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Properties

Label 3333.2.a.b.1.1
Level $3333$
Weight $2$
Character 3333.1
Self dual yes
Analytic conductor $26.614$
Analytic rank $0$
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] = [3333,2,Mod(1,3333)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3333, 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("3333.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3333 = 3 \cdot 11 \cdot 101 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3333.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: \(26.6141389937\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 3333.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -1.00000 q^{13} +4.00000 q^{14} +1.00000 q^{15} -4.00000 q^{16} +3.00000 q^{17} -2.00000 q^{18} +5.00000 q^{19} +2.00000 q^{20} -2.00000 q^{21} -2.00000 q^{22} -1.00000 q^{23} -4.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} -2.00000 q^{30} +7.00000 q^{31} +8.00000 q^{32} +1.00000 q^{33} -6.00000 q^{34} -2.00000 q^{35} +2.00000 q^{36} -2.00000 q^{37} -10.0000 q^{38} -1.00000 q^{39} +2.00000 q^{41} +4.00000 q^{42} +4.00000 q^{43} +2.00000 q^{44} +1.00000 q^{45} +2.00000 q^{46} -7.00000 q^{47} -4.00000 q^{48} -3.00000 q^{49} +8.00000 q^{50} +3.00000 q^{51} -2.00000 q^{52} +4.00000 q^{53} -2.00000 q^{54} +1.00000 q^{55} +5.00000 q^{57} +2.00000 q^{60} +2.00000 q^{61} -14.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} -1.00000 q^{65} -2.00000 q^{66} +8.00000 q^{67} +6.00000 q^{68} -1.00000 q^{69} +4.00000 q^{70} -13.0000 q^{71} +14.0000 q^{73} +4.00000 q^{74} -4.00000 q^{75} +10.0000 q^{76} -2.00000 q^{77} +2.00000 q^{78} +5.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -4.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +3.00000 q^{85} -8.00000 q^{86} -10.0000 q^{89} -2.00000 q^{90} +2.00000 q^{91} -2.00000 q^{92} +7.00000 q^{93} +14.0000 q^{94} +5.00000 q^{95} +8.00000 q^{96} +18.0000 q^{97} +6.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214 0.223607 − 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −2.00000 −0.816497
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511
\(12\) 2.00000 0.577350
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 4.00000 1.06904
\(15\) 1.00000 0.258199
\(16\) −4.00000 −1.00000
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) −2.00000 −0.471405
\(19\) 5.00000 1.14708 0.573539 − 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) 2.00000 0.447214
\(21\) −2.00000 −0.436436
\(22\) −2.00000 −0.426401
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −2.00000 −0.365148
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 8.00000 1.41421
\(33\) 1.00000 0.174078
\(34\) −6.00000 −1.02899
\(35\) −2.00000 −0.338062
\(36\) 2.00000 0.333333
\(37\) −2.00000 −0.328798 −0.164399 − 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −10.0000 −1.62221
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 4.00000 0.617213
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 2.00000 0.301511
\(45\) 1.00000 0.149071
\(46\) 2.00000 0.294884
\(47\) −7.00000 −1.02105 −0.510527 − 0.859861i \(-0.670550\pi\)
−0.510527 + 0.859861i \(0.670550\pi\)
\(48\) −4.00000 −0.577350
\(49\) −3.00000 −0.428571
\(50\) 8.00000 1.13137
\(51\) 3.00000 0.420084
\(52\) −2.00000 −0.277350
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) −2.00000 −0.272166
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 5.00000 0.662266
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 2.00000 0.258199
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −14.0000 −1.77800
\(63\) −2.00000 −0.251976
\(64\) −8.00000 −1.00000
\(65\) −1.00000 −0.124035
\(66\) −2.00000 −0.246183
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 6.00000 0.727607
\(69\) −1.00000 −0.120386
\(70\) 4.00000 0.478091
\(71\) −13.0000 −1.54282 −0.771408 − 0.636341i \(-0.780447\pi\)
−0.771408 + 0.636341i \(0.780447\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 4.00000 0.464991
\(75\) −4.00000 −0.461880
\(76\) 10.0000 1.14708
\(77\) −2.00000 −0.227921
\(78\) 2.00000 0.226455
\(79\) 5.00000 0.562544 0.281272 − 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) −4.00000 −0.441726
\(83\) −6.00000 −0.658586 −0.329293 − 0.944228i \(-0.606810\pi\)
−0.329293 + 0.944228i \(0.606810\pi\)
\(84\) −4.00000 −0.436436
\(85\) 3.00000 0.325396
