↑

Properties

Label 1386.2.a.d.1.1
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(1,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1386.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.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: \(11.0672657201\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1386.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} -1.00000 q^{11} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -6.00000 q^{19} +2.00000 q^{20} +1.00000 q^{22} -2.00000 q^{23} -1.00000 q^{25} +4.00000 q^{26} -1.00000 q^{28} -2.00000 q^{29} -8.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} -2.00000 q^{35} -2.00000 q^{37} +6.00000 q^{38} -2.00000 q^{40} -2.00000 q^{41} -2.00000 q^{43} -1.00000 q^{44} +2.00000 q^{46} +2.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} +6.00000 q^{53} -2.00000 q^{55} +1.00000 q^{56} +2.00000 q^{58} +4.00000 q^{61} +8.00000 q^{62} +1.00000 q^{64} -8.00000 q^{65} +2.00000 q^{68} +2.00000 q^{70} +10.0000 q^{71} -2.00000 q^{73} +2.00000 q^{74} -6.00000 q^{76} +1.00000 q^{77} -12.0000 q^{79} +2.00000 q^{80} +2.00000 q^{82} -12.0000 q^{83} +4.00000 q^{85} +2.00000 q^{86} +1.00000 q^{88} +12.0000 q^{89} +4.00000 q^{91} -2.00000 q^{92} -2.00000 q^{94} -12.0000 q^{95} -2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 − 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −2.00000 −0.316228
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) 2.00000 0.291730 0.145865 − 0.989305i \(-0.453403\pi\)
0.145865 + 0.989305i \(0.453403\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 1.00000 0.133631
\(57\) 0 0
\(58\) 2.00000 0.262613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 − 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 8.00000 1.01600
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 2.00000 0.223607
\(81\) 0 0
\(82\) 2.00000 0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 2.00000 0.215666
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) 12.0000 1.27200 0.635999 − 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) −2.00000 −0.208514
\(93\) 0 0
\(94\) −2.00000 −0.206284
\(95\) −12.0000 −1.23117
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 − 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) −4.00000 −0.376288 −0.188144 − 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) −2.00000 −0.185695
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 − 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 8.00000 0.701646
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 6.00000 0.520266
\(134\) 0 0
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 − 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) −10.0000 −0.839181
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 2.00000 0.165521
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −22.0000 −1.80231 −0.901155 − 0.433497i \(-0.857280\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 − 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) −16.0000 −1.28515
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 − 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) −2.00000 −0.158114
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) −8.00000 −0.626608 −0.313304 − 0.949653i \(-0.601436\pi\)
−0.313304 + 0.949653i \(0.601436\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) −2.00000 −0.152499
\(173\) 22.0000 1.67263 0.836315 − 0.548250i \(-0.184706\pi\)
0.836315 + 0.548250i \(0.184706\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) −12.0000 −0.899438
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 − 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) −4.00000 −0.296500
\(183\) 0 0
\(184\) 2.00000 0.147442
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 2.00000 0.145865
\(189\) 0 0
\(190\) 12.0000 0.870572
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 2.00000 0.140372
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) 0 0
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 − 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 4.00000 0.273434
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) 8.00000 0.543075
\(218\) 8.00000 0.541828
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) 4.00000 0.266076
\(227\) −20.0000 −1.32745 −0.663723 − 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) 2.00000 0.131306
\(233\) 18.0000 1.17922 0.589610 − 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 2.00000 0.129641
\(239\) 28.0000 1.81117 0.905585 − 0.424165i \(-0.139432\pi\)
