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Properties

Label 6762.2.a.b.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6762,2,Mod(1,6762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6762, 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("6762.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.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: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6762.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} +3.00000 q^{13} +3.00000 q^{15} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{18} -3.00000 q^{20} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} -3.00000 q^{26} -1.00000 q^{27} -8.00000 q^{29} -3.00000 q^{30} +10.0000 q^{31} -1.00000 q^{32} +2.00000 q^{33} +3.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} -3.00000 q^{39} +3.00000 q^{40} +8.00000 q^{41} -4.00000 q^{43} -2.00000 q^{44} -3.00000 q^{45} +1.00000 q^{46} +9.00000 q^{47} -1.00000 q^{48} -4.00000 q^{50} +3.00000 q^{51} +3.00000 q^{52} -9.00000 q^{53} +1.00000 q^{54} +6.00000 q^{55} +8.00000 q^{58} +4.00000 q^{59} +3.00000 q^{60} -6.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -9.00000 q^{65} -2.00000 q^{66} +3.00000 q^{67} -3.00000 q^{68} +1.00000 q^{69} +7.00000 q^{71} -1.00000 q^{72} +3.00000 q^{73} +2.00000 q^{74} -4.00000 q^{75} +3.00000 q^{78} +4.00000 q^{79} -3.00000 q^{80} +1.00000 q^{81} -8.00000 q^{82} -14.0000 q^{83} +9.00000 q^{85} +4.00000 q^{86} +8.00000 q^{87} +2.00000 q^{88} +10.0000 q^{89} +3.00000 q^{90} -1.00000 q^{92} -10.0000 q^{93} -9.00000 q^{94} +1.00000 q^{96} +4.00000 q^{97} -2.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 − 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) −1.00000 −0.288675
\(13\) 3.00000 0.832050 0.416025 − 0.909353i \(-0.363423\pi\)
0.416025 + 0.909353i \(0.363423\pi\)
\(14\) 0 0
\(15\) 3.00000 0.774597
\(16\) 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
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 4.00000 0.800000
\(26\) −3.00000 −0.588348
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) −3.00000 −0.547723
\(31\) 10.0000 1.79605 0.898027 − 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) −1.00000 −0.176777
\(33\) 2.00000 0.348155
\(34\) 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 \(0.552568\pi\)
\(38\) 0 0
\(39\) −3.00000 −0.480384
\(40\) 3.00000 0.474342
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) −3.00000 −0.447214
\(46\) 1.00000 0.147442
\(47\) 9.00000 1.31278 0.656392 − 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −4.00000 −0.565685
\(51\) 3.00000 0.420084
\(52\) 3.00000 0.416025
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 1.00000 0.136083
\(55\) 6.00000 0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 8.00000 1.05045
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 3.00000 0.387298
\(61\) −6.00000 −0.768221 −0.384111 − 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −9.00000 −1.11631
\(66\) −2.00000 −0.246183
\(67\) 3.00000 0.366508 0.183254 − 0.983066i \(-0.441337\pi\)
0.183254 + 0.983066i \(0.441337\pi\)
\(68\) −3.00000 −0.363803
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 7.00000 0.830747 0.415374 − 0.909651i \(-0.363651\pi\)
0.415374 + 0.909651i \(0.363651\pi\)
\(72\) −1.00000 −0.117851
\(73\) 3.00000 0.351123 0.175562 − 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 2.00000 0.232495
\(75\) −4.00000 −0.461880
\(76\) 0 0
\(77\) 0 0
\(78\) 3.00000 0.339683
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −3.00000 −0.335410
\(81\) 1.00000 0.111111
\(82\) −8.00000 −0.883452
\(83\) −14.0000 −1.53670 −0.768350 − 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 4.00000 0.431331
\(87\) 8.00000 0.857690
\(88\) 2.00000 0.213201
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) −10.0000 −1.03695
\(94\) −9.00000 −0.928279
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 4.00000 0.406138 0.203069 − 0.979164i \(-0.434908\pi\)
0.203069 + 0.979164i \(0.434908\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 4.00000 0.400000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) −3.00000 −0.297044
\(103\) 17.0000 1.67506 0.837530 − 0.546392i \(-0.183999\pi\)