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 −0.210819
\(91\) 2.00000 0.209657
\(92\) −2.00000 −0.208514
\(93\) 7.00000 0.725866
\(94\) 14.0000 1.44399
\(95\) 5.00000 0.512989
\(96\) 8.00000 0.816497
\(97\) 18.0000 1.82762 0.913812 − 0.406138i \(-0.133125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 6.00000 0.606092
\(99\) 1.00000 0.100504
\(100\) −8.00000 −0.800000
\(101\) 1.00000 0.0995037
\(102\) −6.00000 −0.594089
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) −8.00000 −0.777029
\(107\) 13.0000 1.25676 0.628379 − 0.777908i \(-0.283719\pi\)
0.628379 + 0.777908i \(0.283719\pi\)
\(108\) 2.00000 0.192450
\(109\) −10.0000 −0.957826 −0.478913 − 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) −2.00000 −0.190693
\(111\) −2.00000 −0.189832
\(112\) 8.00000 0.755929
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) −10.0000 −0.936586
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) 2.00000 0.180334
\(124\) 14.0000 1.25724
\(125\) −9.00000 −0.804984
\(126\) 4.00000 0.356348
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 2.00000 0.175412
\(131\) 7.00000 0.611593 0.305796 − 0.952097i \(-0.401077\pi\)
0.305796 + 0.952097i \(0.401077\pi\)
\(132\) 2.00000 0.174078
\(133\) −10.0000 −0.867110
\(134\) −16.0000 −1.38219
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 − 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 2.00000 0.170251
\(139\) −10.0000 −0.848189 −0.424094 − 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) −4.00000 −0.338062
\(141\) −7.00000 −0.589506
\(142\) 26.0000 2.18187
\(143\) −1.00000 −0.0836242
\(144\) −4.00000 −0.333333
\(145\) 0 0
\(146\) −28.0000 −2.31730
\(147\) −3.00000 −0.247436
\(148\) −4.00000 −0.328798
\(149\) 20.0000 1.63846 0.819232 − 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 8.00000 0.653197
\(151\) 2.00000 0.162758 0.0813788 − 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 4.00000 0.322329
\(155\) 7.00000 0.562254
\(156\) −2.00000 −0.160128
\(157\) 3.00000 0.239426 0.119713 − 0.992809i \(-0.461803\pi\)
0.119713 + 0.992809i \(0.461803\pi\)
\(158\) −10.0000 −0.795557
\(159\) 4.00000 0.317221
\(160\) 8.00000 0.632456
\(161\) 2.00000 0.157622
\(162\) −2.00000 −0.157135
\(163\) −16.0000 −1.25322 −0.626608 − 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 4.00000 0.312348
\(165\) 1.00000 0.0778499
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) −6.00000 −0.460179
\(171\) 5.00000 0.382360
\(172\) 8.00000 0.609994
\(173\) 4.00000 0.304114 0.152057 − 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) 8.00000 0.604743
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 20.0000 1.49906
\(179\) −20.0000 −1.49487 −0.747435 − 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 2.00000 0.149071
\(181\) 17.0000 1.26360 0.631800 − 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) −4.00000 −0.296500
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) −2.00000 −0.147043
\(186\) −14.0000 −1.02653
\(187\) 3.00000 0.219382
\(188\) −14.0000 −1.02105
\(189\) −2.00000 −0.145479
\(190\) −10.0000 −0.725476
\(191\) −18.0000 −1.30243 −0.651217 − 0.758891i \(-0.725741\pi\)
−0.651217 + 0.758891i \(0.725741\pi\)
\(192\) −8.00000 −0.577350
\(193\) 19.0000 1.36765 0.683825 − 0.729646i \(-0.260315\pi\)
0.683825 + 0.729646i \(0.260315\pi\)
\(194\) −36.0000 −2.58465
\(195\) −1.00000 −0.0716115
\(196\) −6.00000 −0.428571
\(197\) 3.00000 0.213741 0.106871 − 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) −2.00000 −0.142134
\(199\) 20.0000 1.41776 0.708881 − 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −2.00000 −0.140720
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 2.00000 0.139686
\(206\) −8.00000 −0.557386
\(207\) −1.00000 −0.0695048
\(208\) 4.00000 0.277350
\(209\) 5.00000 0.345857
\(210\) 4.00000 0.276026
\(211\) 27.0000 1.85876 0.929378 − 0.369129i \(-0.120344\pi\)
0.929378 + 0.369129i \(0.120344\pi\)
\(212\) 8.00000 0.549442
\(213\) −13.0000 −0.890745
\(214\) −26.0000 −1.77732
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −14.0000 −0.950382
\(218\) 20.0000 1.35457
\(219\) 14.0000 0.946032
\(220\) 2.00000 0.134840
\(221\) −3.00000 −0.201802
\(222\) 4.00000 0.268462
\(223\) 4.00000 0.267860 0.133930 − 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) −16.0000 −1.06904
\(225\) −4.00000 −0.266667
