self dual yes
Analytic conductor $1.365$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.36544187456\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 171.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} -2.00000 q^{10} +3.00000 q^{11} -6.00000 q^{13} +6.00000 q^{14} -4.00000 q^{16} -3.00000 q^{17} -1.00000 q^{19} -2.00000 q^{20} +6.00000 q^{22} -4.00000 q^{23} -4.00000 q^{25} -12.0000 q^{26} +6.00000 q^{28} +10.0000 q^{29} +2.00000 q^{31} -8.00000 q^{32} -6.00000 q^{34} -3.00000 q^{35} +8.00000 q^{37} -2.00000 q^{38} +8.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} -8.00000 q^{46} -3.00000 q^{47} +2.00000 q^{49} -8.00000 q^{50} -12.0000 q^{52} +6.00000 q^{53} -3.00000 q^{55} +20.0000 q^{58} +7.00000 q^{61} +4.00000 q^{62} -8.00000 q^{64} +6.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} -6.00000 q^{70} -12.0000 q^{71} -11.0000 q^{73} +16.0000 q^{74} -2.00000 q^{76} +9.00000 q^{77} +4.00000 q^{80} +16.0000 q^{82} -4.00000 q^{83} +3.00000 q^{85} -2.00000 q^{86} -10.0000 q^{89} -18.0000 q^{91} -8.00000 q^{92} -6.00000 q^{94} +1.00000 q^{95} -2.00000 q^{97} +4.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −12.0000 −2.35339
\(27\) 0 0
\(28\) 6.00000 1.13389
\(29\) 10.0000 1.85695 0.928477 0.371391i \(-0.121119\pi\)
0.928477 + 0.371391i \(0.121119\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) −8.00000 −1.13137
\(51\) 0 0
\(52\) −12.0000 −1.66410
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −3.00000 −0.404520
\(56\) 0 0
\(57\) 0 0
\(58\) 20.0000 2.62613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) −6.00000 −0.717137
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 16.0000 1.85996
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 9.00000 1.02565
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 4.00000 0.447214
\(81\) 0 0
\(82\) 16.0000 1.76690
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −2.00000 −0.215666
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) −18.0000 −1.88691
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 4.00000 0.404061
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 14.0000 1.37946 0.689730 0.724066i \(-0.257729\pi\)
0.689730 + 0.724066i \(0.257729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) −6.00000 −0.572078
\(111\) 0 0
\(112\) −12.0000 −1.13389
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) 20.0000 1.85695
\(117\) 0 0
\(118\) 0 0
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 14.0000 1.26750
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 12.0000 1.05247
\(131\) 13.0000 1.13582 0.567908 0.823092i \(-0.307753\pi\)
0.567908 + 0.823092i \(0.307753\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) −24.0000 −2.01404
\(143\) −18.0000 −1.50524
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) −22.0000 −1.82073
\(147\) 0 0
\(148\) 16.0000 1.31519
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 18.0000 1.45048
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 8.00000 0.632456
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 16.0000 1.24939
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) −20.0000 −1.49906
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) −36.0000 −2.66850
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 4.00000 0.285714
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −5.00000 −0.354441 −0.177220 0.984171i \(-0.556711\pi\)
−0.177220 + 0.984171i \(0.556711\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −4.00000 −0.281439
\(203\) 30.0000 2.10559
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 28.0000 1.95085
\(207\) 0 0
\(208\) 24.0000 1.66410
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 40.0000 2.70914
\(219\) 0 0
\(220\) −6.00000 −0.404520
\(221\) 18.0000 1.21081
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −24.0000 −1.60357
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) 11.0000 0.720634 0.360317 0.932830i \(-0.382669\pi\)
0.360317 + 0.932830i \(0.382669\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) 0 0
\(237\) 0 0
\(238\) −18.0000 −1.16677
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) −4.00000 −0.257130