0.837530 + 0.546392i \(0.183999\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) 9.00000 0.874157
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 8.00000 0.766261 0.383131 − 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) −6.00000 −0.572078
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) −5.00000 −0.470360 −0.235180 − 0.971952i \(-0.575568\pi\)
−0.235180 + 0.971952i \(0.575568\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) −8.00000 −0.742781
\(117\) 3.00000 0.277350
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) −3.00000 −0.273861
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) −8.00000 −0.721336
\(124\) 10.0000 0.898027
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 9.00000 0.789352
\(131\) 15.0000 1.31056 0.655278 − 0.755388i \(-0.272551\pi\)
0.655278 + 0.755388i \(0.272551\pi\)
\(132\) 2.00000 0.174078
\(133\) 0 0
\(134\) −3.00000 −0.259161
\(135\) 3.00000 0.258199
\(136\) 3.00000 0.257248
\(137\) 19.0000 1.62328 0.811640 − 0.584158i \(-0.198575\pi\)
0.811640 + 0.584158i \(0.198575\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 10.0000 0.848189 0.424094 − 0.905618i \(-0.360592\pi\)
0.424094 + 0.905618i \(0.360592\pi\)
\(140\) 0 0
\(141\) −9.00000 −0.757937
\(142\) −7.00000 −0.587427
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) 24.0000 1.99309
\(146\) −3.00000 −0.248282
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 21.0000 1.72039 0.860194 − 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 4.00000 0.326599
\(151\) −20.0000 −1.62758 −0.813788 − 0.581161i \(-0.802599\pi\)
−0.813788 + 0.581161i \(0.802599\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −30.0000 −2.40966
\(156\) −3.00000 −0.240192
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −4.00000 −0.318223
\(159\) 9.00000 0.713746
\(160\) 3.00000 0.237171
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 8.00000 0.624695
\(165\) −6.00000 −0.467099
\(166\) 14.0000 1.08661
\(167\) −21.0000 −1.62503 −0.812514 − 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) −9.00000 −0.690268
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) 20.0000 1.52057 0.760286 − 0.649589i \(-0.225059\pi\)
0.760286 + 0.649589i \(0.225059\pi\)
\(174\) −8.00000 −0.606478
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) −4.00000 −0.300658
\(178\) −10.0000 −0.749532
\(179\) 15.0000 1.12115 0.560576 − 0.828103i \(-0.310580\pi\)
0.560576 + 0.828103i \(0.310580\pi\)
\(180\) −3.00000 −0.223607
\(181\) 6.00000 0.445976 0.222988 − 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 6.00000 0.443533
\(184\) 1.00000 0.0737210
\(185\) 6.00000 0.441129
\(186\) 10.0000 0.733236
\(187\) 6.00000 0.438763
\(188\) 9.00000 0.656392
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 − 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 5.00000 0.359908 0.179954 − 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) −4.00000 −0.287183
\(195\) 9.00000 0.644503
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 2.00000 0.142134
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) −4.00000 −0.282843
\(201\) −3.00000 −0.211604
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 3.00000 0.210042
\(205\) −24.0000 −1.67623
\(206\) −17.0000 −1.18445
\(207\) −1.00000 −0.0695048
\(208\) 3.00000 0.208013
\(209\) 0 0
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 − 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) −9.00000 −0.618123
\(213\) −7.00000 −0.479632
\(214\) 8.00000 0.546869
\(215\) 12.0000 0.818393
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −8.00000 −0.541828
\(219\) −3.00000 −0.202721
\(220\) 6.00000 0.404520
\(221\) −9.00000 −0.605406
\(222\) −2.00000 −0.134231
\(223\) 8.00000 0.535720 0.267860 − 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 4.00000 0.266667
\(226\) 5.00000 0.332595
\(227\) −14.0000 −0.929213 −0.464606 − 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −20.0000 −1.32164 −0.660819 − 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) −3.00000 −0.197814
\(231\) 0 0
\(232\) 8.00000 0.525226
\(233\) −4.00000 −0.262049 −0.131024 − 0.991379i \(-0.541827\pi\)
−0.131024 + 0.991379i \(0.541827\pi\)