\(226\) −28.0000 −1.86253
\(227\) 13.0000 0.862840 0.431420 − 0.902151i \(-0.358013\pi\)
0.431420 + 0.902151i \(0.358013\pi\)
\(228\) 10.0000 0.662266
\(229\) −10.0000 −0.660819 −0.330409 − 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 2.00000 0.131876
\(231\) −2.00000 −0.131590
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 2.00000 0.130744
\(235\) −7.00000 −0.456630
\(236\) 0 0
\(237\) 5.00000 0.324785
\(238\) 12.0000 0.777844
\(239\) −5.00000 −0.323423 −0.161712 − 0.986838i \(-0.551701\pi\)
−0.161712 + 0.986838i \(0.551701\pi\)
\(240\) −4.00000 −0.258199
\(241\) −8.00000 −0.515325 −0.257663 − 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) −2.00000 −0.128565
\(243\) 1.00000 0.0641500
\(244\) 4.00000 0.256074
\(245\) −3.00000 −0.191663
\(246\) −4.00000 −0.255031
\(247\) −5.00000 −0.318142
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 18.0000 1.13842
\(251\) 7.00000 0.441836 0.220918 − 0.975292i \(-0.429095\pi\)
0.220918 + 0.975292i \(0.429095\pi\)
\(252\) −4.00000 −0.251976
\(253\) −1.00000 −0.0628695
\(254\) −16.0000 −1.00393
\(255\) 3.00000 0.187867
\(256\) 16.0000 1.00000
\(257\) −12.0000 −0.748539 −0.374270 − 0.927320i \(-0.622107\pi\)
−0.374270 + 0.927320i \(0.622107\pi\)
\(258\) −8.00000 −0.498058
\(259\) 4.00000 0.248548
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −14.0000 −0.864923
\(263\) −26.0000 −1.60323 −0.801614 − 0.597841i \(-0.796025\pi\)
−0.801614 + 0.597841i \(0.796025\pi\)
\(264\) 0 0
\(265\) 4.00000 0.245718
\(266\) 20.0000 1.22628
\(267\) −10.0000 −0.611990
\(268\) 16.0000 0.977356
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) −2.00000 −0.121716
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) −12.0000 −0.727607
\(273\) 2.00000 0.121046
\(274\) −6.00000 −0.362473
\(275\) −4.00000 −0.241209
\(276\) −2.00000 −0.120386
\(277\) 18.0000 1.08152 0.540758 − 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 20.0000 1.19952
\(279\) 7.00000 0.419079
\(280\) 0 0
\(281\) 27.0000 1.61068 0.805342 − 0.592810i \(-0.201981\pi\)
0.805342 + 0.592810i \(0.201981\pi\)
\(282\) 14.0000 0.833688
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −26.0000 −1.54282
\(285\) 5.00000 0.296174
\(286\) 2.00000 0.118262
\(287\) −4.00000 −0.236113
\(288\) 8.00000 0.471405
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) 28.0000 1.63858
\(293\) −16.0000 −0.934730 −0.467365 − 0.884064i \(-0.654797\pi\)
−0.467365 + 0.884064i \(0.654797\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) −40.0000 −2.31714
\(299\) 1.00000 0.0578315
\(300\) −8.00000 −0.461880
\(301\) −8.00000 −0.461112
\(302\) −4.00000 −0.230174
\(303\) 1.00000 0.0574485
\(304\) −20.0000 −1.14708
\(305\) 2.00000 0.114520
\(306\) −6.00000 −0.342997
\(307\) −7.00000 −0.399511 −0.199756 − 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) −4.00000 −0.227921
\(309\) 4.00000 0.227552
\(310\) −14.0000 −0.795147
\(311\) 22.0000 1.24751 0.623753 − 0.781622i \(-0.285607\pi\)
0.623753 + 0.781622i \(0.285607\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 − 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −6.00000 −0.338600
\(315\) −2.00000 −0.112687
\(316\) 10.0000 0.562544
\(317\) 13.0000 0.730153 0.365076 − 0.930978i \(-0.381043\pi\)
0.365076 + 0.930978i \(0.381043\pi\)
\(318\) −8.00000 −0.448618
\(319\) 0 0
\(320\) −8.00000 −0.447214
\(321\) 13.0000 0.725589
\(322\) −4.00000 −0.222911
\(323\) 15.0000 0.834622
\(324\) 2.00000 0.111111
\(325\) 4.00000 0.221880
\(326\) 32.0000 1.77232
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 14.0000 0.771845
\(330\) −2.00000 −0.110096
\(331\) 32.0000 1.75888 0.879440 − 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) −12.0000 −0.658586
\(333\) −2.00000 −0.109599
\(334\) 24.0000 1.31322
\(335\) 8.00000 0.437087
\(336\) 8.00000 0.436436
\(337\) 18.0000 0.980522 0.490261 − 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 24.0000 1.30543
\(339\) 14.0000 0.760376
\(340\) 6.00000 0.325396
\(341\) 7.00000 0.379071
\(342\) −10.0000 −0.540738
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) −8.00000 −0.430083
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) 0 0
\(349\) −20.0000 −1.07058 −0.535288 − 0.844670i \(-0.679797\pi\)
−0.535288 + 0.844670i \(0.679797\pi\)
\(350\) −16.0000 −0.855236
\(351\) −1.00000 −0.0533761
\(352\) 8.00000 0.426401
