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Properties

Label 6762.2.a.c.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: yes
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -6.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} +6.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} -2.00000 q^{30} -6.00000 q^{31} -1.00000 q^{32} +6.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} -6.00000 q^{39} +2.00000 q^{40} +2.00000 q^{43} -6.00000 q^{44} -2.00000 q^{45} -1.00000 q^{46} +6.00000 q^{47} -1.00000 q^{48} +1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} +4.00000 q^{53} +1.00000 q^{54} +12.0000 q^{55} +4.00000 q^{57} -2.00000 q^{58} -8.00000 q^{59} +2.00000 q^{60} +2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} -6.00000 q^{66} +6.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} -8.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} -8.00000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +6.00000 q^{78} +4.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -12.0000 q^{83} -4.00000 q^{85} -2.00000 q^{86} -2.00000 q^{87} +6.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} +1.00000 q^{92} +6.00000 q^{93} -6.00000 q^{94} +8.00000 q^{95} +1.00000 q^{96} +10.0000 q^{97} -6.00000 q^{99} +O(q^{100})\)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) −1.00000 −0.288675
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 − 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −6.00000 −1.17670
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −2.00000 −0.365148
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.00000 1.04447
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 4.00000 0.648886
\(39\) −6.00000 −0.960769
\(40\) 2.00000 0.316228
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 − 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) −6.00000 −0.904534
\(45\) −2.00000 −0.298142
\(46\) −1.00000 −0.147442
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 6.00000 0.832050
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 1.00000 0.136083
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) −2.00000 −0.262613
\(59\) −8.00000 −1.04151 −0.520756 − 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 2.00000 0.258199
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −12.0000 −1.48842
\(66\) −6.00000 −0.738549
\(67\) 6.00000 0.733017 0.366508 − 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 2.00000 0.242536
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −8.00000 −0.929981
\(75\) 1.00000 0.115470
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −2.00000 −0.215666
\(87\) −2.00000 −0.214423
\(88\) 6.00000 0.639602
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) 6.00000 0.622171
\(94\) −6.00000 −0.618853
\(95\) 8.00000 0.820783
\(96\) 1.00000 0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) −6.00000 −0.603023
\(100\) −1.00000 −0.100000
\(101\) −6.00000 −0.597022 −0.298511 − 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 2.00000 0.198030
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 12.0000 1.14939 0.574696 − 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) −12.0000 −1.14416
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) −4.00000 −0.374634
\(115\) −2.00000 −0.186501
\(116\) 2.00000 0.185695
\(117\) 6.00000 0.554700
\(118\) 8.00000 0.736460
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 25.0000 2.27273
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 − 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 12.0000 1.05247
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) −6.00000 −0.518321
\(135\) 2.00000 0.172133
\(136\) −2.00000 −0.171499
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 1.00000 0.0851257
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 8.00000 0.671345
\(143\) −36.0000 −3.01047
\(144\) 1.00000 0.0833333
\(145\) −4.00000 −0.332182
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 4.00000 0.324443
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 12.0000 0.963863
\(156\) −6.00000 −0.480384
\(157\) −18.0000 −1.43656 −0.718278 − 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −4.00000 −0.318223