\(234\) −3.00000 −0.196116
\(235\) −27.0000 −1.76129
\(236\) 4.00000 0.260378
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 3.00000 0.193649
\(241\) −6.00000 −0.386494 −0.193247 − 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) 0 0
\(248\) −10.0000 −0.635001
\(249\) 14.0000 0.887214
\(250\) −3.00000 −0.189737
\(251\) −26.0000 −1.64111 −0.820553 − 0.571571i \(-0.806334\pi\)
−0.820553 + 0.571571i \(0.806334\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) 2.00000 0.125491
\(255\) −9.00000 −0.563602
\(256\) 1.00000 0.0625000
\(257\) 14.0000 0.873296 0.436648 − 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −4.00000 −0.249029
\(259\) 0 0
\(260\) −9.00000 −0.558156
\(261\) −8.00000 −0.495188
\(262\) −15.0000 −0.926703
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) −2.00000 −0.123091
\(265\) 27.0000 1.65860
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 3.00000 0.183254
\(269\) −20.0000 −1.21942 −0.609711 − 0.792624i \(-0.708714\pi\)
−0.609711 + 0.792624i \(0.708714\pi\)
\(270\) −3.00000 −0.182574
\(271\) 6.00000 0.364474 0.182237 − 0.983255i \(-0.441666\pi\)
0.182237 + 0.983255i \(0.441666\pi\)
\(272\) −3.00000 −0.181902
\(273\) 0 0
\(274\) −19.0000 −1.14783
\(275\) −8.00000 −0.482418
\(276\) 1.00000 0.0601929
\(277\) 9.00000 0.540758 0.270379 − 0.962754i \(-0.412851\pi\)
0.270379 + 0.962754i \(0.412851\pi\)
\(278\) −10.0000 −0.599760
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 7.00000 0.417585 0.208792 − 0.977960i \(-0.433047\pi\)
0.208792 + 0.977960i \(0.433047\pi\)
\(282\) 9.00000 0.535942
\(283\) −7.00000 −0.416107 −0.208053 − 0.978117i \(-0.566713\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) 7.00000 0.415374
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −8.00000 −0.470588
\(290\) −24.0000 −1.40933
\(291\) −4.00000 −0.234484
\(292\) 3.00000 0.175562
\(293\) −13.0000 −0.759468 −0.379734 − 0.925096i \(-0.623985\pi\)
−0.379734 + 0.925096i \(0.623985\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 2.00000 0.116248
\(297\) 2.00000 0.116052
\(298\) −21.0000 −1.21650
\(299\) −3.00000 −0.173494
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) 20.0000 1.15087
\(303\) 10.0000 0.574485
\(304\) 0 0
\(305\) 18.0000 1.03068
\(306\) 3.00000 0.171499
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) −17.0000 −0.967096
\(310\) 30.0000 1.70389
\(311\) −13.0000 −0.737162 −0.368581 − 0.929596i \(-0.620156\pi\)
−0.368581 + 0.929596i \(0.620156\pi\)
\(312\) 3.00000 0.169842
\(313\) −28.0000 −1.58265 −0.791327 − 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −6.00000 −0.336994 −0.168497 − 0.985702i \(-0.553891\pi\)
−0.168497 + 0.985702i \(0.553891\pi\)
\(318\) −9.00000 −0.504695
\(319\) 16.0000 0.895828
\(320\) −3.00000 −0.167705
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 12.0000 0.665640
\(326\) 0 0
\(327\) −8.00000 −0.442401
\(328\) −8.00000 −0.441726
\(329\) 0 0
\(330\) 6.00000 0.330289
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −14.0000 −0.768350
\(333\) −2.00000 −0.109599
\(334\) 21.0000 1.14907
\(335\) −9.00000 −0.491723
\(336\) 0 0
\(337\) 16.0000 0.871576 0.435788 − 0.900049i \(-0.356470\pi\)
0.435788 + 0.900049i \(0.356470\pi\)
\(338\) 4.00000 0.217571
\(339\) 5.00000 0.271563
\(340\) 9.00000 0.488094
\(341\) −20.0000 −1.08306
\(342\) 0 0
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) −3.00000 −0.161515
\(346\) −20.0000 −1.07521
\(347\) −15.0000 −0.805242 −0.402621 − 0.915367i \(-0.631901\pi\)
−0.402621 + 0.915367i \(0.631901\pi\)
\(348\) 8.00000 0.428845
\(349\) −25.0000 −1.33822 −0.669110 − 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) −3.00000 −0.160128
\(352\) 2.00000 0.106600
\(353\) 4.00000 0.212899 0.106449 − 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 4.00000 0.212598
\(355\) −21.0000 −1.11456
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) −15.0000 −0.792775
\(359\) −6.00000 −0.316668 −0.158334 − 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 3.00000 0.158114
\(361\) −19.0000 −1.00000
\(362\) −6.00000 −0.315353
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −9.00000 −0.471082
\(366\) −6.00000 −0.313625