\(353\) 4.00000 0.212899 0.106449 − 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 0 0
\(355\) −13.0000 −0.689968
\(356\) −20.0000 −1.06000
\(357\) −6.00000 −0.317554
\(358\) 40.0000 2.11407
\(359\) −5.00000 −0.263890 −0.131945 − 0.991257i \(-0.542122\pi\)
−0.131945 + 0.991257i \(0.542122\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) −34.0000 −1.78700
\(363\) 1.00000 0.0524864
\(364\) 4.00000 0.209657
\(365\) 14.0000 0.732793
\(366\) −4.00000 −0.209083
\(367\) 23.0000 1.20059 0.600295 − 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 4.00000 0.208514
\(369\) 2.00000 0.104116
\(370\) 4.00000 0.207950
\(371\) −8.00000 −0.415339
\(372\) 14.0000 0.725866
\(373\) −11.0000 −0.569558 −0.284779 − 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) −6.00000 −0.310253
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 10.0000 0.512989
\(381\) 8.00000 0.409852
\(382\) 36.0000 1.84192
\(383\) 24.0000 1.22634 0.613171 − 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) −2.00000 −0.101929
\(386\) −38.0000 −1.93415
\(387\) 4.00000 0.203331
\(388\) 36.0000 1.82762
\(389\) −30.0000 −1.52106 −0.760530 − 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 2.00000 0.101274
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) 7.00000 0.353103
\(394\) −6.00000 −0.302276
\(395\) 5.00000 0.251577
\(396\) 2.00000 0.100504
\(397\) −22.0000 −1.10415 −0.552074 − 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) −40.0000 −2.00502
\(399\) −10.0000 −0.500626
\(400\) 16.0000 0.800000
\(401\) 12.0000 0.599251 0.299626 − 0.954057i \(-0.403138\pi\)
0.299626 + 0.954057i \(0.403138\pi\)
\(402\) −16.0000 −0.798007
\(403\) −7.00000 −0.348695
\(404\) 2.00000 0.0995037
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −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
\(411\) 3.00000 0.147979
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) −6.00000 −0.294528
\(416\) −8.00000 −0.392232
\(417\) −10.0000 −0.489702
\(418\) −10.0000 −0.489116
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) −4.00000 −0.195180
\(421\) −23.0000 −1.12095 −0.560476 − 0.828171i \(-0.689382\pi\)
−0.560476 + 0.828171i \(0.689382\pi\)
\(422\) −54.0000 −2.62868
\(423\) −7.00000 −0.340352
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 26.0000 1.25970
\(427\) −4.00000 −0.193574
\(428\) 26.0000 1.25676
\(429\) −1.00000 −0.0482805
\(430\) −8.00000 −0.385794
\(431\) 32.0000 1.54139 0.770693 − 0.637207i \(-0.219910\pi\)
0.770693 + 0.637207i \(0.219910\pi\)
\(432\) −4.00000 −0.192450
\(433\) 4.00000 0.192228 0.0961139 − 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 28.0000 1.34404
\(435\) 0 0
\(436\) −20.0000 −0.957826
\(437\) −5.00000 −0.239182
\(438\) −28.0000 −1.33789
\(439\) 30.0000 1.43182 0.715911 − 0.698192i \(-0.246012\pi\)
0.715911 + 0.698192i \(0.246012\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 6.00000 0.285391
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −4.00000 −0.189832
\(445\) −10.0000 −0.474045
\(446\) −8.00000 −0.378811
\(447\) 20.0000 0.945968
\(448\) 16.0000 0.755929
\(449\) 15.0000 0.707894 0.353947 − 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 8.00000 0.377124
\(451\) 2.00000 0.0941763
\(452\) 28.0000 1.31701
\(453\) 2.00000 0.0939682
\(454\) −26.0000 −1.22024
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −12.0000 −0.561336 −0.280668 − 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(458\) 20.0000 0.934539
\(459\) 3.00000 0.140028
\(460\) −2.00000 −0.0932505
\(461\) 32.0000 1.49039 0.745194 − 0.666847i \(-0.232357\pi\)
0.745194 + 0.666847i \(0.232357\pi\)
\(462\) 4.00000 0.186097
\(463\) 24.0000 1.11537 0.557687 − 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) 0 0
\(465\) 7.00000 0.324617
\(466\) 12.0000 0.555889
\(467\) 18.0000 0.832941 0.416470 − 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −16.0000 −0.738811
\(470\) 14.0000 0.645772
\(471\) 3.00000 0.138233
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) −10.0000 −0.459315
\(475\) −20.0000 −0.917663
\(476\) −12.0000 −0.550019
\(477\) 4.00000 0.183147
\(478\) 10.0000 0.457389
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 8.00000 0.365148
\(481\) 2.00000 0.0911922
\(482\) 16.0000 0.728780
\(483\) 2.00000 0.0910032
\(484\) 2.00000 0.0909091
\(485\) 18.0000 0.817338
\(486\) −2.00000 −0.0907218