0.905585 + 0.424165i \(0.139432\pi\)
\(240\) 0 0
\(241\) −30.0000 −1.93247 −0.966235 − 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 8.00000 0.508001
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 2.00000 0.124274
\(260\) −8.00000 −0.496139
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) −6.00000 −0.367884
\(267\) 0 0
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 12.0000 0.721010 0.360505 − 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) −2.00000 −0.119952
\(279\) 0 0
\(280\) 2.00000 0.119523
\(281\) 2.00000 0.119310 0.0596550 − 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 − 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) 0 0
\(292\) −2.00000 −0.117041
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 0 0
\(298\) 22.0000 1.27443
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 2.00000 0.115278
\(302\) 12.0000 0.690522
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 8.00000 0.458079
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 − 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) 16.0000 0.908739
\(311\) 2.00000 0.113410 0.0567048 − 0.998391i \(-0.481941\pi\)
0.0567048 + 0.998391i \(0.481941\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 30.0000 1.68497 0.842484 − 0.538721i \(-0.181092\pi\)
0.842484 + 0.538721i \(0.181092\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) −2.00000 −0.111456
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 8.00000 0.443079
\(327\) 0 0
\(328\) 2.00000 0.110432
\(329\) −2.00000 −0.110264
\(330\) 0 0
\(331\) 16.0000 0.879440 0.439720 − 0.898135i \(-0.355078\pi\)
0.439720 + 0.898135i \(0.355078\pi\)
\(332\) −12.0000 −0.658586
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 − 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 2.00000 0.107833
\(345\) 0 0
\(346\) −22.0000 −1.18273
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 − 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) −36.0000 −1.91609 −0.958043 − 0.286623i \(-0.907467\pi\)
−0.958043 + 0.286623i \(0.907467\pi\)
\(354\) 0 0
\(355\) 20.0000 1.06149
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 32.0000 1.68890 0.844448 − 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −14.0000 −0.735824
\(363\) 0 0
\(364\) 4.00000 0.209657
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 − 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −2.00000 −0.104257
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −4.00000 −0.207112 −0.103556 − 0.994624i \(-0.533022\pi\)
−0.103556 + 0.994624i \(0.533022\pi\)
\(374\) 2.00000 0.103418
\(375\) 0 0
\(376\) −2.00000 −0.103142
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −12.0000 −0.615587
\(381\) 0 0
\(382\) 10.0000 0.511645
\(383\) 2.00000 0.102195 0.0510976 − 0.998694i \(-0.483728\pi\)
0.0510976 + 0.998694i \(0.483728\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 18.0000 0.912636 0.456318 − 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) −18.0000 −0.906827
\(395\) −24.0000 −1.20757
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 − 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 − 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 4.00000 0.197546
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −24.0000 −1.17811
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) −6.00000 −0.293470
\(419\) 4.00000 0.195413 0.0977064 − 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 − 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −10.0000 −0.486792
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) −4.00000 −0.193347
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) −32.0000 −1.54139 −0.770693 − 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 − 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −8.00000 −0.384012
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) 12.0000 0.574038
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 − 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 8.00000 0.380521
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 24.0000 1.13771
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) 36.0000 1.69895 0.849473 − 0.527633i \(-0.176920\pi\)
0.849473 + 0.527633i \(0.176920\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −4.00000 −0.188144
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 − 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) −4.00000 −0.186501