\(367\) −9.00000 −0.469796 −0.234898 − 0.972020i \(-0.575476\pi\)
−0.234898 + 0.972020i \(0.575476\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 8.00000 0.416463
\(370\) −6.00000 −0.311925
\(371\) 0 0
\(372\) −10.0000 −0.518476
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) −6.00000 −0.310253
\(375\) −3.00000 −0.154919
\(376\) −9.00000 −0.464140
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −29.0000 −1.48963 −0.744815 − 0.667271i \(-0.767462\pi\)
−0.744815 + 0.667271i \(0.767462\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) 8.00000 0.409316
\(383\) −22.0000 −1.12415 −0.562074 − 0.827087i \(-0.689996\pi\)
−0.562074 + 0.827087i \(0.689996\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −5.00000 −0.254493
\(387\) −4.00000 −0.203331
\(388\) 4.00000 0.203069
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) −9.00000 −0.455733
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) −18.0000 −0.906827
\(395\) −12.0000 −0.603786
\(396\) −2.00000 −0.100504
\(397\) 9.00000 0.451697 0.225849 − 0.974162i \(-0.427485\pi\)
0.225849 + 0.974162i \(0.427485\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) −39.0000 −1.94757 −0.973784 − 0.227477i \(-0.926952\pi\)
−0.973784 + 0.227477i \(0.926952\pi\)
\(402\) 3.00000 0.149626
\(403\) 30.0000 1.49441
\(404\) −10.0000 −0.497519
\(405\) −3.00000 −0.149071
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) −3.00000 −0.148522
\(409\) 39.0000 1.92843 0.964213 − 0.265129i \(-0.0854146\pi\)
0.964213 + 0.265129i \(0.0854146\pi\)
\(410\) 24.0000 1.18528
\(411\) −19.0000 −0.937201
\(412\) 17.0000 0.837530
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 42.0000 2.06170
\(416\) −3.00000 −0.147087
\(417\) −10.0000 −0.489702
\(418\) 0 0
\(419\) 28.0000 1.36789 0.683945 − 0.729534i \(-0.260263\pi\)
0.683945 + 0.729534i \(0.260263\pi\)
\(420\) 0 0
\(421\) −4.00000 −0.194948 −0.0974740 − 0.995238i \(-0.531076\pi\)
−0.0974740 + 0.995238i \(0.531076\pi\)
\(422\) 6.00000 0.292075
\(423\) 9.00000 0.437595
\(424\) 9.00000 0.437079
\(425\) −12.0000 −0.582086
\(426\) 7.00000 0.339151
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 6.00000 0.289683
\(430\) −12.0000 −0.578691
\(431\) 36.0000 1.73406 0.867029 − 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −18.0000 −0.865025 −0.432512 − 0.901628i \(-0.642373\pi\)
−0.432512 + 0.901628i \(0.642373\pi\)
\(434\) 0 0
\(435\) −24.0000 −1.15071
\(436\) 8.00000 0.383131
\(437\) 0 0
\(438\) 3.00000 0.143346
\(439\) 26.0000 1.24091 0.620456 − 0.784241i \(-0.286947\pi\)
0.620456 + 0.784241i \(0.286947\pi\)
\(440\) −6.00000 −0.286039
\(441\) 0 0
\(442\) 9.00000 0.428086
\(443\) −17.0000 −0.807694 −0.403847 − 0.914826i \(-0.632327\pi\)
−0.403847 + 0.914826i \(0.632327\pi\)
\(444\) 2.00000 0.0949158
\(445\) −30.0000 −1.42214
\(446\) −8.00000 −0.378811
\(447\) −21.0000 −0.993266
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 − 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) −4.00000 −0.188562
\(451\) −16.0000 −0.753411
\(452\) −5.00000 −0.235180
\(453\) 20.0000 0.939682
\(454\) 14.0000 0.657053
\(455\) 0 0
\(456\) 0 0
\(457\) −28.0000 −1.30978 −0.654892 − 0.755722i \(-0.727286\pi\)
−0.654892 + 0.755722i \(0.727286\pi\)
\(458\) 20.0000 0.934539
\(459\) 3.00000 0.140028
\(460\) 3.00000 0.139876
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 − 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) −8.00000 −0.371391
\(465\) 30.0000 1.39122
\(466\) 4.00000 0.185296
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 3.00000 0.138675
\(469\) 0 0
\(470\) 27.0000 1.24542
\(471\) 18.0000 0.829396
\(472\) −4.00000 −0.184115
\(473\) 8.00000 0.367840
\(474\) 4.00000 0.183726
\(475\) 0 0
\(476\) 0 0
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) −3.00000 −0.136931
\(481\) −6.00000 −0.273576
\(482\) 6.00000 0.273293
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −12.0000 −0.544892
\(486\) 1.00000 0.0453609
\(487\) −20.0000 −0.906287 −0.453143 − 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 0 0
\(491\) −29.0000 −1.30875 −0.654376 − 0.756169i \(-0.727069\pi\)
−0.654376 + 0.756169i \(0.727069\pi\)
\(492\) −8.00000 −0.360668
\(493\) 24.0000 1.08091
\(494\) 0 0
\(495\) 6.00000 0.269680