\(487\) −12.0000 −0.543772 −0.271886 − 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 6.00000 0.271052
\(491\) −13.0000 −0.586682 −0.293341 − 0.956008i \(-0.594767\pi\)
−0.293341 + 0.956008i \(0.594767\pi\)
\(492\) 4.00000 0.180334
\(493\) 0 0
\(494\) 10.0000 0.449921
\(495\) 1.00000 0.0449467
\(496\) −28.0000 −1.25724
\(497\) 26.0000 1.16626
\(498\) 12.0000 0.537733
\(499\) 35.0000 1.56682 0.783408 − 0.621508i \(-0.213480\pi\)
0.783408 + 0.621508i \(0.213480\pi\)
\(500\) −18.0000 −0.804984
\(501\) −12.0000 −0.536120
\(502\) −14.0000 −0.624851
\(503\) 34.0000 1.51599 0.757993 − 0.652263i \(-0.226180\pi\)
0.757993 + 0.652263i \(0.226180\pi\)
\(504\) 0 0
\(505\) 1.00000 0.0444994
\(506\) 2.00000 0.0889108
\(507\) −12.0000 −0.532939
\(508\) 16.0000 0.709885
\(509\) −30.0000 −1.32973 −0.664863 − 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) −6.00000 −0.265684
\(511\) −28.0000 −1.23865
\(512\) −32.0000 −1.41421
\(513\) 5.00000 0.220755
\(514\) 24.0000 1.05859
\(515\) 4.00000 0.176261
\(516\) 8.00000 0.352180
\(517\) −7.00000 −0.307860
\(518\) −8.00000 −0.351500
\(519\) 4.00000 0.175581
\(520\) 0 0
\(521\) 17.0000 0.744784 0.372392 − 0.928076i \(-0.378538\pi\)
0.372392 + 0.928076i \(0.378538\pi\)
\(522\) 0 0
\(523\) 24.0000 1.04945 0.524723 − 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 14.0000 0.611593
\(525\) 8.00000 0.349149
\(526\) 52.0000 2.26731
\(527\) 21.0000 0.914774
\(528\) −4.00000 −0.174078
\(529\) −22.0000 −0.956522
\(530\) −8.00000 −0.347498
\(531\) 0 0
\(532\) −20.0000 −0.867110
\(533\) −2.00000 −0.0866296
\(534\) 20.0000 0.865485
\(535\) 13.0000 0.562039
\(536\) 0 0
\(537\) −20.0000 −0.863064
\(538\) 0 0
\(539\) −3.00000 −0.129219
\(540\) 2.00000 0.0860663
\(541\) −23.0000 −0.988847 −0.494424 − 0.869221i \(-0.664621\pi\)
−0.494424 + 0.869221i \(0.664621\pi\)
\(542\) −24.0000 −1.03089
\(543\) 17.0000 0.729540
\(544\) 24.0000 1.02899
\(545\) −10.0000 −0.428353
\(546\) −4.00000 −0.171184
\(547\) −22.0000 −0.940652 −0.470326 − 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) 6.00000 0.256307
\(549\) 2.00000 0.0853579
\(550\) 8.00000 0.341121
\(551\) 0 0
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) −36.0000 −1.52949
\(555\) −2.00000 −0.0848953
\(556\) −20.0000 −0.848189
\(557\) −2.00000 −0.0847427 −0.0423714 − 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) −14.0000 −0.592667
\(559\) −4.00000 −0.169182
\(560\) 8.00000 0.338062
\(561\) 3.00000 0.126660
\(562\) −54.0000 −2.27785
\(563\) −21.0000 −0.885044 −0.442522 − 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) −14.0000 −0.589506
\(565\) 14.0000 0.588984
\(566\) −8.00000 −0.336265
\(567\) −2.00000 −0.0839921
\(568\) 0 0
\(569\) −45.0000 −1.88650 −0.943249 − 0.332086i \(-0.892248\pi\)
−0.943249 + 0.332086i \(0.892248\pi\)
\(570\) −10.0000 −0.418854
\(571\) 22.0000 0.920671 0.460336 − 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) −2.00000 −0.0836242
\(573\) −18.0000 −0.751961
\(574\) 8.00000 0.333914
\(575\) 4.00000 0.166812
\(576\) −8.00000 −0.333333
\(577\) 8.00000 0.333044 0.166522 − 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(578\) 16.0000 0.665512
\(579\) 19.0000 0.789613
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) −36.0000 −1.49225
\(583\) 4.00000 0.165663
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 32.0000 1.32191
\(587\) −47.0000 −1.93990 −0.969949 − 0.243309i \(-0.921767\pi\)
−0.969949 + 0.243309i \(0.921767\pi\)
\(588\) −6.00000 −0.247436
\(589\) 35.0000 1.44215
\(590\) 0 0
\(591\) 3.00000 0.123404
\(592\) 8.00000 0.328798
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) −2.00000 −0.0820610
\(595\) −6.00000 −0.245976
\(596\) 40.0000 1.63846
\(597\) 20.0000 0.818546
\(598\) −2.00000 −0.0817861
\(599\) 30.0000 1.22577 0.612883 − 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 0 0
\(601\) −33.0000 −1.34610 −0.673049 − 0.739598i \(-0.735016\pi\)
−0.673049 + 0.739598i \(0.735016\pi\)
\(602\) 16.0000 0.652111
\(603\) 8.00000 0.325785
\(604\) 4.00000 0.162758
\(605\) 1.00000 0.0406558
\(606\) −2.00000 −0.0812444
\(607\) −7.00000 −0.284121 −0.142061 − 0.989858i \(-0.545373\pi\)
−0.142061 + 0.989858i \(0.545373\pi\)
\(608\) 40.0000 1.62221
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) 7.00000 0.283190
\(612\) 6.00000 0.242536