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 0 0
\(250\) 18.0000 1.13842
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) 0 0
\(259\) 24.0000 1.49129
\(260\) 12.0000 0.744208
\(261\) 0 0
\(262\) 26.0000 1.60629
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) 16.0000 0.977356
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) −10.0000 −0.599760
\(279\) 0 0
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 19.0000 1.12943 0.564716 0.825285i \(-0.308986\pi\)
0.564716 + 0.825285i \(0.308986\pi\)
\(284\) −24.0000 −1.42414
\(285\) 0 0
\(286\) −36.0000 −2.12872
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) −20.0000 −1.17444
\(291\) 0 0
\(292\) −22.0000 −1.28745
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −30.0000 −1.73785
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) −7.00000 −0.400819
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 18.0000 1.02565
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −7.00000 −0.396934 −0.198467 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000 0.673987 0.336994 0.941507i \(-0.390590\pi\)
0.336994 + 0.941507i \(0.390590\pi\)
\(318\) 0 0
\(319\) 30.0000 1.67968
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) −32.0000 −1.77232
\(327\) 0 0
\(328\) 0 0
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) −36.0000 −1.96983
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 46.0000 2.50207
\(339\) 0 0
\(340\) 6.00000 0.325396
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 0 0
\(345\) 0 0
\(346\) −28.0000 −1.50529
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 0 0
\(349\) 25.0000 1.33822 0.669110 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) −24.0000 −1.28285
\(351\) 0 0
\(352\) −24.0000 −1.27920
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −20.0000 −1.06000
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) −25.0000 −1.31945 −0.659725 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 4.00000 0.210235
\(363\) 0 0
\(364\) −36.0000 −1.88691
\(365\) 11.0000 0.575766
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 16.0000 0.834058
\(369\) 0 0
\(370\) −16.0000 −0.831800
\(371\) 18.0000 0.934513
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −18.0000 −0.930758
\(375\) 0 0
\(376\) 0 0
\(377\) −60.0000 −3.09016
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) 6.00000 0.306987
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) −9.00000 −0.458682
\(386\) 8.00000 0.407189
\(387\) 0 0
\(388\) −4.00000 −0.203069
\(389\) 15.0000 0.760530 0.380265 0.924878i \(-0.375833\pi\)
0.380265 + 0.924878i \(0.375833\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 4.00000 0.201517
\(395\) 0 0
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) 28.0000 1.39825 0.699127 0.714998i \(-0.253572\pi\)
0.699127 + 0.714998i \(0.253572\pi\)
\(402\) 0 0
\(403\) −12.0000 −0.597763
\(404\) −4.00000 −0.199007
\(405\) 0 0
\(406\) 60.0000 2.97775
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) −16.0000 −0.790184
\(411\) 0 0
\(412\) 28.0000 1.37946
\(413\) 0 0
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 48.0000 2.35339
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) −56.0000 −2.72604
\(423\) 0 0
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 21.0000 1.01626
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 2.00000 0.0964486
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) 40.0000 1.91565
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 10.0000 0.477274 0.238637 0.971109i \(-0.423299\pi\)
0.238637 + 0.971109i \(0.423299\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 36.0000 1.71235
\(443\) −39.0000 −1.85295 −0.926473 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 8.00000 0.378811
\(447\) 0 0
\(448\) −24.0000 −1.13389
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) −36.0000 −1.68956
\(455\) 18.0000 0.843853
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) −30.0000 −1.40181
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) −40.0000 −1.85695
\(465\) 0 0
\(466\) 22.0000 1.01913
\(467\) 17.0000 0.786666 0.393333 0.919396i \(-0.371322\pi\)
0.393333 + 0.919396i \(0.371322\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) 6.00000 0.276759