\(159\) −4.00000 −0.317221
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 0 0
\(165\) −12.0000 −0.934199
\(166\) 12.0000 0.931381
\(167\) −6.00000 −0.464294 −0.232147 − 0.972681i \(-0.574575\pi\)
−0.232147 + 0.972681i \(0.574575\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 4.00000 0.306786
\(171\) −4.00000 −0.305888
\(172\) 2.00000 0.152499
\(173\) −10.0000 −0.760286 −0.380143 − 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −6.00000 −0.452267
\(177\) 8.00000 0.601317
\(178\) 6.00000 0.449719
\(179\) −24.0000 −1.79384 −0.896922 − 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) −2.00000 −0.149071
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) −1.00000 −0.0737210
\(185\) −16.0000 −1.17634
\(186\) −6.00000 −0.439941
\(187\) −12.0000 −0.877527
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 14.0000 1.00774 0.503871 − 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −10.0000 −0.717958
\(195\) 12.0000 0.859338
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 6.00000 0.426401
\(199\) −20.0000 −1.41776 −0.708881 − 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 1.00000 0.0707107
\(201\) −6.00000 −0.423207
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 1.00000 0.0695048
\(208\) 6.00000 0.416025
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 4.00000 0.274721
\(213\) 8.00000 0.548151
\(214\) −6.00000 −0.410152
\(215\) −4.00000 −0.272798
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −12.0000 −0.812743
\(219\) 4.00000 0.270295
\(220\) 12.0000 0.809040
\(221\) 12.0000 0.807207
\(222\) 8.00000 0.536925
\(223\) −10.0000 −0.669650 −0.334825 − 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −18.0000 −1.19734
\(227\) −4.00000 −0.265489 −0.132745 − 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 4.00000 0.264906
\(229\) −14.0000 −0.925146 −0.462573 − 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 10.0000 0.655122 0.327561 − 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) −6.00000 −0.392232
\(235\) −12.0000 −0.782794
\(236\) −8.00000 −0.520756
\(237\) −4.00000 −0.259828
\(238\) 0 0
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 2.00000 0.129099
\(241\) 14.0000 0.901819 0.450910 − 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) −25.0000 −1.60706
\(243\) −1.00000 −0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) −24.0000 −1.52708
\(248\) 6.00000 0.381000
\(249\) 12.0000 0.760469
\(250\) −12.0000 −0.758947
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −8.00000 −0.501965
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) −12.0000 −0.744208
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −6.00000 −0.369274
\(265\) −8.00000 −0.491436
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 6.00000 0.366508
\(269\) 6.00000 0.365826 0.182913 − 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) −2.00000 −0.121716
\(271\) −26.0000 −1.57939 −0.789694 − 0.613501i \(-0.789761\pi\)
−0.789694 + 0.613501i \(0.789761\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 6.00000 0.361814
\(276\) −1.00000 −0.0601929
\(277\) 2.00000 0.120168 0.0600842 − 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 4.00000 0.239904
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 6.00000 0.357295
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) −8.00000 −0.474713
\(285\) −8.00000 −0.473879
\(286\) 36.0000 2.12872
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) 4.00000 0.234888
\(291\) −10.0000 −0.586210
\(292\) −4.00000 −0.234082
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) −8.00000 −0.464991
\(297\) 6.00000 0.348155
\(298\) 12.0000 0.695141
\(299\) 6.00000 0.346989
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000 0.344691
\(304\) −4.00000 −0.229416
\(305\) −4.00000 −0.229039
\(306\) −2.00000 −0.114332
\(307\) 32.0000 1.82634 0.913168 − 0.407583i \(-0.133628\pi\)
0.913168 + 0.407583i \(0.133628\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) −12.0000 −0.681554
\(311\) −6.00000 −0.340229 −0.170114 − 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 6.00000 0.339683
