Properties

Label 1682.2.a.i.1.1
Level $1682$
Weight $2$
Character 1682.1
Self dual yes
Analytic conductor $13.431$
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] = [1682,2,Mod(1,1682)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1682, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1682.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1682 = 2 \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1682.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: \(13.4308376200\)
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 \(\chi\) \(=\) 1682.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +6.00000 q^{11} +2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{21} +6.00000 q^{22} +9.00000 q^{23} +2.00000 q^{24} -5.00000 q^{25} -4.00000 q^{26} -4.00000 q^{27} -1.00000 q^{28} -5.00000 q^{31} +1.00000 q^{32} +12.0000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} -8.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} +4.00000 q^{43} +6.00000 q^{44} +9.00000 q^{46} +9.00000 q^{47} +2.00000 q^{48} -6.00000 q^{49} -5.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} -12.0000 q^{53} -4.00000 q^{54} -1.00000 q^{56} +8.00000 q^{57} +6.00000 q^{59} -2.00000 q^{61} -5.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +12.0000 q^{66} +8.00000 q^{67} +3.00000 q^{68} +18.0000 q^{69} +3.00000 q^{71} +1.00000 q^{72} -11.0000 q^{73} -2.00000 q^{74} -10.0000 q^{75} +4.00000 q^{76} -6.00000 q^{77} -8.00000 q^{78} -11.0000 q^{79} -11.0000 q^{81} +6.00000 q^{82} -2.00000 q^{84} +4.00000 q^{86} +6.00000 q^{88} -3.00000 q^{89} +4.00000 q^{91} +9.00000 q^{92} -10.0000 q^{93} +9.00000 q^{94} +2.00000 q^{96} -17.0000 q^{97} -6.00000 q^{98} +6.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.00000 0.816497
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 2.00000 0.577350
\(13\) −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
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 6.00000 1.27920
\(23\) 9.00000 1.87663 0.938315 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) 2.00000 0.408248
\(25\) −5.00000 −1.00000
\(26\) −4.00000 −0.784465
\(27\) −4.00000 −0.769800
\(28\) −1.00000 −0.188982
\(29\) 0 0
\(30\) 0 0
\(31\) −5.00000 −0.898027 −0.449013 0.893525i \(-0.648224\pi\)
−0.449013 + 0.893525i \(0.648224\pi\)
\(32\) 1.00000 0.176777
\(33\) 12.0000 2.08893
\(34\) 3.00000 0.514496
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −2.00000 −0.308607
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 6.00000 0.904534
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 2.00000 0.288675
\(49\) −6.00000 −0.857143
\(50\) −5.00000 −0.707107
\(51\) 6.00000 0.840168
\(52\) −4.00000 −0.554700
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −5.00000 −0.635001
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 12.0000 1.47710
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 3.00000 0.363803
\(69\) 18.0000 2.16695
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 1.00000 0.117851
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) −2.00000 −0.232495
\(75\) −10.0000 −1.15470
\(76\) 4.00000 0.458831
\(77\) −6.00000 −0.683763
\(78\) −8.00000 −0.905822
\(79\) −11.0000 −1.23760 −0.618798 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 6.00000 0.662589
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 9.00000 0.938315
\(93\) −10.0000 −1.03695
\(94\) 9.00000 0.928279
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) −6.00000 −0.606092
\(99\) 6.00000 0.603023
\(100\) −5.00000 −0.500000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 6.00000 0.594089
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −4.00000 −0.384900
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) −1.00000 −0.0944911
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 8.00000 0.749269
\(115\) 0 0
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 6.00000 0.552345
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) −2.00000 −0.181071
\(123\) 12.0000 1.08200
\(124\) −5.00000 −0.449013
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) 12.0000 1.04447
