Properties

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

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [171,2,Mod(1,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.36544187456\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} +O(q^{10})\) \(q-2.00000 q^{4} -3.00000 q^{5} -1.00000 q^{7} -3.00000 q^{11} -4.00000 q^{13} +4.00000 q^{16} +3.00000 q^{17} +1.00000 q^{19} +6.00000 q^{20} +4.00000 q^{25} +2.00000 q^{28} -6.00000 q^{29} -4.00000 q^{31} +3.00000 q^{35} +2.00000 q^{37} +6.00000 q^{41} -1.00000 q^{43} +6.00000 q^{44} +3.00000 q^{47} -6.00000 q^{49} +8.00000 q^{52} -12.0000 q^{53} +9.00000 q^{55} +6.00000 q^{59} -1.00000 q^{61} -8.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} -6.00000 q^{68} -6.00000 q^{71} -7.00000 q^{73} -2.00000 q^{76} +3.00000 q^{77} +8.00000 q^{79} -12.0000 q^{80} -12.0000 q^{83} -9.00000 q^{85} -12.0000 q^{89} +4.00000 q^{91} -3.00000 q^{95} +8.00000 q^{97} +O(q^{100})\)

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 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 6.00000 1.34164
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.00000 0.507093
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 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
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 8.00000 1.10940
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 9.00000 1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −7.00000 −0.819288 −0.409644 0.912245i \(-0.634347\pi\)
−0.409644 + 0.912245i \(0.634347\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −12.0000 −1.34164
\(81\) 0 0
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −9.00000 −0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.00000 −0.307794
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −8.00000 −0.800000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 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
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 12.0000 1.11417
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 8.00000 0.718421
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.0000 1.31056 0.655278 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) −6.00000 −0.507093
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) 18.0000 1.49482
\(146\) 0 0
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −21.0000 −1.72039 −0.860194 0.509968i \(-0.829657\pi\)
−0.860194 + 0.509968i \(0.829657\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20.0000 1.56652 0.783260 0.621694i \(-0.213555\pi\)
0.783260 + 0.621694i \(0.213555\pi\)
\(164\) −12.0000 −0.937043
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) −4.00000 −0.302372
\(176\) −12.0000 −0.904534
\(177\) 0 0
\(178\) 0 0
\(179\) 18.0000 1.34538 0.672692 0.739923i \(-0.265138\pi\)
0.672692 + 0.739923i \(0.265138\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 12.0000 0.857143
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.00000 0.421117
\(204\) 0 0
\(205\) −18.0000 −1.25717
\(206\) 0 0
\(207\) 0 0
\(208\) −16.0000 −1.10940
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 24.0000 1.64833
\(213\) 0 0
\(214\) 0 0
\(215\) 3.00000 0.204598
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 0 0
\(219\) 0 0
\(220\) −18.0000 −1.21356
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) 0 0
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.0000 1.37576 0.687878 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) 0 0
\(235\) −9.00000 −0.587095
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −21.0000 −1.32551 −0.662754 0.748837i \(-0.730613\pi\)
−0.662754 + 0.748837i \(0.730613\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) −24.0000 −1.48842
\(261\) 0 0
\(262\) 0 0
