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Properties

Label 6762.2.a.bk.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
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] = [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: \(0\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} +1.00000 q^{16} +5.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} +1.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -5.00000 q^{25} -6.00000 q^{26} +1.00000 q^{27} +9.00000 q^{29} -6.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +5.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} +6.00000 q^{38} -6.00000 q^{39} -6.00000 q^{41} -6.00000 q^{43} +1.00000 q^{44} +1.00000 q^{46} +11.0000 q^{47} +1.00000 q^{48} -5.00000 q^{50} +5.00000 q^{51} -6.00000 q^{52} +14.0000 q^{53} +1.00000 q^{54} +6.00000 q^{57} +9.00000 q^{58} -2.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} +10.0000 q^{67} +5.00000 q^{68} +1.00000 q^{69} -3.00000 q^{71} +1.00000 q^{72} -5.00000 q^{73} +6.00000 q^{74} -5.00000 q^{75} +6.00000 q^{76} -6.00000 q^{78} +5.00000 q^{79} +1.00000 q^{81} -6.00000 q^{82} -8.00000 q^{83} -6.00000 q^{86} +9.00000 q^{87} +1.00000 q^{88} +10.0000 q^{89} +1.00000 q^{92} -6.00000 q^{93} +11.0000 q^{94} +1.00000 q^{96} -4.00000 q^{97} +1.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 1.00000 0.288675
\(13\) −6.00000 −1.66410 −0.832050 − 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.00000 1.21268 0.606339 − 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) −5.00000 −1.00000
\(26\) −6.00000 −1.17670
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 6.00000 0.973329
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 − 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 11.0000 1.60451 0.802257 − 0.596978i \(-0.203632\pi\)
0.802257 + 0.596978i \(0.203632\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −5.00000 −0.707107
\(51\) 5.00000 0.700140
\(52\) −6.00000 −0.832050
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) 9.00000 1.18176
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 10.0000 1.22169 0.610847 − 0.791748i \(-0.290829\pi\)
0.610847 + 0.791748i \(0.290829\pi\)
\(68\) 5.00000 0.606339
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 1.00000 0.117851
\(73\) −5.00000 −0.585206 −0.292603 − 0.956234i \(-0.594521\pi\)
−0.292603 + 0.956234i \(0.594521\pi\)
\(74\) 6.00000 0.697486
\(75\) −5.00000 −0.577350
\(76\) 6.00000 0.688247
\(77\) 0 0
\(78\) −6.00000 −0.679366
\(79\) 5.00000 0.562544 0.281272 − 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −6.00000 −0.662589
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 9.00000 0.964901
\(88\) 1.00000 0.106600
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 1.00000 0.104257
\(93\) −6.00000 −0.622171
\(94\) 11.0000 1.13456
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −5.00000 −0.500000
\(101\) 7.00000 0.696526 0.348263 − 0.937397i \(-0.386772\pi\)
0.348263 + 0.937397i \(0.386772\pi\)
\(102\) 5.00000 0.495074
\(103\) 11.0000 1.08386 0.541931 − 0.840423i \(-0.317693\pi\)
0.541931 + 0.840423i \(0.317693\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) −4.00000 −0.386695 −0.193347 − 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.00000 −0.287348 −0.143674 − 0.989625i \(-0.545892\pi\)
−0.143674 + 0.989625i \(0.545892\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 18.0000 1.69330 0.846649 − 0.532152i \(-0.178617\pi\)
0.846649 + 0.532152i \(0.178617\pi\)
\(114\) 6.00000 0.561951
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −2.00000 −0.181071
\(123\) −6.00000 −0.541002
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) −14.0000 −1.24230 −0.621150 − 0.783692i \(-0.713334\pi\)