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 − 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) 12.0000 0.555294 0.277647 − 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) −2.00000 −0.0916698
\(477\) 0 0
\(478\) −28.0000 −1.28069
\(479\) 20.0000 0.913823 0.456912 − 0.889512i \(-0.348956\pi\)
0.456912 + 0.889512i \(0.348956\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 30.0000 1.36646
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 − 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) −4.00000 −0.181071
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) −10.0000 −0.448561
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) −12.0000 −0.536656
\(501\) 0 0
\(502\) −12.0000 −0.535586
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) −2.00000 −0.0889108
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) −10.0000 −0.443242 −0.221621 − 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) −2.00000 −0.0879599
\(518\) −2.00000 −0.0878750
\(519\) 0 0
\(520\) 8.00000 0.350823
\(521\) 4.00000 0.175243 0.0876216 − 0.996154i \(-0.472073\pi\)
0.0876216 + 0.996154i \(0.472073\pi\)
\(522\) 0 0
\(523\) −22.0000 −0.961993 −0.480996 − 0.876723i \(-0.659725\pi\)
−0.480996 + 0.876723i \(0.659725\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) −16.0000 −0.696971
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) 6.00000 0.260133
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) 0 0
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −40.0000 −1.71973 −0.859867 − 0.510518i \(-0.829454\pi\)
−0.859867 + 0.510518i \(0.829454\pi\)
\(542\) 20.0000 0.859074
\(543\) 0 0
\(544\) −2.00000 −0.0857493
\(545\) −16.0000 −0.685365
\(546\) 0 0
\(547\) 18.0000 0.769624 0.384812 − 0.922995i \(-0.374266\pi\)
0.384812 + 0.922995i \(0.374266\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) −12.0000 −0.509831
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 38.0000 1.61011 0.805056 − 0.593199i \(-0.202135\pi\)
0.805056 + 0.593199i \(0.202135\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) −2.00000 −0.0843649
\(563\) −4.00000 −0.168580 −0.0842900 − 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) 0 0
\(565\) −8.00000 −0.336563
\(566\) 6.00000 0.252199
\(567\) 0 0
\(568\) −10.0000 −0.419591
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 10.0000 0.418487 0.209243 − 0.977864i \(-0.432900\pi\)
0.209243 + 0.977864i \(0.432900\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −2.00000 −0.0834784
\(575\) 2.00000 0.0834058
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 − 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) −4.00000 −0.166091
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −6.00000 −0.248495
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −48.0000 −1.98117 −0.990586 − 0.136892i \(-0.956289\pi\)
−0.990586 + 0.136892i \(0.956289\pi\)
\(588\) 0 0
\(589\) 48.0000 1.97781
\(590\) 0 0
\(591\) 0 0
\(592\) −2.00000 −0.0821995
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) −22.0000 −0.901155
\(597\) 0 0
\(598\) −8.00000 −0.327144
\(599\) 30.0000 1.22577 0.612883 − 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 − 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) −2.00000 −0.0815139
\(603\) 0 0
\(604\) −12.0000 −0.488273
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 − 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −8.00000 −0.323911
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) −8.00000 −0.323117 −0.161558 − 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) −26.0000 −1.04927
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 36.0000 1.44931 0.724653 − 0.689114i \(-0.242000\pi\)
0.724653 + 0.689114i \(0.242000\pi\)
\(618\) 0 0
\(619\) −40.0000 −1.60774 −0.803868 − 0.594808i \(-0.797228\pi\)
−0.803868 + 0.594808i \(0.797228\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) −2.00000 −0.0801927
\(623\) −12.0000 −0.480770
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −48.0000 −1.91085 −0.955425 − 0.295234i \(-0.904602\pi\)
−0.955425 + 0.295234i \(0.904602\pi\)
\(632\) 12.0000 0.477334
\(633\) 0 0
\(634\) −30.0000 −1.19145
\(635\) 40.0000 1.58735
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) −2.00000 −0.0791808
\(639\) 0 0
\(640\) −2.00000 −0.0790569
\(641\) −12.0000 −0.473972 −0.236986 − 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 − 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 42.0000 1.65119 0.825595 − 0.564263i \(-0.190840\pi\)
