Properties

Label 2793.2.a.a.1.1
Level $2793$
Weight $2$
Character 2793.1
Self dual yes
Analytic conductor $22.302$
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] = [2793,2,Mod(1,2793)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2793, 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("2793.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2793 = 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2793.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: \(22.3022172845\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2793.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{9} +O(q^{10})\) \(q-2.00000 q^{2} -1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} +1.00000 q^{9} +2.00000 q^{10} -3.00000 q^{11} -2.00000 q^{12} +6.00000 q^{13} +1.00000 q^{15} -4.00000 q^{16} -3.00000 q^{17} -2.00000 q^{18} +1.00000 q^{19} -2.00000 q^{20} +6.00000 q^{22} +4.00000 q^{23} -4.00000 q^{25} -12.0000 q^{26} -1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{30} -2.00000 q^{31} +8.00000 q^{32} +3.00000 q^{33} +6.00000 q^{34} +2.00000 q^{36} +8.00000 q^{37} -2.00000 q^{38} -6.00000 q^{39} +8.00000 q^{41} -1.00000 q^{43} -6.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} -3.00000 q^{47} +4.00000 q^{48} +8.00000 q^{50} +3.00000 q^{51} +12.0000 q^{52} -6.00000 q^{53} +2.00000 q^{54} +3.00000 q^{55} -1.00000 q^{57} +20.0000 q^{58} +2.00000 q^{60} -7.00000 q^{61} +4.00000 q^{62} -8.00000 q^{64} -6.00000 q^{65} -6.00000 q^{66} +8.00000 q^{67} -6.00000 q^{68} -4.00000 q^{69} +12.0000 q^{71} +11.0000 q^{73} -16.0000 q^{74} +4.00000 q^{75} +2.00000 q^{76} +12.0000 q^{78} +4.00000 q^{80} +1.00000 q^{81} -16.0000 q^{82} -4.00000 q^{83} +3.00000 q^{85} +2.00000 q^{86} +10.0000 q^{87} -10.0000 q^{89} +2.00000 q^{90} +8.00000 q^{92} +2.00000 q^{93} +6.00000 q^{94} -1.00000 q^{95} -8.00000 q^{96} +2.00000 q^{97} -3.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) 2.00000 0.816497
\(7\) 0 0
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) −2.00000 −0.577350
\(13\) 6.00000 1.66410 0.832050 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −2.00000 −0.471405
\(19\) 1.00000 0.229416
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) 6.00000 1.27920
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) −12.0000 −2.35339
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) −2.00000 −0.365148
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 8.00000 1.41421
\(33\) 3.00000 0.522233
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −2.00000 −0.324443
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −6.00000 −0.904534
\(45\) −1.00000 −0.149071
\(46\) −8.00000 −1.17954
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 4.00000 0.577350
\(49\) 0 0
\(50\) 8.00000 1.13137
\(51\) 3.00000 0.420084
\(52\) 12.0000 1.66410
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 2.00000 0.272166
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 20.0000 2.62613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 2.00000 0.258199
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −6.00000 −0.744208
\(66\) −6.00000 −0.738549
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) −16.0000 −1.85996
\(75\) 4.00000 0.461880
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 12.0000 1.35873
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) −16.0000 −1.76690
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 2.00000 0.215666
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 2.00000 0.207390
\(94\) 6.00000 0.618853
\(95\) −1.00000 −0.102598
\(96\) −8.00000 −0.816497
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) −8.00000 −0.800000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) −6.00000 −0.594089
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −2.00000 −0.193347 −0.0966736 0.995316i \(-0.530820\pi\)
−0.0966736 + 0.995316i \(0.530820\pi\)
\(108\) −2.00000 −0.192450
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) −6.00000 −0.572078