\(313\) −34.0000 −1.92179 −0.960897 − 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 4.00000 0.224309
\(319\) −12.0000 −0.671871
\(320\) −2.00000 −0.111803
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) 1.00000 0.0555556
\(325\) −6.00000 −0.332820
\(326\) 4.00000 0.221540
\(327\) −12.0000 −0.663602
\(328\) 0 0
\(329\) 0 0
\(330\) 12.0000 0.660578
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −12.0000 −0.658586
\(333\) 8.00000 0.438397
\(334\) 6.00000 0.328305
\(335\) −12.0000 −0.655630
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 − 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −23.0000 −1.25104
\(339\) −18.0000 −0.977626
\(340\) −4.00000 −0.216930
\(341\) 36.0000 1.94951
\(342\) 4.00000 0.216295
\(343\) 0 0
\(344\) −2.00000 −0.107833
\(345\) 2.00000 0.107676
\(346\) 10.0000 0.537603
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) −2.00000 −0.107211
\(349\) −10.0000 −0.535288 −0.267644 − 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 6.00000 0.319801
\(353\) 16.0000 0.851594 0.425797 − 0.904819i \(-0.359994\pi\)
0.425797 + 0.904819i \(0.359994\pi\)
\(354\) −8.00000 −0.425195
\(355\) 16.0000 0.849192
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) 8.00000 0.418739
\(366\) 2.00000 0.104542
\(367\) −20.0000 −1.04399 −0.521996 − 0.852948i \(-0.674812\pi\)
−0.521996 + 0.852948i \(0.674812\pi\)
\(368\) 1.00000 0.0521286
\(369\) 0 0
\(370\) 16.0000 0.831800
\(371\) 0 0
\(372\) 6.00000 0.311086
\(373\) −8.00000 −0.414224 −0.207112 − 0.978317i \(-0.566407\pi\)
−0.207112 + 0.978317i \(0.566407\pi\)
\(374\) 12.0000 0.620505
\(375\) −12.0000 −0.619677
\(376\) −6.00000 −0.309426
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 26.0000 1.33553 0.667765 − 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 8.00000 0.410391
\(381\) −8.00000 −0.409852
\(382\) 16.0000 0.818631
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 2.00000 0.101666
\(388\) 10.0000 0.507673
\(389\) −24.0000 −1.21685 −0.608424 − 0.793612i \(-0.708198\pi\)
−0.608424 + 0.793612i \(0.708198\pi\)
\(390\) −12.0000 −0.607644
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) −8.00000 −0.402524
\(396\) −6.00000 −0.301511
\(397\) −26.0000 −1.30490 −0.652451 − 0.757831i \(-0.726259\pi\)
−0.652451 + 0.757831i \(0.726259\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 2.00000 0.0998752 0.0499376 − 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 6.00000 0.299253
\(403\) −36.0000 −1.79329
\(404\) −6.00000 −0.298511
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) −48.0000 −2.37927
\(408\) 2.00000 0.0990148
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −1.00000 −0.0491473
\(415\) 24.0000 1.17811
\(416\) −6.00000 −0.294174
\(417\) 4.00000 0.195881
\(418\) −24.0000 −1.17388
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) −24.0000 −1.16969 −0.584844 − 0.811146i \(-0.698844\pi\)
−0.584844 + 0.811146i \(0.698844\pi\)
\(422\) −8.00000 −0.389434
\(423\) 6.00000 0.291730
\(424\) −4.00000 −0.194257
\(425\) −2.00000 −0.0970143
\(426\) −8.00000 −0.387601
\(427\) 0 0
\(428\) 6.00000 0.290021
\(429\) 36.0000 1.73810
\(430\) 4.00000 0.192897
\(431\) 28.0000 1.34871 0.674356 − 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −38.0000 −1.82616 −0.913082 − 0.407777i \(-0.866304\pi\)
−0.913082 + 0.407777i \(0.866304\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 12.0000 0.574696
\(437\) −4.00000 −0.191346
\(438\) −4.00000 −0.191127
\(439\) −14.0000 −0.668184 −0.334092 − 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) −12.0000 −0.572078
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 36.0000 1.71041 0.855206 − 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −8.00000 −0.379663
\(445\) 12.0000 0.568855
\(446\) 10.0000 0.473514
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −38.0000 −1.79333 −0.896665 − 0.442709i \(-0.854018\pi\)
−0.896665 + 0.442709i \(0.854018\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) 18.0000 0.846649
\(453\) 0 0
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −34.0000 −1.59045 −0.795226 − 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 14.0000 0.654177