\(613\) 14.0000 0.565455 0.282727 − 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 14.0000 0.564994
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) −12.0000 −0.483102 −0.241551 − 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) −8.00000 −0.321807
\(619\) −15.0000 −0.602901 −0.301450 − 0.953482i \(-0.597471\pi\)
−0.301450 + 0.953482i \(0.597471\pi\)
\(620\) 14.0000 0.562254
\(621\) −1.00000 −0.0401286
\(622\) −44.0000 −1.76424
\(623\) 20.0000 0.801283
\(624\) 4.00000 0.160128
\(625\) 11.0000 0.440000
\(626\) −28.0000 −1.11911
\(627\) 5.00000 0.199681
\(628\) 6.00000 0.239426
\(629\) −6.00000 −0.239236
\(630\) 4.00000 0.159364
\(631\) −13.0000 −0.517522 −0.258761 − 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) 0 0
\(633\) 27.0000 1.07315
\(634\) −26.0000 −1.03259
\(635\) 8.00000 0.317470
\(636\) 8.00000 0.317221
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −13.0000 −0.514272
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 − 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) −26.0000 −1.02614
\(643\) −1.00000 −0.0394362 −0.0197181 − 0.999806i \(-0.506277\pi\)
−0.0197181 + 0.999806i \(0.506277\pi\)
\(644\) 4.00000 0.157622
\(645\) 4.00000 0.157500
\(646\) −30.0000 −1.18033
\(647\) −12.0000 −0.471769 −0.235884 − 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −8.00000 −0.313786
\(651\) −14.0000 −0.548703
\(652\) −32.0000 −1.25322
\(653\) 34.0000 1.33052 0.665261 − 0.746611i \(-0.268320\pi\)
0.665261 + 0.746611i \(0.268320\pi\)
\(654\) 20.0000 0.782062
\(655\) 7.00000 0.273513
\(656\) −8.00000 −0.312348
\(657\) 14.0000 0.546192
\(658\) −28.0000 −1.09155
\(659\) −30.0000 −1.16863 −0.584317 − 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 2.00000 0.0778499
\(661\) 2.00000 0.0777910 0.0388955 − 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) −64.0000 −2.48743
\(663\) −3.00000 −0.116510
\(664\) 0 0
\(665\) −10.0000 −0.387783
\(666\) 4.00000 0.154997
\(667\) 0 0
\(668\) −24.0000 −0.928588
\(669\) 4.00000 0.154649
\(670\) −16.0000 −0.618134
\(671\) 2.00000 0.0772091
\(672\) −16.0000 −0.617213
\(673\) 24.0000 0.925132 0.462566 − 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) −36.0000 −1.38667
\(675\) −4.00000 −0.153960
\(676\) −24.0000 −0.923077
\(677\) −27.0000 −1.03769 −0.518847 − 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) −28.0000 −1.07533
\(679\) −36.0000 −1.38155
\(680\) 0 0
\(681\) 13.0000 0.498161
\(682\) −14.0000 −0.536088
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 10.0000 0.382360
\(685\) 3.00000 0.114624
\(686\) −40.0000 −1.52721
\(687\) −10.0000 −0.381524
\(688\) −16.0000 −0.609994
\(689\) −4.00000 −0.152388
\(690\) 2.00000 0.0761387
\(691\) −28.0000 −1.06517 −0.532585 − 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) 8.00000 0.304114
\(693\) −2.00000 −0.0759737
\(694\) −36.0000 −1.36654
\(695\) −10.0000 −0.379322
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 40.0000 1.51402
\(699\) −6.00000 −0.226941
\(700\) 16.0000 0.604743
\(701\) −23.0000 −0.868698 −0.434349 − 0.900745i \(-0.643022\pi\)
−0.434349 + 0.900745i \(0.643022\pi\)
\(702\) 2.00000 0.0754851
\(703\) −10.0000 −0.377157
\(704\) −8.00000 −0.301511
\(705\) −7.00000 −0.263635
\(706\) −8.00000 −0.301084
\(707\) −2.00000 −0.0752177
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 − 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 26.0000 0.975763
\(711\) 5.00000 0.187515
\(712\) 0 0
\(713\) −7.00000 −0.262152
\(714\) 12.0000 0.449089
\(715\) −1.00000 −0.0373979
\(716\) −40.0000 −1.49487
\(717\) −5.00000 −0.186728
\(718\) 10.0000 0.373197
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) −4.00000 −0.149071
\(721\) −8.00000 −0.297936
\(722\) −12.0000 −0.446594
\(723\) −8.00000 −0.297523
\(724\) 34.0000 1.26360
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) −27.0000 −1.00137 −0.500687 − 0.865628i \(-0.666919\pi\)
−0.500687 + 0.865628i \(0.666919\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −28.0000 −1.03633
\(731\) 12.0000 0.443836
\(732\) 4.00000 0.147844
\(733\) −36.0000 −1.32969 −0.664845 − 0.746981i \(-0.731502\pi\)
−0.664845 + 0.746981i \(0.731502\pi\)
\(734\) −46.0000 −1.69789
\(735\) −3.00000 −0.110657
\(736\) −8.00000 −0.294884
\(737\) 8.00000 0.294684
\(738\) −4.00000 −0.147242
\(739\) 20.0000 0.735712 0.367856 − 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) −4.00000 −0.147043