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 2.00000 0.187317
\(115\) −4.00000 −0.373002
\(116\) −20.0000 −1.85695
\(117\) 6.00000 0.554700
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 14.0000 1.26750
\(123\) −8.00000 −0.721336
\(124\) −4.00000 −0.359211
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 1.00000 0.0880451
\(130\) 12.0000 1.05247
\(131\) 13.0000 1.13582 0.567908 0.823092i \(-0.307753\pi\)
0.567908 + 0.823092i \(0.307753\pi\)
\(132\) 6.00000 0.522233
\(133\) 0 0
\(134\) −16.0000 −1.38219
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 8.00000 0.681005
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) −24.0000 −2.01404
\(143\) −18.0000 −1.50524
\(144\) −4.00000 −0.333333
\(145\) 10.0000 0.830455
\(146\) −22.0000 −1.82073
\(147\) 0 0
\(148\) 16.0000 1.31519
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) −8.00000 −0.653197
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) −12.0000 −0.960769
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) −8.00000 −0.632456
\(161\) 0 0
\(162\) −2.00000 −0.157135
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 16.0000 1.24939
\(165\) −3.00000 −0.233550
\(166\) 8.00000 0.620920
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) −6.00000 −0.460179
\(171\) 1.00000 0.0764719
\(172\) −2.00000 −0.152499
\(173\) −14.0000 −1.06440 −0.532200 0.846619i \(-0.678635\pi\)
−0.532200 + 0.846619i \(0.678635\pi\)
\(174\) −20.0000 −1.51620
\(175\) 0 0
\(176\) 12.0000 0.904534
\(177\) 0 0
\(178\) 20.0000 1.49906
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) −4.00000 −0.293294
\(187\) 9.00000 0.658145
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) 2.00000 0.145095
\(191\) −3.00000 −0.217072 −0.108536 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 8.00000 0.577350
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −4.00000 −0.287183
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 6.00000 0.426401
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 4.00000 0.281439
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −8.00000 −0.558744
\(206\) 28.0000 1.95085
\(207\) 4.00000 0.278019
\(208\) −24.0000 −1.66410
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −12.0000 −0.824163
\(213\) −12.0000 −0.822226
\(214\) 4.00000 0.273434
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 0 0
\(218\) −40.0000 −2.70914
\(219\) −11.0000 −0.743311
\(220\) 6.00000 0.404520
\(221\) −18.0000 −1.21081
\(222\) 16.0000 1.07385
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 12.0000 0.798228
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) −2.00000 −0.132453
\(229\) 15.0000 0.991228 0.495614 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) −11.0000 −0.720634 −0.360317 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) −12.0000 −0.784465
\(235\) 3.00000 0.195698
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) −4.00000 −0.258199
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 4.00000 0.257130
\(243\) −1.00000 −0.0641500
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 16.0000 1.02012
\(247\) 6.00000 0.381771
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) −18.0000 −1.13842
\(251\) −27.0000 −1.70422 −0.852112 0.523359i \(-0.824679\pi\)
−0.852112 + 0.523359i \(0.824679\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 4.00000 0.250982
\(255\) −3.00000 −0.187867
\(256\) 16.0000 1.00000
\(257\) −8.00000 −0.499026 −0.249513 0.968371i \(-0.580271\pi\)
−0.249513 + 0.968371i \(0.580271\pi\)
\(258\) −2.00000 −0.124515
\(259\) 0 0
\(260\) −12.0000 −0.744208
\(261\) −10.0000 −0.618984
\(262\) −26.0000 −1.60629
\(263\) −21.0000 −1.29492 −0.647458 0.762101i \(-0.724168\pi\)
−0.647458 + 0.762101i \(0.724168\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) 16.0000 0.977356
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) −2.00000 −0.121716
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 12.0000 0.723627
\(276\) −8.00000 −0.481543
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) −10.0000 −0.599760