\(471\) 0 0
\(472\) 0 0
\(473\) −3.00000 −0.137940
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −18.0000 −0.825029
\(477\) 0 0
\(478\) 30.0000 1.37217
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) −4.00000 −0.181818
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −4.00000 −0.180702
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) −30.0000 −1.35113
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −36.0000 −1.61482
\(498\) 0 0
\(499\) −35.0000 −1.56682 −0.783408 0.621508i \(-0.786520\pi\)
−0.783408 + 0.621508i \(0.786520\pi\)
\(500\) 18.0000 0.804984
\(501\) 0 0
\(502\) −54.0000 −2.41014
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) −24.0000 −1.06693
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) −33.0000 −1.45983
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) −16.0000 −0.705730
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 48.0000 2.10900
\(519\) 0 0
\(520\) 0 0
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 0 0
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 26.0000 1.13582
\(525\) 0 0
\(526\) 42.0000 1.83129
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) −6.00000 −0.260133
\(533\) −48.0000 −2.07911
\(534\) 0 0
\(535\) −2.00000 −0.0864675
\(536\) 0 0
\(537\) 0 0
\(538\) 60.0000 2.58678
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 24.0000 1.03089
\(543\) 0 0
\(544\) 24.0000 1.02899
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) −24.0000 −1.02336
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −10.0000 −0.424094
\(557\) −3.00000 −0.127114 −0.0635570 0.997978i \(-0.520244\pi\)
−0.0635570 + 0.997978i \(0.520244\pi\)
\(558\) 0 0
\(559\) 6.00000 0.253773
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) −4.00000 −0.168730
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 38.0000 1.59726
\(567\) 0 0
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −36.0000 −1.50524
\(573\) 0 0
\(574\) 48.0000 2.00348
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) 3.00000 0.124892 0.0624458 0.998048i \(-0.480110\pi\)
0.0624458 + 0.998048i \(0.480110\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) −20.0000 −0.830455
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) 0 0
\(586\) −8.00000 −0.330477
\(587\) 37.0000 1.52715 0.763577 0.645717i \(-0.223441\pi\)
0.763577 + 0.645717i \(0.223441\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) −32.0000 −1.31519
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 9.00000 0.368964
\(596\) −30.0000 −1.22885
\(597\) 0 0
\(598\) 48.0000 1.96287
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) −6.00000 −0.244542
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 18.0000 0.730597 0.365299 0.930890i \(-0.380967\pi\)
0.365299 + 0.930890i \(0.380967\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) 9.00000 0.363507 0.181753 0.983344i \(-0.441823\pi\)
0.181753 + 0.983344i \(0.441823\pi\)
\(614\) −24.0000 −0.968561
\(615\) 0 0
\(616\) 0 0
\(617\) −23.0000 −0.925945 −0.462973 0.886373i \(-0.653217\pi\)
−0.462973 + 0.886373i \(0.653217\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) −4.00000 −0.160644
\(621\) 0 0
\(622\) −14.0000 −0.561349
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 28.0000 1.11911
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 24.0000 0.953162
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 60.0000 2.37542
\(639\) 0 0
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) −1.00000 −0.0394362 −0.0197181 0.999806i \(-0.506277\pi\)
−0.0197181 + 0.999806i \(0.506277\pi\)
\(644\) −24.0000 −0.945732
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 27.0000 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 48.0000 1.88271
\(651\) 0 0
\(652\) −32.0000 −1.25322
\(653\) 1.00000 0.0391330 0.0195665 0.999809i \(-0.493771\pi\)
0.0195665 + 0.999809i \(0.493771\pi\)
\(654\) 0 0
\(655\) −13.0000 −0.507952
\(656\) −32.0000 −1.24939
\(657\) 0 0
\(658\) −18.0000 −0.701713
\(659\) −10.0000 −0.389545 −0.194772 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 24.0000 0.932786
\(663\) 0 0
\(664\) 0 0
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) −40.0000 −1.54881