−0.621150 + 0.783692i \(0.713334\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 10.0000 0.863868
\(135\) 0 0
\(136\) 5.00000 0.428746
\(137\) 3.00000 0.256307 0.128154 − 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 1.00000 0.0851257
\(139\) 15.0000 1.27228 0.636142 − 0.771572i \(-0.280529\pi\)
0.636142 + 0.771572i \(0.280529\pi\)
\(140\) 0 0
\(141\) 11.0000 0.926367
\(142\) −3.00000 −0.251754
\(143\) −6.00000 −0.501745
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −5.00000 −0.413803
\(147\) 0 0
\(148\) 6.00000 0.493197
\(149\) −8.00000 −0.655386 −0.327693 − 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) −5.00000 −0.408248
\(151\) 12.0000 0.976546 0.488273 − 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 6.00000 0.486664
\(153\) 5.00000 0.404226
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) 13.0000 1.03751 0.518756 − 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) 5.00000 0.397779
\(159\) 14.0000 1.11027
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 19.0000 1.48819 0.744097 − 0.668071i \(-0.232880\pi\)
0.744097 + 0.668071i \(0.232880\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −6.00000 −0.457496
\(173\) −21.0000 −1.59660 −0.798300 − 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) 9.00000 0.682288
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 22.0000 1.64436 0.822179 − 0.569230i \(-0.192758\pi\)
0.822179 + 0.569230i \(0.192758\pi\)
\(180\) 0 0
\(181\) −23.0000 −1.70958 −0.854788 − 0.518977i \(-0.826313\pi\)
−0.854788 + 0.518977i \(0.826313\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 5.00000 0.365636
\(188\) 11.0000 0.802257
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 − 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 1.00000 0.0721688
\(193\) −26.0000 −1.87152 −0.935760 − 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) 0 0
\(197\) 13.0000 0.926212 0.463106 − 0.886303i \(-0.346735\pi\)
0.463106 + 0.886303i \(0.346735\pi\)
\(198\) 1.00000 0.0710669
\(199\) 7.00000 0.496217 0.248108 − 0.968732i \(-0.420191\pi\)
0.248108 + 0.968732i \(0.420191\pi\)
\(200\) −5.00000 −0.353553
\(201\) 10.0000 0.705346
\(202\) 7.00000 0.492518
\(203\) 0 0
\(204\) 5.00000 0.350070
\(205\) 0 0
\(206\) 11.0000 0.766406
\(207\) 1.00000 0.0695048
\(208\) −6.00000 −0.416025
\(209\) 6.00000 0.415029
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 − 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 14.0000 0.961524
\(213\) −3.00000 −0.205557
\(214\) −4.00000 −0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −3.00000 −0.203186
\(219\) −5.00000 −0.337869
\(220\) 0 0
\(221\) −30.0000 −2.01802
\(222\) 6.00000 0.402694
\(223\) 10.0000 0.669650 0.334825 − 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 18.0000 1.19734
\(227\) 17.0000 1.12833 0.564165 − 0.825662i \(-0.309198\pi\)
0.564165 + 0.825662i \(0.309198\pi\)
\(228\) 6.00000 0.397360
\(229\) 21.0000 1.38772 0.693860 − 0.720110i \(-0.255909\pi\)
0.693860 + 0.720110i \(0.255909\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 0 0
\(237\) 5.00000 0.324785
\(238\) 0 0
\(239\) 9.00000 0.582162 0.291081 − 0.956698i \(-0.405985\pi\)
0.291081 + 0.956698i \(0.405985\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 − 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) −10.0000 −0.642824
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) −36.0000 −2.29063
\(248\) −6.00000 −0.381000
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −17.0000 −1.07303 −0.536515 − 0.843891i \(-0.680260\pi\)
−0.536515 + 0.843891i \(0.680260\pi\)
\(252\) 0 0
\(253\) 1.00000 0.0628695