\(133\) −4.00000 −0.346844
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 3.00000 0.257248
\(137\) −9.00000 −0.768922 −0.384461 0.923141i \(-0.625613\pi\)
−0.384461 + 0.923141i \(0.625613\pi\)
\(138\) 18.0000 1.53226
\(139\) 14.0000 1.18746 0.593732 0.804663i \(-0.297654\pi\)
0.593732 + 0.804663i \(0.297654\pi\)
\(140\) 0 0
\(141\) 18.0000 1.51587
\(142\) 3.00000 0.251754
\(143\) −24.0000 −2.00698
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −11.0000 −0.910366
\(147\) −12.0000 −0.989743
\(148\) −2.00000 −0.164399
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) −10.0000 −0.816497
\(151\) −19.0000 −1.54620 −0.773099 0.634285i \(-0.781294\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 4.00000 0.324443
\(153\) 3.00000 0.242536
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) −8.00000 −0.640513
\(157\) 4.00000 0.319235 0.159617 0.987179i \(-0.448974\pi\)
0.159617 + 0.987179i \(0.448974\pi\)
\(158\) −11.0000 −0.875113
\(159\) −24.0000 −1.90332
\(160\) 0 0
\(161\) −9.00000 −0.709299
\(162\) −11.0000 −0.864242
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −2.00000 −0.154303
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) 4.00000 0.304997
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 0 0
\(175\) 5.00000 0.377964
\(176\) 6.00000 0.452267
\(177\) 12.0000 0.901975
\(178\) −3.00000 −0.224860
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 4.00000 0.296500
\(183\) −4.00000 −0.295689
\(184\) 9.00000 0.663489
\(185\) 0 0
\(186\) −10.0000 −0.733236
\(187\) 18.0000 1.31629
\(188\) 9.00000 0.656392
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 2.00000 0.144338
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) −17.0000 −1.22053
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −12.0000 −0.854965 −0.427482 0.904024i \(-0.640599\pi\)
−0.427482 + 0.904024i \(0.640599\pi\)
\(198\) 6.00000 0.426401
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) −5.00000 −0.353553
\(201\) 16.0000 1.12855
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) 5.00000 0.348367
\(207\) 9.00000 0.625543
\(208\) −4.00000 −0.277350
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −12.0000 −0.824163
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) 5.00000 0.339422
\(218\) −10.0000 −0.677285
\(219\) −22.0000 −1.48662
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) −4.00000 −0.268462
\(223\) 23.0000 1.54019 0.770097 0.637927i \(-0.220208\pi\)
0.770097 + 0.637927i \(0.220208\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −5.00000 −0.333333
\(226\) 9.00000 0.598671
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 8.00000 0.529813
\(229\) −20.0000 −1.32164 −0.660819 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 0 0
\(233\) 27.0000 1.76883 0.884414 0.466702i \(-0.154558\pi\)
0.884414 + 0.466702i \(0.154558\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 6.00000 0.390567
\(237\) −22.0000 −1.42905
\(238\) −3.00000 −0.194461
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 25.0000 1.60706
\(243\) −10.0000 −0.641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) −16.0000 −1.01806
\(248\) −5.00000 −0.317500
\(249\) 0 0
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 54.0000 3.39495
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.0000 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) 8.00000 0.498058
\(259\) 2.00000 0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 12.0000 0.738549
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) −6.00000 −0.367194
\(268\) 8.00000 0.488678
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 25.0000 1.51864 0.759321 0.650716i \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(272\) 3.00000 0.181902
\(273\) 8.00000 0.484182
\(274\) −9.00000 −0.543710
\(275\) −30.0000 −1.80907
\(276\) 18.0000 1.08347
\(277\) 14.0000 0.841178 0.420589 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) 14.0000 0.839664
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) −9.00000 −0.536895 −0.268447 0.963294i \(-0.586511\pi\)