\(668\) −36.0000 −1.39288
\(669\) 0 0
\(670\) −16.0000 −0.618134
\(671\) 21.0000 0.810696
\(672\) 0 0
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) −44.0000 −1.69482
\(675\) 0 0
\(676\) 46.0000 1.76923
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −6.00000 −0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) −30.0000 −1.14541
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −28.0000 −1.06440
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) 5.00000 0.189661
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 50.0000 1.89253
\(699\) 0 0
\(700\) −24.0000 −0.907115
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) −24.0000 −0.904534
\(705\) 0 0
\(706\) −28.0000 −1.05379
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 24.0000 0.900704
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 18.0000 0.673162
\(716\) 20.0000 0.747435
\(717\) 0 0
\(718\) −50.0000 −1.86598
\(719\) 35.0000 1.30528 0.652640 0.757668i \(-0.273661\pi\)
0.652640 + 0.757668i \(0.273661\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 2.00000 0.0744323
\(723\) 0 0
\(724\) 4.00000 0.148659
\(725\) −40.0000 −1.48556
\(726\) 0 0
\(727\) −7.00000 −0.259616 −0.129808 0.991539i \(-0.541436\pi\)
−0.129808 + 0.991539i \(0.541436\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 22.0000 0.814257
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 32.0000 1.17954
\(737\) 24.0000 0.884051
\(738\) 0 0
\(739\) −45.0000 −1.65535 −0.827676 0.561206i \(-0.810337\pi\)
−0.827676 + 0.561206i \(0.810337\pi\)
\(740\) −16.0000 −0.588172
\(741\) 0 0
\(742\) 36.0000 1.32160
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 15.0000 0.549557
\(746\) −32.0000 −1.17160
\(747\) 0 0
\(748\) −18.0000 −0.658145
\(749\) 6.00000 0.219235
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) −120.000 −4.37014
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) −60.0000 −2.17930
\(759\) 0 0
\(760\) 0 0
\(761\) 13.0000 0.471250 0.235625 0.971844i \(-0.424286\pi\)
0.235625 + 0.971844i \(0.424286\pi\)
\(762\) 0 0
\(763\) 60.0000 2.17215
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) −28.0000 −1.01168
\(767\) 0 0
\(768\) 0 0
\(769\) −45.0000 −1.62274 −0.811371 0.584532i \(-0.801278\pi\)
−0.811371 + 0.584532i \(0.801278\pi\)
\(770\) −18.0000 −0.648675
\(771\) 0 0
\(772\) 8.00000 0.287926
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) −8.00000 −0.285714
\(785\) 2.00000 0.0713831
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) 4.00000 0.142494
\(789\) 0 0
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) −42.0000 −1.49146
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) 32.0000 1.13137
\(801\) 0 0
\(802\) 56.0000 1.97743
\(803\) −33.0000 −1.16454
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) −24.0000 −0.845364
\(807\) 0 0
\(808\) 0 0
\(809\) −5.00000 −0.175791 −0.0878953 0.996130i \(-0.528014\pi\)
−0.0878953 + 0.996130i \(0.528014\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 60.0000 2.10559
\(813\) 0 0
\(814\) 48.0000 1.68240
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 1.00000 0.0349856
\(818\) 20.0000 0.699284
\(819\) 0 0
\(820\) −16.0000 −0.558744
\(821\) 33.0000 1.15171 0.575854 0.817553i \(-0.304670\pi\)
0.575854 + 0.817553i \(0.304670\pi\)
\(822\) 0 0
\(823\) 19.0000 0.662298 0.331149 0.943578i \(-0.392564\pi\)
0.331149 + 0.943578i \(0.392564\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 52.0000 1.80822 0.904109 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 8.00000 0.277684
\(831\) 0 0
\(832\) 48.0000 1.66410
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 18.0000 0.622916
\(836\) −6.00000 −0.207514
\(837\) 0 0
\(838\) −40.0000 −1.38178
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) 4.00000 0.137849
\(843\) 0 0
\(844\) −56.0000 −1.92760
\(845\) −23.0000 −0.791224
\(846\) 0 0
\(847\) −6.00000 −0.206162
\(848\) −24.0000 −0.824163
\(849\) 0 0
\(850\) 24.0000 0.823193
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 42.0000 1.43721
\(855\) 0 0
\(856\) 0 0
\(857\) −48.0000 −1.63965 −0.819824 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) 0 0
\(859\) 35.0000 1.19418 0.597092 0.802173i \(-0.296323\pi\)