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) −14.0000 −0.627355
\(499\) −40.0000 −1.79065 −0.895323 − 0.445418i \(-0.853055\pi\)
−0.895323 + 0.445418i \(0.853055\pi\)
\(500\) 3.00000 0.134164
\(501\) 21.0000 0.938211
\(502\) 26.0000 1.16044
\(503\) −2.00000 −0.0891756 −0.0445878 − 0.999005i \(-0.514197\pi\)
−0.0445878 + 0.999005i \(0.514197\pi\)
\(504\) 0 0
\(505\) 30.0000 1.33498
\(506\) −2.00000 −0.0889108
\(507\) 4.00000 0.177646
\(508\) −2.00000 −0.0887357
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 9.00000 0.398527
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −14.0000 −0.617514
\(515\) −51.0000 −2.24733
\(516\) 4.00000 0.176090
\(517\) −18.0000 −0.791639
\(518\) 0 0
\(519\) −20.0000 −0.877903
\(520\) 9.00000 0.394676
\(521\) −41.0000 −1.79624 −0.898121 − 0.439748i \(-0.855068\pi\)
−0.898121 + 0.439748i \(0.855068\pi\)
\(522\) 8.00000 0.350150
\(523\) 23.0000 1.00572 0.502860 − 0.864368i \(-0.332281\pi\)
0.502860 + 0.864368i \(0.332281\pi\)
\(524\) 15.0000 0.655278
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −30.0000 −1.30682
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) −27.0000 −1.17281
\(531\) 4.00000 0.173585
\(532\) 0 0
\(533\) 24.0000 1.03956
\(534\) 10.0000 0.432742
\(535\) 24.0000 1.03761
\(536\) −3.00000 −0.129580
\(537\) −15.0000 −0.647298
\(538\) 20.0000 0.862261
\(539\) 0 0
\(540\) 3.00000 0.129099
\(541\) −15.0000 −0.644900 −0.322450 − 0.946586i \(-0.604506\pi\)
−0.322450 + 0.946586i \(0.604506\pi\)
\(542\) −6.00000 −0.257722
\(543\) −6.00000 −0.257485
\(544\) 3.00000 0.128624
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 − 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 19.0000 0.811640
\(549\) −6.00000 −0.256074
\(550\) 8.00000 0.341121
\(551\) 0 0
\(552\) −1.00000 −0.0425628
\(553\) 0 0
\(554\) −9.00000 −0.382373
\(555\) −6.00000 −0.254686
\(556\) 10.0000 0.424094
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −10.0000 −0.423334
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) −7.00000 −0.295277
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −9.00000 −0.378968
\(565\) 15.0000 0.631055
\(566\) 7.00000 0.294232
\(567\) 0 0
\(568\) −7.00000 −0.293713
\(569\) 5.00000 0.209611 0.104805 − 0.994493i \(-0.466578\pi\)
0.104805 + 0.994493i \(0.466578\pi\)
\(570\) 0 0
\(571\) 25.0000 1.04622 0.523109 − 0.852266i \(-0.324772\pi\)
0.523109 + 0.852266i \(0.324772\pi\)
\(572\) −6.00000 −0.250873
\(573\) 8.00000 0.334205
\(574\) 0 0
\(575\) −4.00000 −0.166812
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) 8.00000 0.332756
\(579\) −5.00000 −0.207793
\(580\) 24.0000 0.996546
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) 18.0000 0.745484
\(584\) −3.00000 −0.124141
\(585\) −9.00000 −0.372104
\(586\) 13.0000 0.537025
\(587\) 25.0000 1.03186 0.515930 − 0.856631i \(-0.327446\pi\)
0.515930 + 0.856631i \(0.327446\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 12.0000 0.494032
\(591\) −18.0000 −0.740421
\(592\) −2.00000 −0.0821995
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 21.0000 0.860194
\(597\) 8.00000 0.327418
\(598\) 3.00000 0.122679
\(599\) 9.00000 0.367730 0.183865 − 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 4.00000 0.163299
\(601\) 18.0000 0.734235 0.367118 − 0.930175i \(-0.380345\pi\)
0.367118 + 0.930175i \(0.380345\pi\)
\(602\) 0 0
\(603\) 3.00000 0.122169
\(604\) −20.0000 −0.813788
\(605\) 21.0000 0.853771
\(606\) −10.0000 −0.406222
\(607\) −38.0000 −1.54237 −0.771186 − 0.636610i \(-0.780336\pi\)
−0.771186 + 0.636610i \(0.780336\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −18.0000 −0.728799
\(611\) 27.0000 1.09230
\(612\) −3.00000 −0.121268
\(613\) −42.0000 −1.69636 −0.848182 − 0.529705i \(-0.822303\pi\)
−0.848182 + 0.529705i \(0.822303\pi\)
\(614\) −16.0000 −0.645707
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) −9.00000 −0.362326 −0.181163 − 0.983453i \(-0.557986\pi\)
−0.181163 + 0.983453i \(0.557986\pi\)
\(618\) 17.0000 0.683840
\(619\) 19.0000 0.763674 0.381837 − 0.924230i \(-0.375291\pi\)
0.381837 + 0.924230i \(0.375291\pi\)
\(620\) −30.0000 −1.20483
\(621\) 1.00000 0.0401286
\(622\) 13.0000 0.521253
\(623\) 0 0
\(624\) −3.00000 −0.120096