−0.268447 + 0.963294i \(0.586511\pi\)
\(282\) 18.0000 1.07188
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 3.00000 0.178017
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) −6.00000 −0.354169
\(288\) 1.00000 0.0589256
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) −34.0000 −1.99312
\(292\) −11.0000 −0.643726
\(293\) −30.0000 −1.75262 −0.876309 0.481749i \(-0.840002\pi\)
−0.876309 + 0.481749i \(0.840002\pi\)
\(294\) −12.0000 −0.699854
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) −24.0000 −1.39262
\(298\) −18.0000 −1.04271
\(299\) −36.0000 −2.08193
\(300\) −10.0000 −0.577350
\(301\) −4.00000 −0.230556
\(302\) −19.0000 −1.09333
\(303\) 12.0000 0.689382
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) 3.00000 0.171499
\(307\) 28.0000 1.59804 0.799022 0.601302i \(-0.205351\pi\)
0.799022 + 0.601302i \(0.205351\pi\)
\(308\) −6.00000 −0.341882
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −8.00000 −0.452911
\(313\) 2.00000 0.113047 0.0565233 0.998401i \(-0.481998\pi\)
0.0565233 + 0.998401i \(0.481998\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −11.0000 −0.618798
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) −24.0000 −1.34585
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) −9.00000 −0.501550
\(323\) 12.0000 0.667698
\(324\) −11.0000 −0.611111
\(325\) 20.0000 1.10940
\(326\) −14.0000 −0.775388
\(327\) −20.0000 −1.10600
\(328\) 6.00000 0.331295
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) −2.00000 −0.109109
\(337\) 1.00000 0.0544735 0.0272367 0.999629i \(-0.491329\pi\)
0.0272367 + 0.999629i \(0.491329\pi\)
\(338\) 3.00000 0.163178
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) −30.0000 −1.62459
\(342\) 4.00000 0.216295
\(343\) 13.0000 0.701934
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −24.0000 −1.28839 −0.644194 0.764862i \(-0.722807\pi\)
−0.644194 + 0.764862i \(0.722807\pi\)
\(348\) 0 0
\(349\) −16.0000 −0.856460 −0.428230 0.903670i \(-0.640863\pi\)
−0.428230 + 0.903670i \(0.640863\pi\)
\(350\) 5.00000 0.267261
\(351\) 16.0000 0.854017
\(352\) 6.00000 0.319801
\(353\) −9.00000 −0.479022 −0.239511 0.970894i \(-0.576987\pi\)
−0.239511 + 0.970894i \(0.576987\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −3.00000 −0.159000
\(357\) −6.00000 −0.317554
\(358\) 0 0
\(359\) −15.0000 −0.791670 −0.395835 0.918322i \(-0.629545\pi\)
−0.395835 + 0.918322i \(0.629545\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −16.0000 −0.840941
\(363\) 50.0000 2.62432
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) −4.00000 −0.209083
\(367\) 1.00000 0.0521996 0.0260998 0.999659i \(-0.491691\pi\)
0.0260998 + 0.999659i \(0.491691\pi\)
\(368\) 9.00000 0.469157
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) −10.0000 −0.518476
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) 9.00000 0.464140
\(377\) 0 0
\(378\) 4.00000 0.205738
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −3.00000 −0.153493
\(383\) 33.0000 1.68622 0.843111 0.537740i \(-0.180722\pi\)
0.843111 + 0.537740i \(0.180722\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 4.00000 0.203331
\(388\) −17.0000 −0.863044
\(389\) −24.0000 −1.21685 −0.608424 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) 0 0
\(391\) 27.0000 1.36545
\(392\) −6.00000 −0.303046
\(393\) −12.0000 −0.605320
\(394\) −12.0000 −0.604551
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 26.0000 1.30490 0.652451 0.757831i \(-0.273741\pi\)
0.652451 + 0.757831i \(0.273741\pi\)
\(398\) −7.00000 −0.350878
\(399\) −8.00000 −0.400501
\(400\) −5.00000 −0.250000
\(401\) −3.00000 −0.149813 −0.0749064 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 16.0000 0.798007
\(403\) 20.0000 0.996271
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −12.0000 −0.594818
\(408\) 6.00000 0.297044
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 5.00000 0.246332
\(413\) −6.00000 −0.295241
\(414\) 9.00000 0.442326
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) 28.0000 1.37117
\(418\) 24.0000 1.17388
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −8.00000 −0.389434