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) −6.00000 −0.357295
\(283\) −19.0000 −1.12943 −0.564716 0.825285i \(-0.691014\pi\)
−0.564716 + 0.825285i \(0.691014\pi\)
\(284\) 24.0000 1.42414
\(285\) 1.00000 0.0592349
\(286\) 36.0000 2.12872
\(287\) 0 0
\(288\) 8.00000 0.471405
\(289\) −8.00000 −0.470588
\(290\) −20.0000 −1.17444
\(291\) −2.00000 −0.117242
\(292\) 22.0000 1.28745
\(293\) −4.00000 −0.233682 −0.116841 0.993151i \(-0.537277\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.00000 0.174078
\(298\) −30.0000 −1.73785
\(299\) 24.0000 1.38796
\(300\) 8.00000 0.461880
\(301\) 0 0
\(302\) 16.0000 0.920697
\(303\) 2.00000 0.114897
\(304\) −4.00000 −0.229416
\(305\) 7.00000 0.400819
\(306\) 6.00000 0.342997
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 14.0000 0.796432
\(310\) −4.00000 −0.227185
\(311\) −7.00000 −0.396934 −0.198467 0.980108i \(-0.563596\pi\)
−0.198467 + 0.980108i \(0.563596\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) −12.0000 −0.672927
\(319\) 30.0000 1.67968
\(320\) 8.00000 0.447214
\(321\) 2.00000 0.111629
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 2.00000 0.111111
\(325\) −24.0000 −1.33128
\(326\) 32.0000 1.77232
\(327\) −20.0000 −1.10600
\(328\) 0 0
\(329\) 0 0
\(330\) 6.00000 0.330289
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −8.00000 −0.439057
\(333\) 8.00000 0.438397
\(334\) 36.0000 1.96983
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) −46.0000 −2.50207
\(339\) 6.00000 0.325875
\(340\) 6.00000 0.325396
\(341\) 6.00000 0.324918
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 28.0000 1.50529
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 20.0000 1.07211
\(349\) −25.0000 −1.33822 −0.669110 0.743164i \(-0.733324\pi\)
−0.669110 + 0.743164i \(0.733324\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) −24.0000 −1.27920
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) −20.0000 −1.06000
\(357\) 0 0
\(358\) 20.0000 1.05703
\(359\) 25.0000 1.31945 0.659725 0.751507i \(-0.270673\pi\)
0.659725 + 0.751507i \(0.270673\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 4.00000 0.210235
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −11.0000 −0.575766
\(366\) −14.0000 −0.731792
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) −16.0000 −0.834058
\(369\) 8.00000 0.416463
\(370\) 16.0000 0.831800
\(371\) 0 0
\(372\) 4.00000 0.207390
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) −18.0000 −0.930758
\(375\) −9.00000 −0.464758
\(376\) 0 0
\(377\) −60.0000 −3.09016
\(378\) 0 0
\(379\) −30.0000 −1.54100 −0.770498 0.637442i \(-0.779993\pi\)
−0.770498 + 0.637442i \(0.779993\pi\)
\(380\) −2.00000 −0.102598
\(381\) 2.00000 0.102463
\(382\) 6.00000 0.306987
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) −1.00000 −0.0508329
\(388\) 4.00000 0.203069
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) −12.0000 −0.607644
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −13.0000 −0.655763
\(394\) 4.00000 0.201517
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) 7.00000 0.351320 0.175660 0.984451i \(-0.443794\pi\)
0.175660 + 0.984451i \(0.443794\pi\)
\(398\) −10.0000 −0.501255
\(399\) 0 0
\(400\) 16.0000 0.800000
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 16.0000 0.798007
\(403\) −12.0000 −0.597763
\(404\) −4.00000 −0.199007
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 16.0000 0.790184
\(411\) −3.00000 −0.147979
\(412\) −28.0000 −1.37946
\(413\) 0 0
\(414\) −8.00000 −0.393179
\(415\) 4.00000 0.196352
\(416\) 48.0000 2.35339
\(417\) −5.00000 −0.244851
\(418\) 6.00000 0.293470
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 56.0000 2.72604
\(423\) −3.00000 −0.145865
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 24.0000 1.16280
\(427\) 0 0
\(428\) −4.00000 −0.193347
\(429\) 18.0000 0.869048
\(430\) −2.00000 −0.0964486
\(431\) −18.0000 −0.867029 −0.433515 0.901146i \(-0.642727\pi\)
−0.433515 + 0.901146i \(0.642727\pi\)
\(432\) 4.00000 0.192450
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) −10.0000 −0.479463