0.825595 + 0.564263i \(0.190840\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) 10.0000 0.391330 0.195665 − 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) −24.0000 −0.937758
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 2.00000 0.0779681
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 − 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) 12.0000 0.465690
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) 4.00000 0.154881
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 − 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −30.0000 −1.15556
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 26.0000 0.999261 0.499631 − 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) −4.00000 −0.153393
\(681\) 0 0
\(682\) −8.00000 −0.306336
\(683\) −36.0000 −1.37750 −0.688751 − 0.724998i \(-0.741841\pi\)
−0.688751 + 0.724998i \(0.741841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 0 0
\(688\) −2.00000 −0.0762493
\(689\) −24.0000 −0.914327
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 − 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) 22.0000 0.836315
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) −8.00000 −0.302804
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 30.0000 1.13308 0.566542 − 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 36.0000 1.35488
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) −6.00000 −0.225335 −0.112667 − 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) −20.0000 −0.750587
\(711\) 0 0
\(712\) −12.0000 −0.449719
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 8.00000 0.299183
\(716\) −20.0000 −0.747435
\(717\) 0 0
\(718\) −32.0000 −1.19423
\(719\) −50.0000 −1.86469 −0.932343 − 0.361576i \(-0.882239\pi\)
−0.932343 + 0.361576i \(0.882239\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) 0 0
\(724\) 14.0000 0.520306
\(725\) 2.00000 0.0742781
\(726\) 0 0
\(727\) −16.0000 −0.593407 −0.296704 − 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) −4.00000 −0.148250
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) −4.00000 −0.147743 −0.0738717 − 0.997268i \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 0 0
\(738\) 0 0
\(739\) 6.00000 0.220714 0.110357 − 0.993892i \(-0.464801\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) 6.00000 0.220267
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −44.0000 −1.61204
\(746\) 4.00000 0.146450
\(747\) 0 0
\(748\) −2.00000 −0.0731272
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 2.00000 0.0729325
\(753\) 0 0
\(754\) −8.00000 −0.291343
\(755\) −24.0000 −0.873449
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 − 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) −30.0000 −1.08750 −0.543750 − 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) −2.00000 −0.0722629
\(767\) 0 0
\(768\) 0 0
\(769\) −2.00000 −0.0721218 −0.0360609 − 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) −2.00000 −0.0720750
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −18.0000 −0.645331
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −10.0000 −0.357828
\(782\) 4.00000 0.143040
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −4.00000 −0.142766
\(786\) 0 0
\(787\) −22.0000 −0.784215 −0.392108 − 0.919919i \(-0.628254\pi\)
−0.392108 + 0.919919i \(0.628254\pi\)
\(788\) 18.0000 0.641223
\(789\) 0 0
\(790\) 24.0000 0.853882
\(791\) 4.00000 0.142224
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) 18.0000 0.638796
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −22.0000 −0.779280 −0.389640 − 0.920967i \(-0.627401\pi\)
−0.389640 + 0.920967i \(0.627401\pi\)
\(798\) 0 0
\(799\) 4.00000 0.141510
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 0 0
\(803\) 2.00000 0.0705785
\(804\) 0 0
\(805\) 4.00000 0.140981
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −18.0000 −0.632846 −0.316423 − 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 38.0000 1.33436 0.667180 − 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 2.00000 0.0701862
\(813\) 0 0
\(814\) −2.00000 −0.0701000
\(815\) −16.0000 −0.560456
\(816\) 0 0
\(817\) 12.0000 0.419827
\(818\) −10.0000 −0.349642
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 − 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 − 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 24.0000 0.833052
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −16.0000 −0.553703
\(836\) 6.00000 0.207514
\(837\) 0 0
\(838\) −4.00000 −0.138178
\(839\) −18.0000 −0.621429 −0.310715 − 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 10.0000 0.344214