\(459\) −2.00000 −0.0933520
\(460\) −2.00000 −0.0932505
\(461\) 30.0000 1.39724 0.698620 − 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 40.0000 1.85896 0.929479 − 0.368875i \(-0.120257\pi\)
0.929479 + 0.368875i \(0.120257\pi\)
\(464\) 2.00000 0.0928477
\(465\) −12.0000 −0.556487
\(466\) −10.0000 −0.463241
\(467\) −12.0000 −0.555294 −0.277647 − 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 6.00000 0.277350
\(469\) 0 0
\(470\) 12.0000 0.553519
\(471\) 18.0000 0.829396
\(472\) 8.00000 0.368230
\(473\) −12.0000 −0.551761
\(474\) 4.00000 0.183726
\(475\) 4.00000 0.183533
\(476\) 0 0
\(477\) 4.00000 0.183147
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 − 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 48.0000 2.18861
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) −20.0000 −0.908153
\(486\) 1.00000 0.0453609
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −28.0000 −1.26362 −0.631811 − 0.775122i \(-0.717688\pi\)
−0.631811 + 0.775122i \(0.717688\pi\)
\(492\) 0 0
\(493\) 4.00000 0.180151
\(494\) 24.0000 1.07981
\(495\) 12.0000 0.539360
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) −12.0000 −0.537733
\(499\) 28.0000 1.25345 0.626726 − 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 12.0000 0.536656
\(501\) 6.00000 0.268060
\(502\) −12.0000 −0.535586
\(503\) −12.0000 −0.535054 −0.267527 − 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 6.00000 0.266733
\(507\) −23.0000 −1.02147
\(508\) 8.00000 0.354943
\(509\) −42.0000 −1.86162 −0.930809 − 0.365507i \(-0.880896\pi\)
−0.930809 + 0.365507i \(0.880896\pi\)
\(510\) −4.00000 −0.177123
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 4.00000 0.176604
\(514\) −12.0000 −0.529297
\(515\) −16.0000 −0.705044
\(516\) −2.00000 −0.0880451
\(517\) −36.0000 −1.58328
\(518\) 0 0
\(519\) 10.0000 0.438951
\(520\) 12.0000 0.526235
\(521\) 14.0000 0.613351 0.306676 − 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 4.00000 0.174908 0.0874539 − 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) −12.0000 −0.522728
\(528\) 6.00000 0.261116
\(529\) 1.00000 0.0434783
\(530\) 8.00000 0.347498
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) −6.00000 −0.259645
\(535\) −12.0000 −0.518805
\(536\) −6.00000 −0.259161
\(537\) 24.0000 1.03568
\(538\) −6.00000 −0.258678
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 26.0000 1.11680
\(543\) −2.00000 −0.0858282
\(544\) −2.00000 −0.0857493
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 18.0000 0.768922
\(549\) 2.00000 0.0853579
\(550\) −6.00000 −0.255841
\(551\) −8.00000 −0.340811
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) −2.00000 −0.0849719
\(555\) 16.0000 0.679162
\(556\) −4.00000 −0.169638
\(557\) −32.0000 −1.35588 −0.677942 − 0.735116i \(-0.737128\pi\)
−0.677942 + 0.735116i \(0.737128\pi\)
\(558\) 6.00000 0.254000
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 18.0000 0.759284
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) −6.00000 −0.252646
\(565\) −36.0000 −1.51453
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −46.0000 −1.92842 −0.964210 − 0.265139i \(-0.914582\pi\)
−0.964210 + 0.265139i \(0.914582\pi\)
\(570\) 8.00000 0.335083
\(571\) −10.0000 −0.418487 −0.209243 − 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) −36.0000 −1.50524
\(573\) 16.0000 0.668410
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 1.00000 0.0416667
\(577\) −28.0000 −1.16566 −0.582828 − 0.812596i \(-0.698054\pi\)
−0.582828 + 0.812596i \(0.698054\pi\)
\(578\) 13.0000 0.540729
\(579\) −14.0000 −0.581820
\(580\) −4.00000 −0.166091
\(581\) 0 0
\(582\) 10.0000 0.414513
\(583\) −24.0000 −0.993978
\(584\) 4.00000 0.165521
\(585\) −12.0000 −0.496139
\(586\) 6.00000 0.247858
\(587\) −44.0000 −1.81607 −0.908037 − 0.418890i \(-0.862419\pi\)
−0.908037 + 0.418890i \(0.862419\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) −16.0000 −0.658710
\(591\) 18.0000 0.740421
\(592\) 8.00000 0.328798
\(593\) 4.00000 0.164260 0.0821302 − 0.996622i \(-0.473828\pi\)
0.0821302 + 0.996622i \(0.473828\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 20.0000 0.818546
\(598\) −6.00000 −0.245358