\(254\) −14.0000 −0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.0000 −0.873296 −0.436648 − 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) −6.00000 −0.373544
\(259\) 0 0
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) −12.0000 −0.741362
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 1.00000 0.0615457
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 10.0000 0.610847
\(269\) −3.00000 −0.182913 −0.0914566 − 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 − 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 5.00000 0.303170
\(273\) 0 0
\(274\) 3.00000 0.181237
\(275\) −5.00000 −0.301511
\(276\) 1.00000 0.0601929
\(277\) 32.0000 1.92269 0.961347 − 0.275340i \(-0.0887905\pi\)
0.961347 + 0.275340i \(0.0887905\pi\)
\(278\) 15.0000 0.899640
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 5.00000 0.298275 0.149137 − 0.988816i \(-0.452350\pi\)
0.149137 + 0.988816i \(0.452350\pi\)
\(282\) 11.0000 0.655040
\(283\) −12.0000 −0.713326 −0.356663 − 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) −3.00000 −0.178017
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −4.00000 −0.234484
\(292\) −5.00000 −0.292603
\(293\) −12.0000 −0.701047 −0.350524 − 0.936554i \(-0.613996\pi\)
−0.350524 + 0.936554i \(0.613996\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) 1.00000 0.0580259
\(298\) −8.00000 −0.463428
\(299\) −6.00000 −0.346989
\(300\) −5.00000 −0.288675
\(301\) 0 0
\(302\) 12.0000 0.690522
\(303\) 7.00000 0.402139
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) 5.00000 0.285831
\(307\) −33.0000 −1.88341 −0.941705 − 0.336440i \(-0.890777\pi\)
−0.941705 + 0.336440i \(0.890777\pi\)
\(308\) 0 0
\(309\) 11.0000 0.625768
\(310\) 0 0
\(311\) −13.0000 −0.737162 −0.368581 − 0.929596i \(-0.620156\pi\)
−0.368581 + 0.929596i \(0.620156\pi\)
\(312\) −6.00000 −0.339683
\(313\) −20.0000 −1.13047 −0.565233 − 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(314\) 13.0000 0.733632
\(315\) 0 0
\(316\) 5.00000 0.281272
\(317\) 2.00000 0.112331 0.0561656 − 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 14.0000 0.785081
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 30.0000 1.66924
\(324\) 1.00000 0.0555556
\(325\) 30.0000 1.66410
\(326\) 19.0000 1.05231
\(327\) −3.00000 −0.165900
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 − 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) −8.00000 −0.439057
\(333\) 6.00000 0.328798
\(334\) −8.00000 −0.437741
\(335\) 0 0
\(336\) 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
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −21.0000 −1.12897
\(347\) −22.0000 −1.18102 −0.590511 − 0.807030i \(-0.701074\pi\)
−0.590511 + 0.807030i \(0.701074\pi\)
\(348\) 9.00000 0.482451
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 1.00000 0.0533002
\(353\) −14.0000 −0.745145 −0.372572 − 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 22.0000 1.16274
\(359\) −16.0000 −0.844448 −0.422224 − 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −23.0000 −1.20885
\(363\) −10.0000 −0.524864
\(364\) 0 0
\(365\) 0 0
\(366\) −2.00000 −0.104542
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 1.00000 0.0521286
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) −6.00000 −0.311086
\(373\) −23.0000 −1.19089 −0.595447 − 0.803394i \(-0.703025\pi\)
−0.595447 + 0.803394i \(0.703025\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) 11.0000 0.567282
\(377\) −54.0000 −2.78114
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 − 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −14.0000 −0.717242
\(382\) 12.0000 0.613973
\(383\) −18.0000 −0.919757 −0.459879 − 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −26.0000 −1.32337
\(387\) −6.00000 −0.304997
\(388\) −4.00000 −0.203069