\(423\) 9.00000 0.437595
\(424\) −12.0000 −0.582772
\(425\) −15.0000 −0.727607
\(426\) 6.00000 0.290701
\(427\) 2.00000 0.0967868
\(428\) 0 0
\(429\) −48.0000 −2.31746
\(430\) 0 0
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) −4.00000 −0.192450
\(433\) 19.0000 0.913082 0.456541 0.889702i \(-0.349088\pi\)
0.456541 + 0.889702i \(0.349088\pi\)
\(434\) 5.00000 0.240008
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 36.0000 1.72211
\(438\) −22.0000 −1.05120
\(439\) −31.0000 −1.47955 −0.739775 0.672855i \(-0.765068\pi\)
−0.739775 + 0.672855i \(0.765068\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) −12.0000 −0.570782
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) 23.0000 1.08908
\(447\) −36.0000 −1.70274
\(448\) −1.00000 −0.0472456
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) −5.00000 −0.235702
\(451\) 36.0000 1.69517
\(452\) 9.00000 0.423324
\(453\) −38.0000 −1.78540
\(454\) 0 0
\(455\) 0 0
\(456\) 8.00000 0.374634
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −20.0000 −0.934539
\(459\) −12.0000 −0.560112
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −12.0000 −0.558291
\(463\) −1.00000 −0.0464739 −0.0232370 0.999730i \(-0.507397\pi\)
−0.0232370 + 0.999730i \(0.507397\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 27.0000 1.25075
\(467\) −18.0000 −0.832941 −0.416470 0.909149i \(-0.636733\pi\)
−0.416470 + 0.909149i \(0.636733\pi\)
\(468\) −4.00000 −0.184900
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 8.00000 0.368621
\(472\) 6.00000 0.276172
\(473\) 24.0000 1.10352
\(474\) −22.0000 −1.01049
\(475\) −20.0000 −0.917663
\(476\) −3.00000 −0.137505
\(477\) −12.0000 −0.549442
\(478\) −9.00000 −0.411650
\(479\) 15.0000 0.685367 0.342684 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 17.0000 0.774329
\(483\) −18.0000 −0.819028
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) −2.00000 −0.0905357
\(489\) −28.0000 −1.26620
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 12.0000 0.541002
\(493\) 0 0
\(494\) −16.0000 −0.719874
\(495\) 0 0
\(496\) −5.00000 −0.224507
\(497\) −3.00000 −0.134568
\(498\) 0 0
\(499\) −34.0000 −1.52205 −0.761025 0.648723i \(-0.775303\pi\)
−0.761025 + 0.648723i \(0.775303\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 30.0000 1.33897
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 54.0000 2.40059
\(507\) 6.00000 0.266469
\(508\) −8.00000 −0.354943
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) 11.0000 0.486611
\(512\) 1.00000 0.0441942
\(513\) −16.0000 −0.706417
\(514\) 15.0000 0.661622
\(515\) 0 0
\(516\) 8.00000 0.352180
\(517\) 54.0000 2.37492
\(518\) 2.00000 0.0878750
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 21.0000 0.920027 0.460013 0.887912i \(-0.347845\pi\)
0.460013 + 0.887912i \(0.347845\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −6.00000 −0.262111
\(525\) 10.0000 0.436436
\(526\) 12.0000 0.523225
\(527\) −15.0000 −0.653410
\(528\) 12.0000 0.522233
\(529\) 58.0000 2.52174
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) −4.00000 −0.173422
\(533\) −24.0000 −1.03956
\(534\) −6.00000 −0.259645
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) −36.0000 −1.55063
\(540\) 0 0
\(541\) 16.0000 0.687894 0.343947 0.938989i \(-0.388236\pi\)
0.343947 + 0.938989i \(0.388236\pi\)
\(542\) 25.0000 1.07384
\(543\) −32.0000 −1.37325
\(544\) 3.00000 0.128624
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) −9.00000 −0.384461
\(549\) −2.00000 −0.0853579
\(550\) −30.0000 −1.27920
\(551\) 0 0
\(552\) 18.0000 0.766131
\(553\) 11.0000 0.467768
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 14.0000 0.593732
\(557\) 12.0000 0.508456 0.254228 0.967144i \(-0.418179\pi\)
0.254228 + 0.967144i \(0.418179\pi\)
\(558\) −5.00000 −0.211667
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) −9.00000 −0.379642
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 18.0000 0.757937
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) 11.0000 0.461957
\(568\) 3.00000 0.125877