\(845\) 6.00000 0.206406
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 4.00000 0.137118
\(852\) 0 0
\(853\) −52.0000 −1.78045 −0.890223 − 0.455525i \(-0.849452\pi\)
−0.890223 + 0.455525i \(0.849452\pi\)
\(854\) 4.00000 0.136877
\(855\) 0 0
\(856\) 4.00000 0.136717
\(857\) 2.00000 0.0683187 0.0341593 − 0.999416i \(-0.489125\pi\)
0.0341593 + 0.999416i \(0.489125\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 − 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) 10.0000 0.340404 0.170202 − 0.985409i \(-0.445558\pi\)
0.170202 + 0.985409i \(0.445558\pi\)
\(864\) 0 0
\(865\) 44.0000 1.49604
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) 0 0
\(872\) 8.00000 0.270914
\(873\) 0 0
\(874\) −12.0000 −0.405906
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) 4.00000 0.135070 0.0675352 − 0.997717i \(-0.478487\pi\)
0.0675352 + 0.997717i \(0.478487\pi\)
\(878\) 8.00000 0.269987
\(879\) 0 0
\(880\) −2.00000 −0.0674200
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 − 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −24.0000 −0.806296
\(887\) −20.0000 −0.671534 −0.335767 − 0.941945i \(-0.608996\pi\)
−0.335767 + 0.941945i \(0.608996\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) −24.0000 −0.804482
\(891\) 0 0
\(892\) 16.0000 0.535720
\(893\) −12.0000 −0.401565
\(894\) 0 0
\(895\) −40.0000 −1.33705
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −36.0000 −1.20134
\(899\) 16.0000 0.533630
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −2.00000 −0.0665927
\(903\) 0 0
\(904\) 4.00000 0.133038
\(905\) 28.0000 0.930751
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) −8.00000 −0.265197
\(911\) 18.0000 0.596367 0.298183 − 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 22.0000 0.727695
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 12.0000 0.396275
\(918\) 0 0
\(919\) −12.0000 −0.395843 −0.197922 − 0.980218i \(-0.563419\pi\)
−0.197922 + 0.980218i \(0.563419\pi\)
\(920\) 4.00000 0.131876
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) −40.0000 −1.31662
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) −16.0000 −0.525793
\(927\) 0 0
\(928\) 2.00000 0.0656532
\(929\) 12.0000 0.393707 0.196854 − 0.980433i \(-0.436928\pi\)
0.196854 + 0.980433i \(0.436928\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 18.0000 0.589610
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 − 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 4.00000 0.130466
\(941\) −14.0000 −0.456387 −0.228193 − 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 0 0
\(943\) 4.00000 0.130258
\(944\) 0 0
\(945\) 0 0
\(946\) −2.00000 −0.0650256
\(947\) −16.0000 −0.519930 −0.259965 − 0.965618i \(-0.583711\pi\)
−0.259965 + 0.965618i \(0.583711\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) −6.00000 −0.194666
\(951\) 0 0
\(952\) 2.00000 0.0648204
\(953\) 38.0000 1.23094 0.615470 − 0.788160i \(-0.288966\pi\)
0.615470 + 0.788160i \(0.288966\pi\)
\(954\) 0 0
\(955\) −20.0000 −0.647185
\(956\) 28.0000 0.905585
\(957\) 0 0
\(958\) −20.0000 −0.646171
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −8.00000 −0.257930
\(963\) 0 0
\(964\) −30.0000 −0.966235
\(965\) 28.0000 0.901352
\(966\) 0 0
\(967\) 20.0000 0.643157 0.321578 − 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 4.00000 0.128432
\(971\) −16.0000 −0.513464 −0.256732 − 0.966483i \(-0.582646\pi\)
−0.256732 + 0.966483i \(0.582646\pi\)
\(972\) 0 0
\(973\) −2.00000 −0.0641171
\(974\) −32.0000 −1.02535
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) 32.0000 1.02377 0.511885 − 0.859054i \(-0.328947\pi\)
0.511885 + 0.859054i \(0.328947\pi\)
\(978\) 0 0
\(979\) −12.0000 −0.383522
\(980\) 2.00000 0.0638877
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 42.0000 1.33959 0.669796 − 0.742545i \(-0.266382\pi\)
0.669796 + 0.742545i \(0.266382\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 − 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 8.00000 0.254000
\(993\) 0 0
\(994\) 10.0000 0.317181
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) 48.0000 1.52018 0.760088 − 0.649821i \(-0.225156\pi\)
0.760088 + 0.649821i \(0.225156\pi\)
\(998\) 4.00000 0.126618
\(999\) 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 1386.2.a.d.1.1 ✓ 1
3.2 odd 2 1386.2.a.h.1.1 yes 1
7.6 odd 2 9702.2.a.h.1.1 1
21.20 even 2 9702.2.a.cc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.a.d.1.1 ✓ 1 1.1 even 1 trivial
1386.2.a.h.1.1 yes 1 3.2 odd 2
9702.2.a.h.1.1 1 7.6 odd 2
9702.2.a.cc.1.1 1 21.20 even 2