\(741\) −5.00000 −0.183680
\(742\) 16.0000 0.587378
\(743\) 29.0000 1.06391 0.531953 − 0.846774i \(-0.321458\pi\)
0.531953 + 0.846774i \(0.321458\pi\)
\(744\) 0 0
\(745\) 20.0000 0.732743
\(746\) 22.0000 0.805477
\(747\) −6.00000 −0.219529
\(748\) 6.00000 0.219382
\(749\) −26.0000 −0.950019
\(750\) 18.0000 0.657267
\(751\) −38.0000 −1.38664 −0.693320 − 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 28.0000 1.02105
\(753\) 7.00000 0.255094
\(754\) 0 0
\(755\) 2.00000 0.0727875
\(756\) −4.00000 −0.145479
\(757\) 28.0000 1.01768 0.508839 − 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) −40.0000 −1.45287
\(759\) −1.00000 −0.0362977
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) −16.0000 −0.579619
\(763\) 20.0000 0.724049
\(764\) −36.0000 −1.30243
\(765\) 3.00000 0.108465
\(766\) −48.0000 −1.73431
\(767\) 0 0
\(768\) 16.0000 0.577350
\(769\) −10.0000 −0.360609 −0.180305 − 0.983611i \(-0.557708\pi\)
−0.180305 + 0.983611i \(0.557708\pi\)
\(770\) 4.00000 0.144150
\(771\) −12.0000 −0.432169
\(772\) 38.0000 1.36765
\(773\) 4.00000 0.143870 0.0719350 − 0.997409i \(-0.477083\pi\)
0.0719350 + 0.997409i \(0.477083\pi\)
\(774\) −8.00000 −0.287554
\(775\) −28.0000 −1.00579
\(776\) 0 0
\(777\) 4.00000 0.143499
\(778\) 60.0000 2.15110
\(779\) 10.0000 0.358287
\(780\) −2.00000 −0.0716115
\(781\) −13.0000 −0.465177
\(782\) 6.00000 0.214560
\(783\) 0 0
\(784\) 12.0000 0.428571
\(785\) 3.00000 0.107075
\(786\) −14.0000 −0.499363
\(787\) −17.0000 −0.605985 −0.302992 − 0.952993i \(-0.597986\pi\)
−0.302992 + 0.952993i \(0.597986\pi\)
\(788\) 6.00000 0.213741
\(789\) −26.0000 −0.925625
\(790\) −10.0000 −0.355784
\(791\) −28.0000 −0.995565
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 44.0000 1.56150
\(795\) 4.00000 0.141865
\(796\) 40.0000 1.41776
\(797\) 8.00000 0.283375 0.141687 − 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 20.0000 0.707992
\(799\) −21.0000 −0.742927
\(800\) −32.0000 −1.13137
\(801\) −10.0000 −0.353333
\(802\) −24.0000 −0.847469
\(803\) 14.0000 0.494049
\(804\) 16.0000 0.564276
\(805\) 2.00000 0.0704907
\(806\) 14.0000 0.493129
\(807\) 0 0
\(808\) 0 0
\(809\) 10.0000 0.351581 0.175791 − 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 32.0000 1.12367 0.561836 − 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) 4.00000 0.140200
\(815\) −16.0000 −0.560456
\(816\) −12.0000 −0.420084
\(817\) 20.0000 0.699711
\(818\) −20.0000 −0.699284
\(819\) 2.00000 0.0698857
\(820\) 4.00000 0.139686
\(821\) −38.0000 −1.32621 −0.663105 − 0.748527i \(-0.730762\pi\)
−0.663105 + 0.748527i \(0.730762\pi\)
\(822\) −6.00000 −0.209274
\(823\) −36.0000 −1.25488 −0.627441 − 0.778664i \(-0.715897\pi\)
−0.627441 + 0.778664i \(0.715897\pi\)
\(824\) 0 0
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 − 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −45.0000 −1.56291 −0.781457 − 0.623959i \(-0.785523\pi\)
−0.781457 + 0.623959i \(0.785523\pi\)
\(830\) 12.0000 0.416526
\(831\) 18.0000 0.624413
\(832\) 8.00000 0.277350
\(833\) −9.00000 −0.311832
\(834\) 20.0000 0.692543
\(835\) −12.0000 −0.415277
\(836\) 10.0000 0.345857
\(837\) 7.00000 0.241955
\(838\) 0 0
\(839\) 20.0000 0.690477 0.345238 − 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 46.0000 1.58526
\(843\) 27.0000 0.929929
\(844\) 54.0000 1.85876
\(845\) −12.0000 −0.412813
\(846\) 14.0000 0.481330
\(847\) −2.00000 −0.0687208
\(848\) −16.0000 −0.549442
\(849\) 4.00000 0.137280
\(850\) 24.0000 0.823193
\(851\) 2.00000 0.0685591
\(852\) −26.0000 −0.890745
\(853\) −46.0000 −1.57501 −0.787505 − 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 8.00000 0.273754
\(855\) 5.00000 0.170996
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 − 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 2.00000 0.0682789
\(859\) −20.0000 −0.682391 −0.341196 − 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 8.00000 0.272798
\(861\) −4.00000 −0.136320
\(862\) −64.0000 −2.17985
\(863\) −6.00000 −0.204242 −0.102121 − 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 8.00000 0.272166
\(865\) 4.00000 0.136004
\(866\) −8.00000 −0.271851
\(867\) −8.00000 −0.271694
\(868\) −28.0000 −0.950382
\(869\) 5.00000 0.169613
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 18.0000 0.609208
\(874\) 10.0000 0.338255
\(875\) 18.0000 0.608511
\(876\) 28.0000 0.946032