\(263\) −9.00000 −0.554964 −0.277482 0.960731i \(-0.589500\pi\)
−0.277482 + 0.960731i \(0.589500\pi\)
\(264\) 0 0
\(265\) 36.0000 2.21146
\(266\) 0 0
\(267\) 0 0
\(268\) 8.00000 0.488678
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) −19.0000 −1.14160 −0.570800 0.821089i \(-0.693367\pi\)
−0.570800 + 0.821089i \(0.693367\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −13.0000 −0.772770 −0.386385 0.922338i \(-0.626276\pi\)
−0.386385 + 0.922338i \(0.626276\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000 0.819288
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) −18.0000 −1.04800
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 0 0
\(303\) 0 0
\(304\) 4.00000 0.229416
\(305\) 3.00000 0.171780
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) −6.00000 −0.341882
\(309\) 0 0
\(310\) 0 0
\(311\) 3.00000 0.170114 0.0850572 0.996376i \(-0.472893\pi\)
0.0850572 + 0.996376i \(0.472893\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) −6.00000 −0.336994 −0.168497 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 24.0000 1.34164
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 24.0000 1.31717
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 32.0000 1.74315 0.871576 0.490261i \(-0.163099\pi\)
0.871576 + 0.490261i \(0.163099\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 18.0000 0.976187
\(341\) 12.0000 0.649836
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −21.0000 −1.12734 −0.563670 0.826000i \(-0.690611\pi\)
−0.563670 + 0.826000i \(0.690611\pi\)
\(348\) 0 0
\(349\) 17.0000 0.909989 0.454995 0.890494i \(-0.349641\pi\)
0.454995 + 0.890494i \(0.349641\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 18.0000 0.955341
\(356\) 24.0000 1.27200
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 21.0000 1.09919
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) −34.0000 −1.74646 −0.873231 0.487306i \(-0.837980\pi\)
−0.873231 + 0.487306i \(0.837980\pi\)
\(380\) 6.00000 0.307794
\(381\) 0 0
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) −9.00000 −0.458682
\(386\) 0 0
\(387\) 0 0
\(388\) −16.0000 −0.812277
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −24.0000 −1.20757
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) −12.0000 −0.599251 −0.299626 0.954057i \(-0.596862\pi\)
−0.299626 + 0.954057i \(0.596862\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 12.0000 0.597022
\(405\) 0 0
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −28.0000 −1.37946
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 36.0000 1.76717
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) 1.00000 0.0483934
\(428\) −36.0000 −1.74013
\(429\) 0 0
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 32.0000 1.53252
\(437\) 0 0
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.00000 0.142534 0.0712672 0.997457i \(-0.477296\pi\)
0.0712672 + 0.997457i \(0.477296\pi\)
\(444\) 0 0
\(445\) 36.0000 1.70656
\(446\) 0 0
\(447\) 0 0
\(448\) 8.00000 0.377964
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −18.0000 −0.847587
\(452\) 12.0000 0.564433
\(453\) 0 0
\(454\) 0 0
\(455\) −12.0000 −0.562569
\(456\) 0 0
\(457\) −37.0000 −1.73079 −0.865393 0.501093i \(-0.832931\pi\)
−0.865393 + 0.501093i \(0.832931\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.00000 −0.419172 −0.209586 0.977790i \(-0.567212\pi\)
−0.209586 + 0.977790i \(0.567212\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) −24.0000 −1.11417
\(465\) 0 0
\(466\) 0 0
\(467\) 27.0000 1.24941 0.624705 0.780860i \(-0.285219\pi\)
0.624705 + 0.780860i \(0.285219\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 4.00000 0.181818
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) −16.0000 −0.718421