\(599\) 40.0000 1.63436 0.817178 − 0.576386i \(-0.195537\pi\)
0.817178 + 0.576386i \(0.195537\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 4.00000 0.163163 0.0815817 − 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) 0 0
\(605\) −50.0000 −2.03279
\(606\) −6.00000 −0.243733
\(607\) 42.0000 1.70473 0.852364 − 0.522949i \(-0.175168\pi\)
0.852364 + 0.522949i \(0.175168\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 36.0000 1.45640
\(612\) 2.00000 0.0808452
\(613\) −44.0000 −1.77714 −0.888572 − 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) −32.0000 −1.29141
\(615\) 0 0
\(616\) 0 0
\(617\) −2.00000 −0.0805170 −0.0402585 − 0.999189i \(-0.512818\pi\)
−0.0402585 + 0.999189i \(0.512818\pi\)
\(618\) 8.00000 0.321807
\(619\) −36.0000 −1.44696 −0.723481 − 0.690344i \(-0.757459\pi\)
−0.723481 + 0.690344i \(0.757459\pi\)
\(620\) 12.0000 0.481932
\(621\) −1.00000 −0.0401286
\(622\) 6.00000 0.240578
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) −19.0000 −0.760000
\(626\) 34.0000 1.35891
\(627\) −24.0000 −0.958468
\(628\) −18.0000 −0.718278
\(629\) 16.0000 0.637962
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) −4.00000 −0.159111
\(633\) −8.00000 −0.317971
\(634\) 18.0000 0.714871
\(635\) −16.0000 −0.634941
\(636\) −4.00000 −0.158610
\(637\) 0 0
\(638\) 12.0000 0.475085
\(639\) −8.00000 −0.316475
\(640\) 2.00000 0.0790569
\(641\) 26.0000 1.02694 0.513469 − 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 6.00000 0.236801
\(643\) −28.0000 −1.10421 −0.552106 − 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 8.00000 0.314756
\(647\) 30.0000 1.17942 0.589711 − 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 48.0000 1.88416
\(650\) 6.00000 0.235339
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −2.00000 −0.0782660 −0.0391330 − 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) 12.0000 0.469237
\(655\) 0 0
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 10.0000 0.389545 0.194772 − 0.980848i \(-0.437603\pi\)
0.194772 + 0.980848i \(0.437603\pi\)
\(660\) −12.0000 −0.467099
\(661\) 10.0000 0.388955 0.194477 − 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 12.0000 0.466393
\(663\) −12.0000 −0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −8.00000 −0.309994
\(667\) 2.00000 0.0774403
\(668\) −6.00000 −0.232147
\(669\) 10.0000 0.386622
\(670\) 12.0000 0.463600
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 − 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 18.0000 0.693334
\(675\) 1.00000 0.0384900
\(676\) 23.0000 0.884615
\(677\) 38.0000 1.46046 0.730229 − 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 18.0000 0.691286
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 4.00000 0.153280
\(682\) −36.0000 −1.37851
\(683\) −48.0000 −1.83667 −0.918334 − 0.395805i \(-0.870466\pi\)
−0.918334 + 0.395805i \(0.870466\pi\)
\(684\) −4.00000 −0.152944
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 2.00000 0.0762493
\(689\) 24.0000 0.914327
\(690\) −2.00000 −0.0761387
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −10.0000 −0.380143
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 8.00000 0.303457
\(696\) 2.00000 0.0758098
\(697\) 0 0
\(698\) 10.0000 0.378506
\(699\) −10.0000 −0.378235
\(700\) 0 0
\(701\) −40.0000 −1.51078 −0.755390 − 0.655276i \(-0.772552\pi\)
−0.755390 + 0.655276i \(0.772552\pi\)
\(702\) 6.00000 0.226455
\(703\) −32.0000 −1.20690
\(704\) −6.00000 −0.226134
\(705\) 12.0000 0.451946
\(706\) −16.0000 −0.602168
\(707\) 0 0
\(708\) 8.00000 0.300658
\(709\) −8.00000 −0.300446 −0.150223 − 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) −16.0000 −0.600469
\(711\) 4.00000 0.150012
\(712\) 6.00000 0.224860
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 72.0000 2.69265
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) 0 0
\(719\) −42.0000 −1.56634 −0.783168 − 0.621810i \(-0.786397\pi\)
−0.783168 + 0.621810i \(0.786397\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −14.0000 −0.520666
\(724\) 2.00000 0.0743294
\(725\) −2.00000 −0.0742781
\(726\) 25.0000 0.927837
\(727\) 12.0000 0.445055 0.222528 − 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −8.00000 −0.296093