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 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
\(573\) −6.00000 −0.250654
\(574\) −6.00000 −0.250435
\(575\) −45.0000 −1.87663
\(576\) 1.00000 0.0416667
\(577\) −11.0000 −0.457936 −0.228968 0.973434i \(-0.573535\pi\)
−0.228968 + 0.973434i \(0.573535\pi\)
\(578\) −8.00000 −0.332756
\(579\) −28.0000 −1.16364
\(580\) 0 0
\(581\) 0 0
\(582\) −34.0000 −1.40935
\(583\) −72.0000 −2.98194
\(584\) −11.0000 −0.455183
\(585\) 0 0
\(586\) −30.0000 −1.23929
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −12.0000 −0.494872
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) −2.00000 −0.0821995
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) −24.0000 −0.984732
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) −14.0000 −0.572982
\(598\) −36.0000 −1.47215
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −10.0000 −0.408248
\(601\) −23.0000 −0.938190 −0.469095 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) −4.00000 −0.163028
\(603\) 8.00000 0.325785
\(604\) −19.0000 −0.773099
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 0 0
\(611\) −36.0000 −1.45640
\(612\) 3.00000 0.121268
\(613\) 20.0000 0.807792 0.403896 0.914805i \(-0.367656\pi\)
0.403896 + 0.914805i \(0.367656\pi\)
\(614\) 28.0000 1.12999
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 10.0000 0.402259
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) −36.0000 −1.44463
\(622\) 0 0
\(623\) 3.00000 0.120192
\(624\) −8.00000 −0.320256
\(625\) 25.0000 1.00000
\(626\) 2.00000 0.0799361
\(627\) 48.0000 1.91694
\(628\) 4.00000 0.159617
\(629\) −6.00000 −0.239236
\(630\) 0 0
\(631\) 41.0000 1.63218 0.816092 0.577922i \(-0.196136\pi\)
0.816092 + 0.577922i \(0.196136\pi\)
\(632\) −11.0000 −0.437557
\(633\) −16.0000 −0.635943
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) −24.0000 −0.951662
\(637\) 24.0000 0.950915
\(638\) 0 0
\(639\) 3.00000 0.118678
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −9.00000 −0.354650
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −21.0000 −0.825595 −0.412798 0.910823i \(-0.635448\pi\)
−0.412798 + 0.910823i \(0.635448\pi\)
\(648\) −11.0000 −0.432121
\(649\) 36.0000 1.41312
\(650\) 20.0000 0.784465
\(651\) 10.0000 0.391931
\(652\) −14.0000 −0.548282
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −20.0000 −0.782062
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −11.0000 −0.429151
\(658\) −9.00000 −0.350857
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) −8.00000 −0.310929
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 0 0
\(669\) 46.0000 1.77846
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) −2.00000 −0.0771517
\(673\) −43.0000 −1.65753 −0.828764 0.559598i \(-0.810955\pi\)
−0.828764 + 0.559598i \(0.810955\pi\)
\(674\) 1.00000 0.0385186
\(675\) 20.0000 0.769800
\(676\) 3.00000 0.115385
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 18.0000 0.691286
\(679\) 17.0000 0.652400
\(680\) 0 0
\(681\) 0 0
\(682\) −30.0000 −1.14876
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) −40.0000 −1.52610
\(688\) 4.00000 0.152499
\(689\) 48.0000 1.82865
\(690\) 0 0
\(691\) 50.0000 1.90209 0.951045 0.309053i \(-0.100012\pi\)
0.951045 + 0.309053i \(0.100012\pi\)
\(692\) 6.00000 0.228086
\(693\) −6.00000 −0.227921
\(694\) −24.0000 −0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) −16.0000 −0.605609
\(699\) 54.0000 2.04247
\(700\) 5.00000 0.188982
\(701\) −12.0000 −0.453234 −0.226617 0.973984i \(-0.572767\pi\)
−0.226617 + 0.973984i \(0.572767\pi\)
\(702\) 16.0000 0.603881
\(703\) −8.00000 −0.301726
\(704\) 6.00000 0.226134
\(705\) 0 0
\(706\) −9.00000 −0.338719
\(707\) −6.00000 −0.225653
\(708\) 12.0000 0.450988
\(709\) −16.0000 −0.600893 −0.300446 0.953799i \(-0.597136\pi\)
−0.300446 + 0.953799i \(0.597136\pi\)
\(710\) 0 0
\(711\) −11.0000 −0.412532
\(712\) −3.00000 −0.112430
\(713\) −45.0000 −1.68526
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 0 0
\(717\) −18.0000 −0.672222
\(718\) −15.0000 −0.559795
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −5.00000 −0.186210
\(722\) −3.00000 −0.111648