0.597092 + 0.802173i \(0.296323\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) −52.0000 −1.76703
\(867\) 0 0
\(868\) 12.0000 0.407307
\(869\) 0 0
\(870\) 0 0
\(871\) −48.0000 −1.62642
\(872\) 0 0
\(873\) 0 0
\(874\) 8.00000 0.270604
\(875\) 27.0000 0.912767
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 20.0000 0.674967
\(879\) 0 0
\(880\) 12.0000 0.404520
\(881\) −7.00000 −0.235836 −0.117918 0.993023i \(-0.537622\pi\)
−0.117918 + 0.993023i \(0.537622\pi\)
\(882\) 0 0
\(883\) −21.0000 −0.706706 −0.353353 0.935490i \(-0.614959\pi\)
−0.353353 + 0.935490i \(0.614959\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) −78.0000 −2.62046
\(887\) 52.0000 1.74599 0.872995 0.487730i \(-0.162175\pi\)
0.872995 + 0.487730i \(0.162175\pi\)
\(888\) 0 0
\(889\) −6.00000 −0.201234
\(890\) 20.0000 0.670402
\(891\) 0 0
\(892\) 8.00000 0.267860
\(893\) 3.00000 0.100391
\(894\) 0 0
\(895\) −10.0000 −0.334263
\(896\) 0 0
\(897\) 0 0
\(898\) 40.0000 1.33482
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 48.0000 1.59823
\(903\) 0 0
\(904\) 0 0
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) −2.00000 −0.0664089 −0.0332045 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(908\) −36.0000 −1.19470
\(909\) 0 0
\(910\) 36.0000 1.19339
\(911\) −2.00000 −0.0662630 −0.0331315 0.999451i \(-0.510548\pi\)
−0.0331315 + 0.999451i \(0.510548\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 6.00000 0.198462
\(915\) 0 0
\(916\) −30.0000 −0.991228
\(917\) 39.0000 1.28789
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 66.0000 2.17359
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) −62.0000 −2.03745
\(927\) 0 0
\(928\) −80.0000 −2.62613
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) 22.0000 0.720634
\(933\) 0 0
\(934\) 34.0000 1.11251
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) 53.0000 1.73143 0.865717 0.500533i \(-0.166863\pi\)
0.865717 + 0.500533i \(0.166863\pi\)
\(938\) 48.0000 1.56726
\(939\) 0 0
\(940\) 6.00000 0.195698
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 0 0
\(943\) −32.0000 −1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) 66.0000 2.14245
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) 0 0
\(953\) 16.0000 0.518291 0.259145 0.965838i \(-0.416559\pi\)
0.259145 + 0.965838i \(0.416559\pi\)
\(954\) 0 0
\(955\) −3.00000 −0.0970777
\(956\) 30.0000 0.970269
\(957\) 0 0
\(958\) 80.0000 2.58468
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) −96.0000 −3.09516
\(963\) 0 0
\(964\) 24.0000 0.772988
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) −15.0000 −0.480878
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) −28.0000 −0.896258
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) −30.0000 −0.958804
\(980\) −4.00000 −0.127775
\(981\) 0 0
\(982\) 16.0000 0.510581
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) −60.0000 −1.91079
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −16.0000 −0.508001
\(993\) 0 0
\(994\) −72.0000 −2.28370
\(995\) 5.00000 0.158511
\(996\) 0 0
\(997\) −7.00000 −0.221692 −0.110846 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(998\) −70.0000 −2.21581
\(999\) 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 171.2.a.c.1.1 1
3.2 odd 2 57.2.a.b.1.1 1
4.3 odd 2 2736.2.a.h.1.1 1
5.4 even 2 4275.2.a.a.1.1 1
7.6 odd 2 8379.2.a.q.1.1 1
12.11 even 2 912.2.a.d.1.1 1
15.2 even 4 1425.2.c.a.799.1 2
15.8 even 4 1425.2.c.a.799.2 2
15.14 odd 2 1425.2.a.i.1.1 1
19.18 odd 2 3249.2.a.a.1.1 1
21.20 even 2 2793.2.a.a.1.1 1
24.5 odd 2 3648.2.a.h.1.1 1
24.11 even 2 3648.2.a.y.1.1 1
33.32 even 2 6897.2.a.g.1.1 1
39.38 odd 2 9633.2.a.p.1.1 1
57.56 even 2 1083.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 1 3.2 odd 2
171.2.a.c.1.1 1 1.1 even 1 trivial
912.2.a.d.1.1 1 12.11 even 2
1083.2.a.d.1.1 1 57.56 even 2
1425.2.a.i.1.1 1 15.14 odd 2
1425.2.c.a.799.1 2 15.2 even 4
1425.2.c.a.799.2 2 15.8 even 4
2736.2.a.h.1.1 1 4.3 odd 2
2793.2.a.a.1.1 1 21.20 even 2
3249.2.a.a.1.1 1 19.18 odd 2
3648.2.a.h.1.1 1 24.5 odd 2
3648.2.a.y.1.1 1 24.11 even 2
4275.2.a.a.1.1 1 5.4 even 2
6897.2.a.g.1.1 1 33.32 even 2
8379.2.a.q.1.1 1 7.6 odd 2
9633.2.a.p.1.1 1 39.38 odd 2