\(389\) −12.0000 −0.608424 −0.304212 − 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 13.0000 0.654931
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) −38.0000 −1.90717 −0.953583 − 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −31.0000 −1.54807 −0.774033 − 0.633145i \(-0.781764\pi\)
−0.774033 + 0.633145i \(0.781764\pi\)
\(402\) 10.0000 0.498755
\(403\) 36.0000 1.79329
\(404\) 7.00000 0.348263
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 0.297409
\(408\) 5.00000 0.247537
\(409\) 13.0000 0.642809 0.321404 − 0.946942i \(-0.395845\pi\)
0.321404 + 0.946942i \(0.395845\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 11.0000 0.541931
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 15.0000 0.734553
\(418\) 6.00000 0.293470
\(419\) −25.0000 −1.22133 −0.610665 − 0.791889i \(-0.709098\pi\)
−0.610665 + 0.791889i \(0.709098\pi\)
\(420\) 0 0
\(421\) −21.0000 −1.02348 −0.511739 − 0.859141i \(-0.670998\pi\)
−0.511739 + 0.859141i \(0.670998\pi\)
\(422\) 9.00000 0.438113
\(423\) 11.0000 0.534838
\(424\) 14.0000 0.679900
\(425\) −25.0000 −1.21268
\(426\) −3.00000 −0.145350
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) −6.00000 −0.289683
\(430\) 0 0
\(431\) 6.00000 0.289010 0.144505 − 0.989504i \(-0.453841\pi\)
0.144505 + 0.989504i \(0.453841\pi\)
\(432\) 1.00000 0.0481125
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.00000 −0.143674
\(437\) 6.00000 0.287019
\(438\) −5.00000 −0.238909
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −30.0000 −1.42695
\(443\) −20.0000 −0.950229 −0.475114 − 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 10.0000 0.473514
\(447\) −8.00000 −0.378387
\(448\) 0 0
\(449\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) −5.00000 −0.235702
\(451\) −6.00000 −0.282529
\(452\) 18.0000 0.846649
\(453\) 12.0000 0.563809
\(454\) 17.0000 0.797850
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) 12.0000 0.561336 0.280668 − 0.959805i \(-0.409444\pi\)
0.280668 + 0.959805i \(0.409444\pi\)
\(458\) 21.0000 0.981266
\(459\) 5.00000 0.233380
\(460\) 0 0
\(461\) 14.0000 0.652045 0.326023 − 0.945362i \(-0.394291\pi\)
0.326023 + 0.945362i \(0.394291\pi\)
\(462\) 0 0
\(463\) −20.0000 −0.929479 −0.464739 − 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) 31.0000 1.43451 0.717254 − 0.696811i \(-0.245399\pi\)
0.717254 + 0.696811i \(0.245399\pi\)
\(468\) −6.00000 −0.277350
\(469\) 0 0
\(470\) 0 0
\(471\) 13.0000 0.599008
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 5.00000 0.229658
\(475\) −30.0000 −1.37649
\(476\) 0 0
\(477\) 14.0000 0.641016
\(478\) 9.00000 0.411650
\(479\) 30.0000 1.37073 0.685367 − 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) −14.0000 −0.637683
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −14.0000 −0.634401 −0.317200 − 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 19.0000 0.859210
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 − 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) −6.00000 −0.270501
\(493\) 45.0000 2.02670
\(494\) −36.0000 −1.61972
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) 0 0
\(498\) −8.00000 −0.358489
\(499\) 7.00000 0.313363 0.156682 − 0.987649i \(-0.449920\pi\)
0.156682 + 0.987649i \(0.449920\pi\)
\(500\) 0 0
\(501\) −8.00000 −0.357414
\(502\) −17.0000 −0.758747
\(503\) 18.0000 0.802580 0.401290 − 0.915951i \(-0.368562\pi\)
0.401290 + 0.915951i \(0.368562\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.00000 0.0444554
\(507\) 23.0000 1.02147
\(508\) −14.0000 −0.621150
\(509\) 41.0000 1.81729 0.908647 − 0.417566i \(-0.137117\pi\)
0.908647 + 0.417566i \(0.137117\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 6.00000 0.264906