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) 5.00000 0.223831 0.111915 0.993718i \(-0.464301\pi\)
0.111915 + 0.993718i \(0.464301\pi\)
\(500\) −6.00000 −0.268328
\(501\) 0 0
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −42.0000 −1.85074
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 38.0000 1.66162 0.830812 0.556553i \(-0.187876\pi\)
0.830812 + 0.556553i \(0.187876\pi\)
\(524\) −30.0000 −1.31056
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) −54.0000 −2.33462
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −25.0000 −1.07483 −0.537417 0.843317i \(-0.680600\pi\)
−0.537417 + 0.843317i \(0.680600\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 48.0000 2.05609
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 26.0000 1.10265
\(557\) −21.0000 −0.889799 −0.444899 0.895581i \(-0.646761\pi\)
−0.444899 + 0.895581i \(0.646761\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 12.0000 0.507093
\(561\) 0 0
\(562\) 0 0
\(563\) −6.00000 −0.252870 −0.126435 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 24.0000 1.00613 0.503066 0.864248i \(-0.332205\pi\)
0.503066 + 0.864248i \(0.332205\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −36.0000 −1.49482
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −45.0000 −1.85735 −0.928674 0.370896i \(-0.879051\pi\)
−0.928674 + 0.370896i \(0.879051\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 8.00000 0.328798
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 9.00000 0.368964
\(596\) 42.0000 1.72039
\(597\) 0 0
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 6.00000 0.243935
\(606\) 0 0
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −9.00000 −0.362326 −0.181163 0.983453i \(-0.557986\pi\)
−0.181163 + 0.983453i \(0.557986\pi\)
\(618\) 0 0
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) −24.0000 −0.963863
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0000 0.480770
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) −28.0000 −1.11732
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 11.0000 0.437903 0.218952 0.975736i \(-0.429736\pi\)
0.218952 + 0.975736i \(0.429736\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 24.0000 0.950915
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −13.0000 −0.512670 −0.256335 0.966588i \(-0.582515\pi\)
−0.256335 + 0.966588i \(0.582515\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −27.0000 −1.06148 −0.530740 0.847535i \(-0.678086\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 0 0
\(649\) −18.0000 −0.706562
\(650\) 0 0
\(651\) 0 0
\(652\) −40.0000 −1.56652
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) −45.0000 −1.75830
\(656\) 24.0000 0.937043
\(657\) 0 0
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.00000 0.116335
\(666\) 0 0
\(667\) 0 0
\(668\) −36.0000 −1.39288
\(669\) 0 0
\(670\) 0 0
\(671\) 3.00000 0.115814
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −6.00000 −0.230769
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −36.0000 −1.37750 −0.688751 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) 48.0000 1.82865
\(690\) 0 0
\(691\) 17.0000 0.646710 0.323355 0.946278i \(-0.395189\pi\)
0.323355 + 0.946278i \(0.395189\pi\)
\(692\) −36.0000 −1.36851
\(693\) 0 0
\(694\) 0 0
\(695\) 39.0000 1.47935
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 8.00000 0.302372
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 2.00000 0.0754314
\(704\) 24.0000 0.904534
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 26.0000 0.976450 0.488225 0.872718i \(-0.337644\pi\)
0.488225 + 0.872718i \(0.337644\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −36.0000 −1.34632
\(716\) −36.0000 −1.34538
\(717\) 0 0
\(718\) 0 0
\(719\) −15.0000 −0.559406 −0.279703 0.960087i \(-0.590236\pi\)