\(731\) 4.00000 0.147945
\(732\) −2.00000 −0.0739221
\(733\) 34.0000 1.25582 0.627909 − 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 20.0000 0.738213
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −36.0000 −1.32608
\(738\) 0 0
\(739\) −28.0000 −1.03000 −0.514998 − 0.857191i \(-0.672207\pi\)
−0.514998 + 0.857191i \(0.672207\pi\)
\(740\) −16.0000 −0.588172
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 − 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) −6.00000 −0.219971
\(745\) 24.0000 0.879292
\(746\) 8.00000 0.292901
\(747\) −12.0000 −0.439057
\(748\) −12.0000 −0.438763
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) −4.00000 −0.145962 −0.0729810 − 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 6.00000 0.218797
\(753\) −12.0000 −0.437304
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 44.0000 1.59921 0.799604 − 0.600528i \(-0.205043\pi\)
0.799604 + 0.600528i \(0.205043\pi\)
\(758\) −26.0000 −0.944363
\(759\) 6.00000 0.217786
\(760\) −8.00000 −0.290191
\(761\) 36.0000 1.30500 0.652499 − 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 8.00000 0.289809
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) −48.0000 −1.73318
\(768\) −1.00000 −0.0360844
\(769\) 30.0000 1.08183 0.540914 − 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 14.0000 0.503871
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −2.00000 −0.0718885
\(775\) 6.00000 0.215526
\(776\) −10.0000 −0.358979
\(777\) 0 0
\(778\) 24.0000 0.860442
\(779\) 0 0
\(780\) 12.0000 0.429669
\(781\) 48.0000 1.71758
\(782\) −2.00000 −0.0715199
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 36.0000 1.28490
\(786\) 0 0
\(787\) −44.0000 −1.56843 −0.784215 − 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) −18.0000 −0.641223
\(789\) −16.0000 −0.569615
\(790\) 8.00000 0.284627
\(791\) 0 0
\(792\) 6.00000 0.213201
\(793\) 12.0000 0.426132
\(794\) 26.0000 0.922705
\(795\) 8.00000 0.283731
\(796\) −20.0000 −0.708881
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) −2.00000 −0.0706225
\(803\) 24.0000 0.846942
\(804\) −6.00000 −0.211604
\(805\) 0 0
\(806\) 36.0000 1.26805
\(807\) −6.00000 −0.211210
\(808\) 6.00000 0.211079
\(809\) −26.0000 −0.914111 −0.457056 − 0.889438i \(-0.651096\pi\)
−0.457056 + 0.889438i \(0.651096\pi\)
\(810\) 2.00000 0.0702728
\(811\) −40.0000 −1.40459 −0.702295 − 0.711886i \(-0.747841\pi\)
−0.702295 + 0.711886i \(0.747841\pi\)
\(812\) 0 0
\(813\) 26.0000 0.911860
\(814\) 48.0000 1.68240
\(815\) 8.00000 0.280228
\(816\) −2.00000 −0.0700140
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 18.0000 0.627822
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −8.00000 −0.278693
\(825\) −6.00000 −0.208893
\(826\) 0 0
\(827\) −2.00000 −0.0695468 −0.0347734 − 0.999395i \(-0.511071\pi\)
−0.0347734 + 0.999395i \(0.511071\pi\)
\(828\) 1.00000 0.0347524
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) −24.0000 −0.833052
\(831\) −2.00000 −0.0693792
\(832\) 6.00000 0.208013
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) 12.0000 0.415277
\(836\) 24.0000 0.830057
\(837\) 6.00000 0.207390
\(838\) −20.0000 −0.690889
\(839\) 32.0000 1.10476 0.552381 − 0.833592i \(-0.313719\pi\)
0.552381 + 0.833592i \(0.313719\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 24.0000 0.827095
\(843\) 18.0000 0.619953
\(844\) 8.00000 0.275371
\(845\) −46.0000 −1.58245
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) 4.00000 0.137361
\(849\) −28.0000 −0.960958
\(850\) 2.00000 0.0685994
\(851\) 8.00000 0.274236
\(852\) 8.00000 0.274075
\(853\) −42.0000 −1.43805 −0.719026 − 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) 8.00000 0.273594
\(856\) −6.00000 −0.205076
\(857\) −16.0000 −0.546550 −0.273275 − 0.961936i \(-0.588107\pi\)
−0.273275 + 0.961936i \(0.588107\pi\)
\(858\) −36.0000 −1.22902
\(859\) −48.0000 −1.63774 −0.818869 − 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −28.0000 −0.953684
\(863\) −24.0000 −0.816970 −0.408485 − 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 20.0000 0.680020
\(866\) 38.0000 1.29129