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 11.0000 0.483779
\(518\) 0 0
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 9.00000 0.393919
\(523\) 18.0000 0.787085 0.393543 − 0.919306i \(-0.371249\pi\)
0.393543 + 0.919306i \(0.371249\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −30.0000 −1.30682
\(528\) 1.00000 0.0435194
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) 10.0000 0.431934
\(537\) 22.0000 0.949370
\(538\) −3.00000 −0.129339
\(539\) 0 0
\(540\) 0 0
\(541\) 8.00000 0.343947 0.171973 − 0.985102i \(-0.444986\pi\)
0.171973 + 0.985102i \(0.444986\pi\)
\(542\) −22.0000 −0.944981
\(543\) −23.0000 −0.987024
\(544\) 5.00000 0.214373
\(545\) 0 0
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 − 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) 3.00000 0.128154
\(549\) −2.00000 −0.0853579
\(550\) −5.00000 −0.213201
\(551\) 54.0000 2.30048
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 32.0000 1.35955
\(555\) 0 0
\(556\) 15.0000 0.636142
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) −6.00000 −0.254000
\(559\) 36.0000 1.52264
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) 5.00000 0.210912
\(563\) −21.0000 −0.885044 −0.442522 − 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 11.0000 0.463184
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) −3.00000 −0.125877
\(569\) 2.00000 0.0838444 0.0419222 − 0.999121i \(-0.486652\pi\)
0.0419222 + 0.999121i \(0.486652\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 − 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) −6.00000 −0.250873
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −5.00000 −0.208514
\(576\) 1.00000 0.0416667
\(577\) 33.0000 1.37381 0.686904 − 0.726748i \(-0.258969\pi\)
0.686904 + 0.726748i \(0.258969\pi\)
\(578\) 8.00000 0.332756
\(579\) −26.0000 −1.08052
\(580\) 0 0
\(581\) 0 0
\(582\) −4.00000 −0.165805
\(583\) 14.0000 0.579821
\(584\) −5.00000 −0.206901
\(585\) 0 0
\(586\) −12.0000 −0.495715
\(587\) −24.0000 −0.990586 −0.495293 − 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 0 0
\(589\) −36.0000 −1.48335
\(590\) 0 0
\(591\) 13.0000 0.534749
\(592\) 6.00000 0.246598
\(593\) −22.0000 −0.903432 −0.451716 − 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −8.00000 −0.327693
\(597\) 7.00000 0.286491
\(598\) −6.00000 −0.245358
\(599\) −7.00000 −0.286012 −0.143006 − 0.989722i \(-0.545677\pi\)
−0.143006 + 0.989722i \(0.545677\pi\)
\(600\) −5.00000 −0.204124
\(601\) 2.00000 0.0815817 0.0407909 − 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 10.0000 0.407231
\(604\) 12.0000 0.488273
\(605\) 0 0
\(606\) 7.00000 0.284356
\(607\) 6.00000 0.243532 0.121766 − 0.992559i \(-0.461144\pi\)
0.121766 + 0.992559i \(0.461144\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) 0 0
\(611\) −66.0000 −2.67007
\(612\) 5.00000 0.202113
\(613\) 7.00000 0.282727 0.141364 − 0.989958i \(-0.454851\pi\)
0.141364 + 0.989958i \(0.454851\pi\)
\(614\) −33.0000 −1.33177
\(615\) 0 0
\(616\) 0 0
\(617\) 33.0000 1.32853 0.664265 − 0.747497i \(-0.268745\pi\)
0.664265 + 0.747497i \(0.268745\pi\)
\(618\) 11.0000 0.442485
\(619\) 8.00000 0.321547 0.160774 − 0.986991i \(-0.448601\pi\)
0.160774 + 0.986991i \(0.448601\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −13.0000 −0.521253
\(623\) 0 0
\(624\) −6.00000 −0.240192
\(625\) 25.0000 1.00000
\(626\) −20.0000 −0.799361
\(627\) 6.00000 0.239617
\(628\) 13.0000 0.518756
\(629\) 30.0000 1.19618
\(630\) 0 0
\(631\) −41.0000 −1.63218 −0.816092 − 0.577922i \(-0.803864\pi\)
−0.816092 + 0.577922i \(0.803864\pi\)
\(632\) 5.00000 0.198889
\(633\) 9.00000 0.357718
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) 14.0000 0.555136
\(637\) 0 0