−0.279703 + 0.960087i \(0.590236\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 0 0
\(723\) 0 0
\(724\) −4.00000 −0.148659
\(725\) −24.0000 −0.891338
\(726\) 0 0
\(727\) −19.0000 −0.704671 −0.352335 0.935874i \(-0.614612\pi\)
−0.352335 + 0.935874i \(0.614612\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) 11.0000 0.404642 0.202321 0.979319i \(-0.435152\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(740\) 12.0000 0.441129
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) 63.0000 2.30814
\(746\) 0 0
\(747\) 0 0
\(748\) 18.0000 0.658145
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) 12.0000 0.437595
\(753\) 0 0
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) −25.0000 −0.908640 −0.454320 0.890838i \(-0.650118\pi\)
−0.454320 + 0.890838i \(0.650118\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −33.0000 −1.19625 −0.598125 0.801403i \(-0.704087\pi\)
−0.598125 + 0.801403i \(0.704087\pi\)
\(762\) 0 0
\(763\) 16.0000 0.579239
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) 23.0000 0.829401 0.414701 0.909958i \(-0.363886\pi\)
0.414701 + 0.909958i \(0.363886\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 8.00000 0.287926
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) −16.0000 −0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 0 0
\(783\) 0 0
\(784\) −24.0000 −0.857143
\(785\) −42.0000 −1.49904
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 36.0000 1.28245
\(789\) 0 0
\(790\) 0 0
\(791\) 6.00000 0.213335
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 0 0
\(796\) −22.0000 −0.779769
\(797\) 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
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 21.0000 0.741074
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) 0 0
\(815\) −60.0000 −2.10171
\(816\) 0 0
\(817\) −1.00000 −0.0349856
\(818\) 0 0
\(819\) 0 0
\(820\) 36.0000 1.25717
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 0 0
\(823\) −49.0000 −1.70803 −0.854016 0.520246i \(-0.825840\pi\)
−0.854016 + 0.520246i \(0.825840\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −16.0000 −0.555703 −0.277851 0.960624i \(-0.589622\pi\)
−0.277851 + 0.960624i \(0.589622\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 32.0000 1.10940
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) −54.0000 −1.86875
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) −28.0000 −0.963800
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) −48.0000 −1.64833
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −49.0000 −1.67186 −0.835929 0.548837i \(-0.815071\pi\)
−0.835929 + 0.548837i \(0.815071\pi\)
\(860\) −6.00000 −0.204598
\(861\) 0 0
\(862\) 0 0
\(863\) −18.0000 −0.612727 −0.306364 0.951915i \(-0.599112\pi\)
−0.306364 + 0.951915i \(0.599112\pi\)
\(864\) 0 0
\(865\) −54.0000 −1.83606
\(866\) 0 0
\(867\) 0 0
\(868\) −8.00000 −0.271538
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 36.0000 1.21356
\(881\) 27.0000 0.909653 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) 47.0000 1.58168 0.790838 0.612026i \(-0.209645\pi\)
0.790838 + 0.612026i \(0.209645\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 0 0
\(887\) −18.0000 −0.604381 −0.302190 0.953248i \(-0.597718\pi\)
−0.302190 + 0.953248i \(0.597718\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 0 0
\(892\) 20.0000 0.669650
\(893\) 3.00000 0.100391
\(894\) 0 0
\(895\) −54.0000 −1.80502
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 24.0000 0.800445
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 24.0000 0.796468
\(909\) 0 0
\(910\) 0 0
\(911\) 6.00000 0.198789 0.0993944 0.995048i \(-0.468309\pi\)
0.0993944 + 0.995048i \(0.468309\pi\)
\(912\) 0 0
\(913\) 36.0000 1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) −15.0000 −0.495344