\(625\) −29.0000 −1.16000
\(626\) 28.0000 1.11911
\(627\) 0 0
\(628\) −18.0000 −0.718278
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) −23.0000 −0.915616 −0.457808 − 0.889051i \(-0.651365\pi\)
−0.457808 + 0.889051i \(0.651365\pi\)
\(632\) −4.00000 −0.159111
\(633\) 6.00000 0.238479
\(634\) 6.00000 0.238290
\(635\) 6.00000 0.238103
\(636\) 9.00000 0.356873
\(637\) 0 0
\(638\) −16.0000 −0.633446
\(639\) 7.00000 0.276916
\(640\) 3.00000 0.118585
\(641\) 37.0000 1.46141 0.730706 − 0.682692i \(-0.239191\pi\)
0.730706 + 0.682692i \(0.239191\pi\)
\(642\) −8.00000 −0.315735
\(643\) −20.0000 −0.788723 −0.394362 − 0.918955i \(-0.629034\pi\)
−0.394362 + 0.918955i \(0.629034\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) 0 0
\(647\) −48.0000 −1.88707 −0.943537 − 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −8.00000 −0.314027
\(650\) −12.0000 −0.470679
\(651\) 0 0
\(652\) 0 0
\(653\) 10.0000 0.391330 0.195665 − 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 8.00000 0.312825
\(655\) −45.0000 −1.75830
\(656\) 8.00000 0.312348
\(657\) 3.00000 0.117041
\(658\) 0 0
\(659\) −2.00000 −0.0779089 −0.0389545 − 0.999241i \(-0.512403\pi\)
−0.0389545 + 0.999241i \(0.512403\pi\)
\(660\) −6.00000 −0.233550
\(661\) 40.0000 1.55582 0.777910 − 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) −10.0000 −0.388661
\(663\) 9.00000 0.349531
\(664\) 14.0000 0.543305
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 8.00000 0.309761
\(668\) −21.0000 −0.812514
\(669\) −8.00000 −0.309298
\(670\) 9.00000 0.347700
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) 34.0000 1.31060 0.655302 − 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) −16.0000 −0.616297
\(675\) −4.00000 −0.153960
\(676\) −4.00000 −0.153846
\(677\) −27.0000 −1.03769 −0.518847 − 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) −5.00000 −0.192024
\(679\) 0 0
\(680\) −9.00000 −0.345134
\(681\) 14.0000 0.536481
\(682\) 20.0000 0.765840
\(683\) −37.0000 −1.41577 −0.707883 − 0.706330i \(-0.750350\pi\)
−0.707883 + 0.706330i \(0.750350\pi\)
\(684\) 0 0
\(685\) −57.0000 −2.17786
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) −4.00000 −0.152499
\(689\) −27.0000 −1.02862
\(690\) 3.00000 0.114208
\(691\) −6.00000 −0.228251 −0.114125 − 0.993466i \(-0.536407\pi\)
−0.114125 + 0.993466i \(0.536407\pi\)
\(692\) 20.0000 0.760286
\(693\) 0 0
\(694\) 15.0000 0.569392
\(695\) −30.0000 −1.13796
\(696\) −8.00000 −0.303239
\(697\) −24.0000 −0.909065
\(698\) 25.0000 0.946264
\(699\) 4.00000 0.151294
\(700\) 0 0
\(701\) 33.0000 1.24639 0.623196 − 0.782065i \(-0.285834\pi\)
0.623196 + 0.782065i \(0.285834\pi\)
\(702\) 3.00000 0.113228
\(703\) 0 0
\(704\) −2.00000 −0.0753778
\(705\) 27.0000 1.01688
\(706\) −4.00000 −0.150542
\(707\) 0 0
\(708\) −4.00000 −0.150329
\(709\) 4.00000 0.150223 0.0751116 − 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 21.0000 0.788116
\(711\) 4.00000 0.150012
\(712\) −10.0000 −0.374766
\(713\) −10.0000 −0.374503
\(714\) 0 0
\(715\) 18.0000 0.673162
\(716\) 15.0000 0.560576
\(717\) 0 0
\(718\) 6.00000 0.223918
\(719\) 41.0000 1.52904 0.764521 − 0.644599i \(-0.222976\pi\)
0.764521 + 0.644599i \(0.222976\pi\)
\(720\) −3.00000 −0.111803
\(721\) 0 0
\(722\) 19.0000 0.707107
\(723\) 6.00000 0.223142
\(724\) 6.00000 0.222988
\(725\) −32.0000 −1.18845
\(726\) −7.00000 −0.259794
\(727\) −40.0000 −1.48352 −0.741759 − 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 9.00000 0.333105
\(731\) 12.0000 0.443836
\(732\) 6.00000 0.221766
\(733\) −24.0000 −0.886460 −0.443230 − 0.896408i \(-0.646168\pi\)
−0.443230 + 0.896408i \(0.646168\pi\)
\(734\) 9.00000 0.332196
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −6.00000 −0.221013
\(738\) −8.00000 −0.294484
\(739\) −24.0000 −0.882854 −0.441427 − 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) 6.00000 0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) 14.0000 0.513610 0.256805 − 0.966463i \(-0.417330\pi\)
0.256805 + 0.966463i \(0.417330\pi\)
\(744\) 10.0000 0.366618
\(745\) −63.0000 −2.30814
\(746\) −10.0000 −0.366126
\(747\) −14.0000 −0.512233
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 3.00000 0.109545