\(638\) 9.00000 0.356313
\(639\) −3.00000 −0.118678
\(640\) 0 0
\(641\) 21.0000 0.829450 0.414725 − 0.909947i \(-0.363878\pi\)
0.414725 + 0.909947i \(0.363878\pi\)
\(642\) −4.00000 −0.157867
\(643\) 4.00000 0.157745 0.0788723 − 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 30.0000 1.18033
\(647\) 15.0000 0.589711 0.294855 − 0.955542i \(-0.404729\pi\)
0.294855 + 0.955542i \(0.404729\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 30.0000 1.17670
\(651\) 0 0
\(652\) 19.0000 0.744097
\(653\) 49.0000 1.91752 0.958759 − 0.284220i \(-0.0917346\pi\)
0.958759 + 0.284220i \(0.0917346\pi\)
\(654\) −3.00000 −0.117309
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) −5.00000 −0.195069
\(658\) 0 0
\(659\) −33.0000 −1.28550 −0.642749 − 0.766077i \(-0.722206\pi\)
−0.642749 + 0.766077i \(0.722206\pi\)
\(660\) 0 0
\(661\) 7.00000 0.272268 0.136134 − 0.990690i \(-0.456532\pi\)
0.136134 + 0.990690i \(0.456532\pi\)
\(662\) 32.0000 1.24372
\(663\) −30.0000 −1.16510
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 9.00000 0.348481
\(668\) −8.00000 −0.309529
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −35.0000 −1.34915 −0.674575 − 0.738206i \(-0.735673\pi\)
−0.674575 + 0.738206i \(0.735673\pi\)
\(674\) −18.0000 −0.693334
\(675\) −5.00000 −0.192450
\(676\) 23.0000 0.884615
\(677\) −14.0000 −0.538064 −0.269032 − 0.963131i \(-0.586704\pi\)
−0.269032 + 0.963131i \(0.586704\pi\)
\(678\) 18.0000 0.691286
\(679\) 0 0
\(680\) 0 0
\(681\) 17.0000 0.651441
\(682\) −6.00000 −0.229752
\(683\) −40.0000 −1.53056 −0.765279 − 0.643699i \(-0.777399\pi\)
−0.765279 + 0.643699i \(0.777399\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) 0 0
\(687\) 21.0000 0.801200
\(688\) −6.00000 −0.228748
\(689\) −84.0000 −3.20015
\(690\) 0 0
\(691\) 13.0000 0.494543 0.247272 − 0.968946i \(-0.420466\pi\)
0.247272 + 0.968946i \(0.420466\pi\)
\(692\) −21.0000 −0.798300
\(693\) 0 0
\(694\) −22.0000 −0.835109
\(695\) 0 0
\(696\) 9.00000 0.341144
\(697\) −30.0000 −1.13633
\(698\) 10.0000 0.378506
\(699\) 8.00000 0.302588
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 − 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) −6.00000 −0.226455
\(703\) 36.0000 1.35777
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 0 0
\(708\) 0 0
\(709\) −37.0000 −1.38956 −0.694782 − 0.719220i \(-0.744499\pi\)
−0.694782 + 0.719220i \(0.744499\pi\)
\(710\) 0 0
\(711\) 5.00000 0.187515
\(712\) 10.0000 0.374766
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 22.0000 0.822179
\(717\) 9.00000 0.336111
\(718\) −16.0000 −0.597115
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 17.0000 0.632674
\(723\) −14.0000 −0.520666
\(724\) −23.0000 −0.854788
\(725\) −45.0000 −1.67126
\(726\) −10.0000 −0.371135
\(727\) 25.0000 0.927199 0.463599 − 0.886045i \(-0.346558\pi\)
0.463599 + 0.886045i \(0.346558\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.0000 −1.10959
\(732\) −2.00000 −0.0739221
\(733\) −13.0000 −0.480166 −0.240083 − 0.970752i \(-0.577175\pi\)
−0.240083 + 0.970752i \(0.577175\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 10.0000 0.368355
\(738\) −6.00000 −0.220863
\(739\) −37.0000 −1.36107 −0.680534 − 0.732717i \(-0.738252\pi\)
−0.680534 + 0.732717i \(0.738252\pi\)
\(740\) 0 0
\(741\) −36.0000 −1.32249
\(742\) 0 0
\(743\) −20.0000 −0.733729 −0.366864 − 0.930274i \(-0.619569\pi\)
−0.366864 + 0.930274i \(0.619569\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) −23.0000 −0.842090
\(747\) −8.00000 −0.292705
\(748\) 5.00000 0.182818
\(749\) 0 0
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 − 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 11.0000 0.401129
\(753\) −17.0000 −0.619514
\(754\) −54.0000 −1.96656