\(877\) 28.0000 0.945493 0.472746 − 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) −60.0000 −2.02490
\(879\) −16.0000 −0.539667
\(880\) −4.00000 −0.134840
\(881\) −48.0000 −1.61716 −0.808581 − 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 6.00000 0.202031
\(883\) −26.0000 −0.874970 −0.437485 − 0.899226i \(-0.644131\pi\)
−0.437485 + 0.899226i \(0.644131\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) −48.0000 −1.61259
\(887\) −32.0000 −1.07445 −0.537227 − 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 20.0000 0.670402
\(891\) 1.00000 0.0335013
\(892\) 8.00000 0.267860
\(893\) −35.0000 −1.17123
\(894\) −40.0000 −1.33780
\(895\) −20.0000 −0.668526
\(896\) 0 0
\(897\) 1.00000 0.0333890
\(898\) −30.0000 −1.00111
\(899\) 0 0
\(900\) −8.00000 −0.266667
\(901\) 12.0000 0.399778
\(902\) −4.00000 −0.133185
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) 17.0000 0.565099
\(906\) −4.00000 −0.132891
\(907\) 28.0000 0.929725 0.464862 − 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 26.0000 0.862840
\(909\) 1.00000 0.0331679
\(910\) −4.00000 −0.132599
\(911\) −18.0000 −0.596367 −0.298183 − 0.954509i \(-0.596381\pi\)
−0.298183 + 0.954509i \(0.596381\pi\)
\(912\) −20.0000 −0.662266
\(913\) −6.00000 −0.198571
\(914\) 24.0000 0.793849
\(915\) 2.00000 0.0661180
\(916\) −20.0000 −0.660819
\(917\) −14.0000 −0.462321
\(918\) −6.00000 −0.198030
\(919\) 30.0000 0.989609 0.494804 − 0.869004i \(-0.335240\pi\)
0.494804 + 0.869004i \(0.335240\pi\)
\(920\) 0 0
\(921\) −7.00000 −0.230658
\(922\) −64.0000 −2.10773
\(923\) 13.0000 0.427900
\(924\) −4.00000 −0.131590
\(925\) 8.00000 0.263038
\(926\) −48.0000 −1.57738
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) −45.0000 −1.47640 −0.738201 − 0.674581i \(-0.764324\pi\)
−0.738201 + 0.674581i \(0.764324\pi\)
\(930\) −14.0000 −0.459078
\(931\) −15.0000 −0.491605
\(932\) −12.0000 −0.393073
\(933\) 22.0000 0.720248
\(934\) −36.0000 −1.17796
\(935\) 3.00000 0.0981105
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 − 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 32.0000 1.04484
\(939\) 14.0000 0.456873
\(940\) −14.0000 −0.456630
\(941\) −8.00000 −0.260793 −0.130396 − 0.991462i \(-0.541625\pi\)
−0.130396 + 0.991462i \(0.541625\pi\)
\(942\) −6.00000 −0.195491
\(943\) −2.00000 −0.0651290
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) −8.00000 −0.260102
\(947\) 38.0000 1.23483 0.617417 − 0.786636i \(-0.288179\pi\)
0.617417 + 0.786636i \(0.288179\pi\)
\(948\) 10.0000 0.324785
\(949\) −14.0000 −0.454459
\(950\) 40.0000 1.29777
\(951\) 13.0000 0.421554
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −8.00000 −0.259010
\(955\) −18.0000 −0.582466
\(956\) −10.0000 −0.323423
\(957\) 0 0
\(958\) 0 0
\(959\) −6.00000 −0.193750
\(960\) −8.00000 −0.258199
\(961\) 18.0000 0.580645
\(962\) −4.00000 −0.128965
\(963\) 13.0000 0.418919
\(964\) −16.0000 −0.515325
\(965\) 19.0000 0.611632
\(966\) −4.00000 −0.128698
\(967\) −7.00000 −0.225105 −0.112552 − 0.993646i \(-0.535903\pi\)
−0.112552 + 0.993646i \(0.535903\pi\)
\(968\) 0 0
\(969\) 15.0000 0.481869
\(970\) −36.0000 −1.15589
\(971\) −18.0000 −0.577647 −0.288824 − 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 2.00000 0.0641500
\(973\) 20.0000 0.641171
\(974\) 24.0000 0.769010
\(975\) 4.00000 0.128103
\(976\) −8.00000 −0.256074
\(977\) −27.0000 −0.863807 −0.431903 − 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) 32.0000 1.02325
\(979\) −10.0000 −0.319601
\(980\) −6.00000 −0.191663
\(981\) −10.0000 −0.319275
\(982\) 26.0000 0.829693
\(983\) −6.00000 −0.191370 −0.0956851 − 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) 3.00000 0.0955879
\(986\) 0 0
\(987\) 14.0000 0.445625
\(988\) −10.0000 −0.318142
\(989\) −4.00000 −0.127193
\(990\) −2.00000 −0.0635642
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) 56.0000 1.77800
\(993\) 32.0000 1.01549
\(994\) −52.0000 −1.64934
\(995\) 20.0000 0.634043
\(996\) −12.0000 −0.380235
\(997\) −17.0000 −0.538395 −0.269198 − 0.963085i \(-0.586759\pi\)
−0.269198 + 0.963085i \(0.586759\pi\)
\(998\) −70.0000 −2.21581
\(999\) −2.00000 −0.0632772
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 3333.2.a.b.1.1 ✓ 1
3.2 odd 2 9999.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3333.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
9999.2.a.k.1.1 1 3.2 odd 2