\(751\) −24.0000 −0.875772 −0.437886 − 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 9.00000 0.328196
\(753\) 26.0000 0.947493
\(754\) 24.0000 0.874028
\(755\) 60.0000 2.18362
\(756\) 0 0
\(757\) −40.0000 −1.45382 −0.726912 − 0.686730i \(-0.759045\pi\)
−0.726912 + 0.686730i \(0.759045\pi\)
\(758\) 29.0000 1.05333
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) −4.00000 −0.145000 −0.0724999 − 0.997368i \(-0.523098\pi\)
−0.0724999 + 0.997368i \(0.523098\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 9.00000 0.325396
\(766\) 22.0000 0.794892
\(767\) 12.0000 0.433295
\(768\) −1.00000 −0.0360844
\(769\) −34.0000 −1.22607 −0.613036 − 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 5.00000 0.179954
\(773\) 17.0000 0.611448 0.305724 − 0.952120i \(-0.401102\pi\)
0.305724 + 0.952120i \(0.401102\pi\)
\(774\) 4.00000 0.143777
\(775\) 40.0000 1.43684
\(776\) −4.00000 −0.143592
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) 0 0
\(780\) 9.00000 0.322252
\(781\) −14.0000 −0.500959
\(782\) −3.00000 −0.107280
\(783\) 8.00000 0.285897
\(784\) 0 0
\(785\) 54.0000 1.92734
\(786\) 15.0000 0.535032
\(787\) 53.0000 1.88925 0.944623 − 0.328158i \(-0.106428\pi\)
0.944623 + 0.328158i \(0.106428\pi\)
\(788\) 18.0000 0.641223
\(789\) 12.0000 0.427211
\(790\) 12.0000 0.426941
\(791\) 0 0
\(792\) 2.00000 0.0710669
\(793\) −18.0000 −0.639199
\(794\) −9.00000 −0.319398
\(795\) −27.0000 −0.957591
\(796\) −8.00000 −0.283552
\(797\) 11.0000 0.389640 0.194820 − 0.980839i \(-0.437588\pi\)
0.194820 + 0.980839i \(0.437588\pi\)
\(798\) 0 0
\(799\) −27.0000 −0.955191
\(800\) −4.00000 −0.141421
\(801\) 10.0000 0.353333
\(802\) 39.0000 1.37714
\(803\) −6.00000 −0.211735
\(804\) −3.00000 −0.105802
\(805\) 0 0
\(806\) −30.0000 −1.05670
\(807\) 20.0000 0.704033
\(808\) 10.0000 0.351799
\(809\) −6.00000 −0.210949 −0.105474 − 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 3.00000 0.105409
\(811\) −46.0000 −1.61528 −0.807639 − 0.589677i \(-0.799255\pi\)
−0.807639 + 0.589677i \(0.799255\pi\)
\(812\) 0 0
\(813\) −6.00000 −0.210429
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) 0 0
\(818\) −39.0000 −1.36360
\(819\) 0 0
\(820\) −24.0000 −0.838116
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 19.0000 0.662701
\(823\) −26.0000 −0.906303 −0.453152 − 0.891434i \(-0.649700\pi\)
−0.453152 + 0.891434i \(0.649700\pi\)
\(824\) −17.0000 −0.592223
\(825\) 8.00000 0.278524
\(826\) 0 0
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 25.0000 0.868286 0.434143 − 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) −42.0000 −1.45784
\(831\) −9.00000 −0.312207
\(832\) 3.00000 0.104006
\(833\) 0 0
\(834\) 10.0000 0.346272
\(835\) 63.0000 2.18020
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) −28.0000 −0.967244
\(839\) −12.0000 −0.414286 −0.207143 − 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 4.00000 0.137849
\(843\) −7.00000 −0.241093
\(844\) −6.00000 −0.206529
\(845\) 12.0000 0.412813
\(846\) −9.00000 −0.309426
\(847\) 0 0
\(848\) −9.00000 −0.309061
\(849\) 7.00000 0.240239
\(850\) 12.0000 0.411597
\(851\) 2.00000 0.0685591
\(852\) −7.00000 −0.239816
\(853\) 38.0000 1.30110 0.650548 − 0.759465i \(-0.274539\pi\)
0.650548 + 0.759465i \(0.274539\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 8.00000 0.273434
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) −6.00000 −0.204837
\(859\) 40.0000 1.36478 0.682391 − 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 12.0000 0.409197
\(861\) 0 0
\(862\) −36.0000 −1.22616
\(863\) −51.0000 −1.73606 −0.868030 − 0.496512i \(-0.834614\pi\)
−0.868030 + 0.496512i \(0.834614\pi\)
\(864\) 1.00000 0.0340207
\(865\) −60.0000 −2.04006
\(866\) 18.0000 0.611665
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −8.00000 −0.271381
\(870\) 24.0000 0.813676
\(871\) 9.00000 0.304953
\(872\) −8.00000 −0.270914
\(873\) 4.00000 0.135379
\(874\) 0 0
\(875\) 0 0
\(876\) −3.00000 −0.101361
\(877\) −41.0000 −1.38447 −0.692236 − 0.721671i \(-0.743374\pi\)
−0.692236 + 0.721671i \(0.743374\pi\)
\(878\) −26.0000 −0.877457
\(879\) 13.0000 0.438479
\(880\) 6.00000 0.202260
\(881\) 35.0000 1.17918 0.589590 − 0.807703i \(-0.299289\pi\)