\(755\) 0 0
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 − 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 20.0000 0.726433
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 − 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) −14.0000 −0.507166
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) 0 0
\(766\) −18.0000 −0.650366
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 26.0000 0.937584 0.468792 − 0.883309i \(-0.344689\pi\)
0.468792 + 0.883309i \(0.344689\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) −26.0000 −0.935760
\(773\) 40.0000 1.43870 0.719350 − 0.694648i \(-0.244440\pi\)
0.719350 + 0.694648i \(0.244440\pi\)
\(774\) −6.00000 −0.215666
\(775\) 30.0000 1.07763
\(776\) −4.00000 −0.143592
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) −3.00000 −0.107348
\(782\) 5.00000 0.178800
\(783\) 9.00000 0.321634
\(784\) 0 0
\(785\) 0 0
\(786\) −12.0000 −0.428026
\(787\) −28.0000 −0.998092 −0.499046 − 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) 13.0000 0.463106
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 12.0000 0.426132
\(794\) −38.0000 −1.34857
\(795\) 0 0
\(796\) 7.00000 0.248108
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 55.0000 1.94576
\(800\) −5.00000 −0.176777
\(801\) 10.0000 0.353333
\(802\) −31.0000 −1.09465
\(803\) −5.00000 −0.176446
\(804\) 10.0000 0.352673
\(805\) 0 0
\(806\) 36.0000 1.26805
\(807\) −3.00000 −0.105605
\(808\) 7.00000 0.246259
\(809\) −34.0000 −1.19538 −0.597688 − 0.801729i \(-0.703914\pi\)
−0.597688 + 0.801729i \(0.703914\pi\)
\(810\) 0 0
\(811\) −27.0000 −0.948098 −0.474049 − 0.880498i \(-0.657208\pi\)
−0.474049 + 0.880498i \(0.657208\pi\)
\(812\) 0 0
\(813\) −22.0000 −0.771574
\(814\) 6.00000 0.210300
\(815\) 0 0
\(816\) 5.00000 0.175035
\(817\) −36.0000 −1.25948
\(818\) 13.0000 0.454534
\(819\) 0 0
\(820\) 0 0
\(821\) 3.00000 0.104701 0.0523504 − 0.998629i \(-0.483329\pi\)
0.0523504 + 0.998629i \(0.483329\pi\)
\(822\) 3.00000 0.104637
\(823\) −38.0000 −1.32460 −0.662298 − 0.749240i \(-0.730419\pi\)
−0.662298 + 0.749240i \(0.730419\pi\)
\(824\) 11.0000 0.383203
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) −9.00000 −0.312961 −0.156480 − 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) 1.00000 0.0347524
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 32.0000 1.11007
\(832\) −6.00000 −0.208013
\(833\) 0 0
\(834\) 15.0000 0.519408
\(835\) 0 0
\(836\) 6.00000 0.207514
\(837\) −6.00000 −0.207390
\(838\) −25.0000 −0.863611
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) −21.0000 −0.723708
\(843\) 5.00000 0.172209
\(844\) 9.00000 0.309793
\(845\) 0 0
\(846\) 11.0000 0.378188
\(847\) 0 0
\(848\) 14.0000 0.480762
\(849\) −12.0000 −0.411839
\(850\) −25.0000 −0.857493
\(851\) 6.00000 0.205677
\(852\) −3.00000 −0.102778
\(853\) −14.0000 −0.479351 −0.239675 − 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) −6.00000 −0.204837
\(859\) −17.0000 −0.580033 −0.290016 − 0.957022i \(-0.593661\pi\)
−0.290016 + 0.957022i \(0.593661\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6.00000 0.204361
\(863\) 37.0000 1.25949 0.629747 − 0.776800i \(-0.283158\pi\)
0.629747 + 0.776800i \(0.283158\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 2.00000 0.0679628
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 5.00000 0.169613
\(870\) 0 0
\(871\) −60.0000 −2.03302
\(872\) −3.00000 −0.101593
\(873\) −4.00000 −0.135379
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) −5.00000 −0.168934
\(877\) −24.0000 −0.810422 −0.405211 − 0.914223i \(-0.632802\pi\)
−0.405211 + 0.914223i \(0.632802\pi\)
\(878\) 20.0000 0.674967
\(879\) −12.0000 −0.404750
\(880\) 0 0
\(881\) −39.0000 −1.31394 −0.656972 − 0.753915i \(-0.728163\pi\)