\(723\) 34.0000 1.26447
\(724\) −16.0000 −0.594635
\(725\) 0 0
\(726\) 50.0000 1.85567
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 4.00000 0.148250
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) −4.00000 −0.147844
\(733\) 4.00000 0.147743 0.0738717 0.997268i \(-0.476464\pi\)
0.0738717 + 0.997268i \(0.476464\pi\)
\(734\) 1.00000 0.0369107
\(735\) 0 0
\(736\) 9.00000 0.331744
\(737\) 48.0000 1.76810
\(738\) 6.00000 0.220863
\(739\) 10.0000 0.367856 0.183928 0.982940i \(-0.441119\pi\)
0.183928 + 0.982940i \(0.441119\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) 12.0000 0.440534
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −10.0000 −0.366618
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) 18.0000 0.658145
\(749\) 0 0
\(750\) 0 0
\(751\) −35.0000 −1.27717 −0.638584 0.769552i \(-0.720480\pi\)
−0.638584 + 0.769552i \(0.720480\pi\)
\(752\) 9.00000 0.328196
\(753\) 60.0000 2.18652
\(754\) 0 0
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 52.0000 1.88997 0.944986 0.327111i \(-0.106075\pi\)
0.944986 + 0.327111i \(0.106075\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 108.000 3.92015
\(760\) 0 0
\(761\) −3.00000 −0.108750 −0.0543750 0.998521i \(-0.517317\pi\)
−0.0543750 + 0.998521i \(0.517317\pi\)
\(762\) −16.0000 −0.579619
\(763\) 10.0000 0.362024
\(764\) −3.00000 −0.108536
\(765\) 0 0
\(766\) 33.0000 1.19234
\(767\) −24.0000 −0.866590
\(768\) 2.00000 0.0721688
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) −14.0000 −0.503871
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 4.00000 0.143777
\(775\) 25.0000 0.898027
\(776\) −17.0000 −0.610264
\(777\) 4.00000 0.143499
\(778\) −24.0000 −0.860442
\(779\) 24.0000 0.859889
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) 27.0000 0.965518
\(783\) 0 0
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) 26.0000 0.926800 0.463400 0.886149i \(-0.346629\pi\)
0.463400 + 0.886149i \(0.346629\pi\)
\(788\) −12.0000 −0.427482
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 6.00000 0.213201
\(793\) 8.00000 0.284088
\(794\) 26.0000 0.922705
\(795\) 0 0
\(796\) −7.00000 −0.248108
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) −8.00000 −0.283197
\(799\) 27.0000 0.955191
\(800\) −5.00000 −0.176777
\(801\) −3.00000 −0.106000
\(802\) −3.00000 −0.105934
\(803\) −66.0000 −2.32909
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 20.0000 0.704470
\(807\) 12.0000 0.422420
\(808\) 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 \(0.267501\pi\)
\(812\) 0 0
\(813\) 50.0000 1.75358
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 16.0000 0.559769
\(818\) 10.0000 0.349642
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) −18.0000 −0.627822
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) 5.00000 0.174183
\(825\) −60.0000 −2.08893
\(826\) −6.00000 −0.208767
\(827\) 42.0000 1.46048 0.730242 0.683189i \(-0.239408\pi\)
0.730242 + 0.683189i \(0.239408\pi\)
\(828\) 9.00000 0.312772
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) −4.00000 −0.138675
\(833\) −18.0000 −0.623663
\(834\) 28.0000 0.969561
\(835\) 0 0
\(836\) 24.0000 0.830057
\(837\) 20.0000 0.691301
\(838\) −12.0000 −0.414533
\(839\) 12.0000 0.414286 0.207143 0.978311i \(-0.433583\pi\)
0.207143 + 0.978311i \(0.433583\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) −26.0000 −0.896019
\(843\) −18.0000 −0.619953
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 9.00000 0.309426
\(847\) −25.0000 −0.859010
\(848\) −12.0000 −0.412082
\(849\) 28.0000 0.960958
\(850\) −15.0000 −0.514496
\(851\) −18.0000 −0.617032
\(852\) 6.00000 0.205557
\(853\) 28.0000 0.958702 0.479351 0.877623i \(-0.340872\pi\)
0.479351 + 0.877623i \(0.340872\pi\)
\(854\) 2.00000 0.0684386
\(855\) 0 0
\(856\) 0 0
\(857\) −3.00000 −0.102478 −0.0512390 0.998686i \(-0.516317\pi\)
−0.0512390 + 0.998686i \(0.516317\pi\)
\(858\) −48.0000 −1.63869
\(859\) −14.0000 −0.477674 −0.238837 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) 3.00000 0.102180
\(863\) 45.0000 1.53182 0.765909 0.642949i \(-0.222289\pi\)
