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Properties

Label 1682.2.a.b.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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -6.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} +6.00000 q^{22} +9.00000 q^{23} +2.00000 q^{24} -5.00000 q^{25} +4.00000 q^{26} +4.00000 q^{27} -1.00000 q^{28} +5.00000 q^{31} -1.00000 q^{32} +12.0000 q^{33} +3.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} +4.00000 q^{38} +8.00000 q^{39} -6.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} -6.00000 q^{44} -9.00000 q^{46} -9.00000 q^{47} -2.00000 q^{48} -6.00000 q^{49} +5.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} -12.0000 q^{53} -4.00000 q^{54} +1.00000 q^{56} +8.00000 q^{57} +6.00000 q^{59} +2.00000 q^{61} -5.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -12.0000 q^{66} +8.00000 q^{67} -3.00000 q^{68} -18.0000 q^{69} +3.00000 q^{71} -1.00000 q^{72} +11.0000 q^{73} -2.00000 q^{74} +10.0000 q^{75} -4.00000 q^{76} +6.00000 q^{77} -8.00000 q^{78} +11.0000 q^{79} -11.0000 q^{81} +6.00000 q^{82} +2.00000 q^{84} +4.00000 q^{86} +6.00000 q^{88} +3.00000 q^{89} +4.00000 q^{91} +9.00000 q^{92} -10.0000 q^{93} +9.00000 q^{94} +2.00000 q^{96} +17.0000 q^{97} +6.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) −2.00000 −1.15470 −0.577350 − 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 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 \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −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 \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 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 \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −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 \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 4.00000 0.648886
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) −2.00000 −0.308607
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) −9.00000 −1.32698
\(47\) −9.00000 −1.31278 −0.656392 − 0.754420i \(-0.727918\pi\)
−0.656392 + 0.754420i \(0.727918\pi\)
\(48\) −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 \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −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 \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −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 \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 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 \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 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 \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 6.00000 0.606092
\(99\) −6.00000 −0.603023
\(100\) −5.00000 −0.500000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) −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 \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) −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 \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 6.00000 0.524222 0.262111 − 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 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 \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 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 \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −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 \(-0.315282\pi\)
0.548282 + 0.836293i \(0.315282\pi\)
\(164\) −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 \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) −2.00000 −0.144338
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −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 \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) −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 \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 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 \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) −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 \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −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 \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 − 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) −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 \(-0.159998\pi\)
0.876309 + 0.481749i \(0.159998\pi\)
\(294\) −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 \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 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 \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −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 \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 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 \(-0.508671\pi\)
−0.0272367 + 0.999629i \(0.508671\pi\)
\(338\) −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 \(-0.370455\pi\)
0.395835 + 0.918322i \(0.370455\pi\)
\(360\) 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 \(-0.508309\pi\)
−0.0260998 + 0.999659i \(0.508309\pi\)
\(368\) 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 \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 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 \(-0.291802\pi\)
0.608424 + 0.793612i \(0.291802\pi\)
\(390\) 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 \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 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 \(-0.281584\pi\)
0.633581 + 0.773676i \(0.281584\pi\)
\(422\) −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 \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 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 \(-0.545525\pi\)
−0.142534 + 0.989790i \(0.545525\pi\)
\(444\) −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 \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 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 \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 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 \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) −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 \(-0.611336\pi\)
−0.342684 + 0.939451i \(0.611336\pi\)
\(480\) 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 \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 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 \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 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 \(-0.611764\pi\)
−0.343947 + 0.938989i \(0.611764\pi\)
\(542\) 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 \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 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 \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) 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 \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 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 \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) −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 \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 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 \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 10.0000 0.402259
\(619\) 26.0000 1.04503 0.522514 − 0.852631i \(-0.324994\pi\)
0.522514 + 0.852631i \(0.324994\pi\)
\(620\) 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 \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 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 \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) −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 \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 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 \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) −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 \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) −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 \(-0.523536\pi\)
−0.0738717 + 0.997268i \(0.523536\pi\)
\(734\) 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 \(-0.558881\pi\)
−0.183928 + 0.982940i \(0.558881\pi\)
\(740\) 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 \(-0.279520\pi\)
0.638584 + 0.769552i \(0.279520\pi\)
\(752\) −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 \(-0.893925\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) −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 \(-0.142450\pi\)
0.901523 + 0.432731i \(0.142450\pi\)
\(770\) 0 0
\(771\) −30.0000 −1.08042
\(772\) 14.0000 0.503871
\(773\) 36.0000 1.29483 0.647415 − 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 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 \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 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 \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 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 \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −5.00000 −0.174183
\(825\) −60.0000 −2.08893
\(826\) 6.00000 0.208767
\(827\) −42.0000 −1.46048 −0.730242 − 0.683189i \(-0.760592\pi\)
−0.730242 + 0.683189i \(0.760592\pi\)
\(828\) 9.00000 0.312772
\(829\) −34.0000 −1.18087 −0.590434 − 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 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 \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 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 \(-0.659128\pi\)
−0.479351 + 0.877623i \(0.659128\pi\)
\(854\) 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 \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 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 \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 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 \(-0.772594\pi\)
−0.755476 + 0.655176i \(0.772594\pi\)
\(888\) 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 \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 15.0000 0.496972 0.248486 − 0.968635i \(-0.420067\pi\)
0.248486 + 0.968635i \(0.420067\pi\)
\(912\) 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 \(-0.301125\pi\)
0.584921 + 0.811090i \(0.301125\pi\)
\(948\) −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 \(-0.567029\pi\)
−0.209026 + 0.977910i \(0.567029\pi\)
\(968\) −25.0000 −0.803530
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 − 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 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 \(-0.576889\pi\)
−0.239213 + 0.970967i \(0.576889\pi\)
\(984\) −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 \(-0.330856\pi\)
0.506725 + 0.862108i \(0.330856\pi\)
\(998\) 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.b.1.1 ✓ 1
29.12 odd 4 1682.2.b.b.1681.1 2
29.17 odd 4 1682.2.b.b.1681.2 2
29.28 even 2 1682.2.a.i.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1682.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
1682.2.a.i.1.1 yes 1 29.28 even 2
1682.2.b.b.1681.1 2 29.12 odd 4
1682.2.b.b.1681.2 2 29.17 odd 4