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) −4.00000 −0.135613
\(871\) 36.0000 1.21981
\(872\) −12.0000 −0.406371
\(873\) 10.0000 0.338449
\(874\) 4.00000 0.135302
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) −2.00000 −0.0675352 −0.0337676 − 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 14.0000 0.472477
\(879\) 6.00000 0.202375
\(880\) 12.0000 0.404520
\(881\) −42.0000 −1.41502 −0.707508 − 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 − 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 12.0000 0.403604
\(885\) −16.0000 −0.537834
\(886\) −36.0000 −1.20944
\(887\) −38.0000 −1.27592 −0.637958 − 0.770072i \(-0.720220\pi\)
−0.637958 + 0.770072i \(0.720220\pi\)
\(888\) 8.00000 0.268462
\(889\) 0 0
\(890\) −12.0000 −0.402241
\(891\) −6.00000 −0.201008
\(892\) −10.0000 −0.334825
\(893\) −24.0000 −0.803129
\(894\) −12.0000 −0.401340
\(895\) 48.0000 1.60446
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 38.0000 1.26808
\(899\) −12.0000 −0.400222
\(900\) −1.00000 −0.0333333
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) −18.0000 −0.598671
\(905\) −4.00000 −0.132964
\(906\) 0 0
\(907\) 38.0000 1.26177 0.630885 − 0.775877i \(-0.282692\pi\)
0.630885 + 0.775877i \(0.282692\pi\)
\(908\) −4.00000 −0.132745
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 4.00000 0.132453
\(913\) 72.0000 2.38285
\(914\) 34.0000 1.12462
\(915\) 4.00000 0.132236
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 2.00000 0.0659380
\(921\) −32.0000 −1.05444
\(922\) −30.0000 −0.987997
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) −8.00000 −0.263038
\(926\) −40.0000 −1.31448
\(927\) 8.00000 0.262754
\(928\) −2.00000 −0.0656532
\(929\) 16.0000 0.524943 0.262471 − 0.964940i \(-0.415462\pi\)
0.262471 + 0.964940i \(0.415462\pi\)
\(930\) 12.0000 0.393496
\(931\) 0 0
\(932\) 10.0000 0.327561
\(933\) 6.00000 0.196431
\(934\) 12.0000 0.392652
\(935\) 24.0000 0.784884
\(936\) −6.00000 −0.196116
\(937\) −46.0000 −1.50275 −0.751377 − 0.659873i \(-0.770610\pi\)
−0.751377 + 0.659873i \(0.770610\pi\)
\(938\) 0 0
\(939\) 34.0000 1.10955
\(940\) −12.0000 −0.391397
\(941\) 54.0000 1.76035 0.880175 − 0.474650i \(-0.157425\pi\)
0.880175 + 0.474650i \(0.157425\pi\)
\(942\) −18.0000 −0.586472
\(943\) 0 0
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) −4.00000 −0.129983 −0.0649913 − 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) −4.00000 −0.129914
\(949\) −24.0000 −0.779073
\(950\) −4.00000 −0.129777
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 − 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) −4.00000 −0.129505
\(955\) 32.0000 1.03550
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 20.0000 0.646171
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 5.00000 0.161290
\(962\) −48.0000 −1.54758
\(963\) 6.00000 0.193347
\(964\) 14.0000 0.450910
\(965\) −28.0000 −0.901352
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 − 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −25.0000 −0.803530
\(969\) 8.00000 0.256997
\(970\) 20.0000 0.642161
\(971\) −20.0000 −0.641831 −0.320915 − 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 0 0
\(975\) 6.00000 0.192154
\(976\) 2.00000 0.0640184
\(977\) 42.0000 1.34370 0.671850 − 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) −4.00000 −0.127906
\(979\) 36.0000 1.15056
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) 28.0000 0.893516
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 36.0000 1.14706
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) −24.0000 −0.763542
\(989\) 2.00000 0.0635963
\(990\) −12.0000 −0.381385
\(991\) −32.0000 −1.01651 −0.508257 − 0.861206i \(-0.669710\pi\)
−0.508257 + 0.861206i \(0.669710\pi\)
\(992\) 6.00000 0.190500
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) 40.0000 1.26809
\(996\) 12.0000 0.380235
\(997\) −46.0000 −1.45683 −0.728417 − 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −28.0000 −0.886325
\(999\) −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 6762.2.a.c.1.1 ✓ 1
7.6 odd 2 6762.2.a.p.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6762.2.a.c.1.1 ✓ 1 1.1 even 1 trivial
6762.2.a.p.1.1 yes 1 7.6 odd 2