0.765909 + 0.642949i \(0.222289\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 19.0000 0.645646
\(867\) −16.0000 −0.543388
\(868\) 5.00000 0.169711
\(869\) −66.0000 −2.23890
\(870\) 0 0
\(871\) −32.0000 −1.08428
\(872\) −10.0000 −0.338643
\(873\) −17.0000 −0.575363
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) −22.0000 −0.743311
\(877\) −4.00000 −0.135070 −0.0675352 0.997717i \(-0.521513\pi\)
−0.0675352 + 0.997717i \(0.521513\pi\)
\(878\) −31.0000 −1.04620
\(879\) −60.0000 −2.02375
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) −6.00000 −0.202031
\(883\) −34.0000 −1.14419 −0.572096 0.820187i \(-0.693869\pi\)
−0.572096 + 0.820187i \(0.693869\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 6.00000 0.201574
\(887\) 45.0000 1.51095 0.755476 0.655176i \(-0.227406\pi\)
0.755476 + 0.655176i \(0.227406\pi\)
\(888\) −4.00000 −0.134231
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −66.0000 −2.21108
\(892\) 23.0000 0.770097
\(893\) 36.0000 1.20469
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −72.0000 −2.40401
\(898\) 30.0000 1.00111
\(899\) 0 0
\(900\) −5.00000 −0.166667
\(901\) −36.0000 −1.19933
\(902\) 36.0000 1.19867
\(903\) −8.00000 −0.266223
\(904\) 9.00000 0.299336
\(905\) 0 0
\(906\) −38.0000 −1.26247
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) 8.00000 0.264906
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −20.0000 −0.660819
\(917\) 6.00000 0.198137
\(918\) −12.0000 −0.396059
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 56.0000 1.84526
\(922\) 18.0000 0.592798
\(923\) −12.0000 −0.394985
\(924\) −12.0000 −0.394771
\(925\) 10.0000 0.328798
\(926\) −1.00000 −0.0328620
\(927\) 5.00000 0.164222
\(928\) 0 0
\(929\) 45.0000 1.47640 0.738201 0.674581i \(-0.235676\pi\)
0.738201 + 0.674581i \(0.235676\pi\)
\(930\) 0 0
\(931\) −24.0000 −0.786568
\(932\) 27.0000 0.884414
\(933\) 0 0
\(934\) −18.0000 −0.588978
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 23.0000 0.751377 0.375689 0.926746i \(-0.377406\pi\)
0.375689 + 0.926746i \(0.377406\pi\)
\(938\) −8.00000 −0.261209
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 42.0000 1.36916 0.684580 0.728937i \(-0.259985\pi\)
0.684580 + 0.728937i \(0.259985\pi\)
\(942\) 8.00000 0.260654
\(943\) 54.0000 1.75848
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −22.0000 −0.714527
\(949\) 44.0000 1.42830
\(950\) −20.0000 −0.648886
\(951\) 12.0000 0.389127
\(952\) −3.00000 −0.0972306
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) −9.00000 −0.291081
\(957\) 0 0
\(958\) 15.0000 0.484628
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) 8.00000 0.257930
\(963\) 0 0
\(964\) 17.0000 0.547533
\(965\) 0 0
\(966\) −18.0000 −0.579141
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) 25.0000 0.803530
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −42.0000 −1.34784 −0.673922 0.738802i \(-0.735392\pi\)
−0.673922 + 0.738802i \(0.735392\pi\)
\(972\) −10.0000 −0.320750
\(973\) −14.0000 −0.448819
\(974\) 20.0000 0.640841
\(975\) 40.0000 1.28103
\(976\) −2.00000 −0.0640184
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) −28.0000 −0.895341
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −6.00000 −0.191468
\(983\) 15.0000 0.478426 0.239213 0.970967i \(-0.423111\pi\)
0.239213 + 0.970967i \(0.423111\pi\)
\(984\) 12.0000 0.382546
\(985\) 0 0
\(986\) 0 0
\(987\) −18.0000 −0.572946
\(988\) −16.0000 −0.509028
\(989\) 36.0000 1.14473
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) −5.00000 −0.158750
\(993\) −16.0000 −0.507745
\(994\) −3.00000 −0.0951542
\(995\) 0 0
\(996\) 0 0
\(997\) −32.0000 −1.01345 −0.506725 0.862108i \(-0.669144\pi\)
−0.506725 + 0.862108i \(0.669144\pi\)
\(998\) −34.0000 −1.07625
\(999\) 8.00000 0.253109
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 1682.2.a.i.1.1 yes 1
29.12 odd 4 1682.2.b.b.1681.2 2
29.17 odd 4 1682.2.b.b.1681.1 2
29.28 even 2 1682.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1682.2.a.b.1.1 1 29.28 even 2
1682.2.a.i.1.1 yes 1 1.1 even 1 trivial
1682.2.b.b.1681.1 2 29.17 odd 4
1682.2.b.b.1681.2 2 29.12 odd 4