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) −42.0000 −1.37576
\(933\) 0 0
\(934\) 0 0
\(935\) 27.0000 0.882994
\(936\) 0 0
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 18.0000 0.587095
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 24.0000 0.781133
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000 1.16984 0.584921 0.811090i \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 48.0000 1.55487 0.777436 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(954\) 0 0
\(955\) 9.00000 0.291233
\(956\) 30.0000 0.970269
\(957\) 0 0
\(958\) 0 0
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 20.0000 0.644157
\(965\) 12.0000 0.386294
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) 13.0000 0.416761
\(974\) 0 0
\(975\) 0 0
\(976\) −4.00000 −0.128037
\(977\) −24.0000 −0.767828 −0.383914 0.923369i \(-0.625424\pi\)
−0.383914 + 0.923369i \(0.625424\pi\)
\(978\) 0 0
\(979\) 36.0000 1.15056
\(980\) −36.0000 −1.14998
\(981\) 0 0
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 54.0000 1.72058
\(986\) 0 0
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −33.0000 −1.04617
\(996\) 0 0
\(997\) 17.0000 0.538395 0.269198 0.963085i \(-0.413241\pi\)
0.269198 + 0.963085i \(0.413241\pi\)
\(998\) 0 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 171.2.a.b.1.1 1
3.2 odd 2 19.2.a.a.1.1 1
4.3 odd 2 2736.2.a.c.1.1 1
5.4 even 2 4275.2.a.i.1.1 1
7.6 odd 2 8379.2.a.j.1.1 1
12.11 even 2 304.2.a.f.1.1 1
15.2 even 4 475.2.b.a.324.2 2
15.8 even 4 475.2.b.a.324.1 2
15.14 odd 2 475.2.a.b.1.1 1
19.18 odd 2 3249.2.a.d.1.1 1
21.2 odd 6 931.2.f.c.704.1 2
21.5 even 6 931.2.f.b.704.1 2
21.11 odd 6 931.2.f.c.324.1 2
21.17 even 6 931.2.f.b.324.1 2
21.20 even 2 931.2.a.a.1.1 1
24.5 odd 2 1216.2.a.o.1.1 1
24.11 even 2 1216.2.a.b.1.1 1
33.32 even 2 2299.2.a.b.1.1 1
39.38 odd 2 3211.2.a.a.1.1 1
51.50 odd 2 5491.2.a.b.1.1 1
57.2 even 18 361.2.e.e.99.1 6
57.5 odd 18 361.2.e.d.234.1 6
57.8 even 6 361.2.c.a.292.1 2
57.11 odd 6 361.2.c.c.292.1 2
57.14 even 18 361.2.e.e.234.1 6
57.17 odd 18 361.2.e.d.99.1 6
57.23 odd 18 361.2.e.d.54.1 6
57.26 odd 6 361.2.c.c.68.1 2
57.29 even 18 361.2.e.e.62.1 6
57.32 even 18 361.2.e.e.245.1 6
57.35 odd 18 361.2.e.d.28.1 6
57.41 even 18 361.2.e.e.28.1 6
57.44 odd 18 361.2.e.d.245.1 6
57.47 odd 18 361.2.e.d.62.1 6
57.50 even 6 361.2.c.a.68.1 2
57.53 even 18 361.2.e.e.54.1 6
57.56 even 2 361.2.a.b.1.1 1
60.59 even 2 7600.2.a.c.1.1 1
228.227 odd 2 5776.2.a.c.1.1 1
285.284 even 2 9025.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.2.a.a.1.1 1 3.2 odd 2
171.2.a.b.1.1 1 1.1 even 1 trivial
304.2.a.f.1.1 1 12.11 even 2
361.2.a.b.1.1 1 57.56 even 2
361.2.c.a.68.1 2 57.50 even 6
361.2.c.a.292.1 2 57.8 even 6
361.2.c.c.68.1 2 57.26 odd 6
361.2.c.c.292.1 2 57.11 odd 6
361.2.e.d.28.1 6 57.35 odd 18
361.2.e.d.54.1 6 57.23 odd 18
361.2.e.d.62.1 6 57.47 odd 18
361.2.e.d.99.1 6 57.17 odd 18
361.2.e.d.234.1 6 57.5 odd 18
361.2.e.d.245.1 6 57.44 odd 18
361.2.e.e.28.1 6 57.41 even 18
361.2.e.e.54.1 6 57.53 even 18
361.2.e.e.62.1 6 57.29 even 18
361.2.e.e.99.1 6 57.2 even 18
361.2.e.e.234.1 6 57.14 even 18
361.2.e.e.245.1 6 57.32 even 18
475.2.a.b.1.1 1 15.14 odd 2
475.2.b.a.324.1 2 15.8 even 4
475.2.b.a.324.2 2 15.2 even 4
931.2.a.a.1.1 1 21.20 even 2
931.2.f.b.324.1 2 21.17 even 6
931.2.f.b.704.1 2 21.5 even 6
931.2.f.c.324.1 2 21.11 odd 6
931.2.f.c.704.1 2 21.2 odd 6
1216.2.a.b.1.1 1 24.11 even 2
1216.2.a.o.1.1 1 24.5 odd 2
2299.2.a.b.1.1 1 33.32 even 2
2736.2.a.c.1.1 1 4.3 odd 2
3211.2.a.a.1.1 1 39.38 odd 2
3249.2.a.d.1.1 1 19.18 odd 2
4275.2.a.i.1.1 1 5.4 even 2
5491.2.a.b.1.1 1 51.50 odd 2
5776.2.a.c.1.1 1 228.227 odd 2
7600.2.a.c.1.1 1 60.59 even 2
8379.2.a.j.1.1 1 7.6 odd 2
9025.2.a.d.1.1 1 285.284 even 2