0.589590 + 0.807703i \(0.299289\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 − 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −9.00000 −0.302703
\(885\) 12.0000 0.403376
\(886\) 17.0000 0.571126
\(887\) 8.00000 0.268614 0.134307 − 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 30.0000 1.00560
\(891\) −2.00000 −0.0670025
\(892\) 8.00000 0.267860
\(893\) 0 0
\(894\) 21.0000 0.702345
\(895\) −45.0000 −1.50418
\(896\) 0 0
\(897\) 3.00000 0.100167
\(898\) −6.00000 −0.200223
\(899\) −80.0000 −2.66815
\(900\) 4.00000 0.133333
\(901\) 27.0000 0.899500
\(902\) 16.0000 0.532742
\(903\) 0 0
\(904\) 5.00000 0.166298
\(905\) −18.0000 −0.598340
\(906\) −20.0000 −0.664455
\(907\) 29.0000 0.962929 0.481465 − 0.876466i \(-0.340105\pi\)
0.481465 + 0.876466i \(0.340105\pi\)
\(908\) −14.0000 −0.464606
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −4.00000 −0.132526 −0.0662630 − 0.997802i \(-0.521108\pi\)
−0.0662630 + 0.997802i \(0.521108\pi\)
\(912\) 0 0
\(913\) 28.0000 0.926665
\(914\) 28.0000 0.926158
\(915\) −18.0000 −0.595062
\(916\) −20.0000 −0.660819
\(917\) 0 0
\(918\) −3.00000 −0.0990148
\(919\) 45.0000 1.48441 0.742207 − 0.670171i \(-0.233779\pi\)
0.742207 + 0.670171i \(0.233779\pi\)
\(920\) −3.00000 −0.0989071
\(921\) −16.0000 −0.527218
\(922\) 18.0000 0.592798
\(923\) 21.0000 0.691223
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) 4.00000 0.131448
\(927\) 17.0000 0.558353
\(928\) 8.00000 0.262613
\(929\) −36.0000 −1.18112 −0.590561 − 0.806993i \(-0.701093\pi\)
−0.590561 + 0.806993i \(0.701093\pi\)
\(930\) −30.0000 −0.983739
\(931\) 0 0
\(932\) −4.00000 −0.131024
\(933\) 13.0000 0.425601
\(934\) 12.0000 0.392652
\(935\) −18.0000 −0.588663
\(936\) −3.00000 −0.0980581
\(937\) 52.0000 1.69877 0.849383 − 0.527777i \(-0.176974\pi\)
0.849383 + 0.527777i \(0.176974\pi\)
\(938\) 0 0
\(939\) 28.0000 0.913745
\(940\) −27.0000 −0.880643
\(941\) 18.0000 0.586783 0.293392 − 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) −18.0000 −0.586472
\(943\) −8.00000 −0.260516
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −8.00000 −0.260102
\(947\) −3.00000 −0.0974869 −0.0487435 − 0.998811i \(-0.515522\pi\)
−0.0487435 + 0.998811i \(0.515522\pi\)
\(948\) −4.00000 −0.129914
\(949\) 9.00000 0.292152
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −46.0000 −1.49009 −0.745043 − 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 9.00000 0.291386
\(955\) 24.0000 0.776622
\(956\) 0 0
\(957\) −16.0000 −0.517207
\(958\) 0 0
\(959\) 0 0
\(960\) 3.00000 0.0968246
\(961\) 69.0000 2.22581
\(962\) 6.00000 0.193448
\(963\) −8.00000 −0.257796
\(964\) −6.00000 −0.193247
\(965\) −15.0000 −0.482867
\(966\) 0 0
\(967\) 16.0000 0.514525 0.257263 − 0.966342i \(-0.417179\pi\)
0.257263 + 0.966342i \(0.417179\pi\)
\(968\) 7.00000 0.224989
\(969\) 0 0
\(970\) 12.0000 0.385297
\(971\) −40.0000 −1.28366 −0.641831 − 0.766846i \(-0.721825\pi\)
−0.641831 + 0.766846i \(0.721825\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 20.0000 0.640841
\(975\) −12.0000 −0.384308
\(976\) −6.00000 −0.192055
\(977\) −37.0000 −1.18373 −0.591867 − 0.806035i \(-0.701609\pi\)
−0.591867 + 0.806035i \(0.701609\pi\)
\(978\) 0 0
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) 8.00000 0.255420
\(982\) 29.0000 0.925427
\(983\) −10.0000 −0.318950 −0.159475 − 0.987202i \(-0.550980\pi\)
−0.159475 + 0.987202i \(0.550980\pi\)
\(984\) 8.00000 0.255031
\(985\) −54.0000 −1.72058
\(986\) −24.0000 −0.764316
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) −6.00000 −0.190693
\(991\) 14.0000 0.444725 0.222362 − 0.974964i \(-0.428623\pi\)
0.222362 + 0.974964i \(0.428623\pi\)
\(992\) −10.0000 −0.317500
\(993\) −10.0000 −0.317340
\(994\) 0 0
\(995\) 24.0000 0.760851
\(996\) 14.0000 0.443607
\(997\) −38.0000 −1.20347 −0.601736 − 0.798695i \(-0.705524\pi\)
−0.601736 + 0.798695i \(0.705524\pi\)
\(998\) 40.0000 1.26618
\(999\) 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 6762.2.a.b.1.1 1
7.2 even 3 966.2.i.g.277.1 ✓ 2
7.4 even 3 966.2.i.g.415.1 yes 2
7.6 odd 2 6762.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.g.277.1 ✓ 2 7.2 even 3
966.2.i.g.415.1 yes 2 7.4 even 3
6762.2.a.b.1.1 1 1.1 even 1 trivial
6762.2.a.u.1.1 1 7.6 odd 2