−0.656972 + 0.753915i \(0.728163\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 − 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) −3.00000 −0.100730 −0.0503651 − 0.998731i \(-0.516038\pi\)
−0.0503651 + 0.998731i \(0.516038\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 10.0000 0.334825
\(893\) 66.0000 2.20861
\(894\) −8.00000 −0.267560
\(895\) 0 0
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) 0 0
\(899\) −54.0000 −1.80100
\(900\) −5.00000 −0.166667
\(901\) 70.0000 2.33204
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 12.0000 0.398673
\(907\) 4.00000 0.132818 0.0664089 − 0.997792i \(-0.478846\pi\)
0.0664089 + 0.997792i \(0.478846\pi\)
\(908\) 17.0000 0.564165
\(909\) 7.00000 0.232175
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 6.00000 0.198680
\(913\) −8.00000 −0.264761
\(914\) 12.0000 0.396925
\(915\) 0 0
\(916\) 21.0000 0.693860
\(917\) 0 0
\(918\) 5.00000 0.165025
\(919\) 51.0000 1.68233 0.841167 − 0.540775i \(-0.181869\pi\)
0.841167 + 0.540775i \(0.181869\pi\)
\(920\) 0 0
\(921\) −33.0000 −1.08739
\(922\) 14.0000 0.461065
\(923\) 18.0000 0.592477
\(924\) 0 0
\(925\) −30.0000 −0.986394
\(926\) −20.0000 −0.657241
\(927\) 11.0000 0.361287
\(928\) 9.00000 0.295439
\(929\) −6.00000 −0.196854 −0.0984268 − 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 8.00000 0.262049
\(933\) −13.0000 −0.425601
\(934\) 31.0000 1.01435
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 34.0000 1.11073 0.555366 − 0.831606i \(-0.312578\pi\)
0.555366 + 0.831606i \(0.312578\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 13.0000 0.423563
\(943\) −6.00000 −0.195387
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) −10.0000 −0.324956 −0.162478 − 0.986712i \(-0.551949\pi\)
−0.162478 + 0.986712i \(0.551949\pi\)
\(948\) 5.00000 0.162392
\(949\) 30.0000 0.973841
\(950\) −30.0000 −0.973329
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 1.00000 0.0323932 0.0161966 − 0.999869i \(-0.494844\pi\)
0.0161966 + 0.999869i \(0.494844\pi\)
\(954\) 14.0000 0.453267
\(955\) 0 0
\(956\) 9.00000 0.291081
\(957\) 9.00000 0.290929
\(958\) 30.0000 0.969256
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) −36.0000 −1.16069
\(963\) −4.00000 −0.128898
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 22.0000 0.707472 0.353736 − 0.935345i \(-0.384911\pi\)
0.353736 + 0.935345i \(0.384911\pi\)
\(968\) −10.0000 −0.321412
\(969\) 30.0000 0.963739
\(970\) 0 0
\(971\) −15.0000 −0.481373 −0.240686 − 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −14.0000 −0.448589
\(975\) 30.0000 0.960769
\(976\) −2.00000 −0.0640184
\(977\) 34.0000 1.08776 0.543878 − 0.839164i \(-0.316955\pi\)
0.543878 + 0.839164i \(0.316955\pi\)
\(978\) 19.0000 0.607553
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) −3.00000 −0.0957826
\(982\) −24.0000 −0.765871
\(983\) 6.00000 0.191370 0.0956851 − 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 45.0000 1.43309
\(987\) 0 0
\(988\) −36.0000 −1.14531
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 − 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) −6.00000 −0.190500
\(993\) 32.0000 1.01549
\(994\) 0 0
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) 32.0000 1.01345 0.506725 − 0.862108i \(-0.330856\pi\)
0.506725 + 0.862108i \(0.330856\pi\)
\(998\) 7.00000 0.221581
\(999\) 6.00000 0.189832
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.bk.1.1 1
7.2 even 3 966.2.i.b.277.1 ✓ 2
7.4 even 3 966.2.i.b.415.1 yes 2
7.6 odd 2 6762.2.a.ba.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.b.277.1 ✓ 2 7.2 even 3
966.2.i.b.415.1 yes 2 7.4 even 3
6762.2.a.ba.1.1 1 7.6 odd 2
6762.2.a.bk.1.1 1 1.1 even 1 trivial