\(436\) 40.0000 1.91565
\(437\) 4.00000 0.191346
\(438\) 22.0000 1.05120
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 36.0000 1.71235
\(443\) 39.0000 1.85295 0.926473 0.376361i \(-0.122825\pi\)
0.926473 + 0.376361i \(0.122825\pi\)
\(444\) −16.0000 −0.759326
\(445\) 10.0000 0.474045
\(446\) 8.00000 0.378811
\(447\) −15.0000 −0.709476
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 8.00000 0.377124
\(451\) −24.0000 −1.13012
\(452\) −12.0000 −0.564433
\(453\) 8.00000 0.375873
\(454\) 36.0000 1.68956
\(455\) 0 0
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) −30.0000 −1.40181
\(459\) 3.00000 0.140028
\(460\) −8.00000 −0.373002
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 40.0000 1.85695
\(465\) −2.00000 −0.0927478
\(466\) 22.0000 1.01913
\(467\) 17.0000 0.786666 0.393333 0.919396i \(-0.371322\pi\)
0.393333 + 0.919396i \(0.371322\pi\)
\(468\) 12.0000 0.554700
\(469\) 0 0
\(470\) −6.00000 −0.276759
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 3.00000 0.137940
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 30.0000 1.37217
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 8.00000 0.365148
\(481\) 48.0000 2.18861
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) −4.00000 −0.181818
\(485\) −2.00000 −0.0908153
\(486\) 2.00000 0.0907218
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) −16.0000 −0.721336
\(493\) 30.0000 1.35113
\(494\) −12.0000 −0.539906
\(495\) 3.00000 0.134840
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) −8.00000 −0.358489
\(499\) −35.0000 −1.56682 −0.783408 0.621508i \(-0.786520\pi\)
−0.783408 + 0.621508i \(0.786520\pi\)
\(500\) 18.0000 0.804984
\(501\) 18.0000 0.804181
\(502\) 54.0000 2.41014
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 24.0000 1.06693
\(507\) −23.0000 −1.02147
\(508\) −4.00000 −0.177471
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 6.00000 0.265684
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) −1.00000 −0.0441511
\(514\) 16.0000 0.705730
\(515\) 14.0000 0.616914
\(516\) 2.00000 0.0880451
\(517\) 9.00000 0.395820
\(518\) 0 0
\(519\) 14.0000 0.614532
\(520\) 0 0
\(521\) 28.0000 1.22670 0.613351 0.789810i \(-0.289821\pi\)
0.613351 + 0.789810i \(0.289821\pi\)
\(522\) 20.0000 0.875376
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 26.0000 1.13582
\(525\) 0 0
\(526\) 42.0000 1.83129
\(527\) 6.00000 0.261364
\(528\) −12.0000 −0.522233
\(529\) −7.00000 −0.304348
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) 0 0
\(533\) 48.0000 2.07911
\(534\) −20.0000 −0.865485
\(535\) 2.00000 0.0864675
\(536\) 0 0
\(537\) 10.0000 0.431532
\(538\) −60.0000 −2.58678
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 24.0000 1.03089
\(543\) 2.00000 0.0858282
\(544\) −24.0000 −1.02899
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) 6.00000 0.256307
\(549\) −7.00000 −0.298753
\(550\) −24.0000 −1.02336
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 0 0
\(554\) −26.0000 −1.10463
\(555\) 8.00000 0.339581
\(556\) 10.0000 0.424094
\(557\) 3.00000 0.127114 0.0635570 0.997978i \(-0.479756\pi\)
0.0635570 + 0.997978i \(0.479756\pi\)
\(558\) 4.00000 0.169334
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) −4.00000 −0.168730
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 6.00000 0.252646
\(565\) 6.00000 0.252422
\(566\) 38.0000 1.59726
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) −2.00000 −0.0837708
\(571\) 32.0000 1.33916 0.669579 0.742741i \(-0.266474\pi\)
0.669579 + 0.742741i \(0.266474\pi\)
\(572\) −36.0000 −1.50524
\(573\) 3.00000 0.125327
\(574\) 0 0
\(575\) −16.0000 −0.667246
\(576\) −8.00000 −0.333333
\(577\) −3.00000 −0.124892 −0.0624458 0.998048i \(-0.519890\pi\)
−0.0624458 + 0.998048i \(0.519890\pi\)
\(578\) 16.0000 0.665512
\(579\) −4.00000 −0.166234
\(580\) 20.0000 0.830455
\(581\) 0 0
\(582\) 4.00000 0.165805
\(583\) 18.0000 0.745484
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 8.00000 0.330477
\(587\) 37.0000 1.52715 0.763577 0.645717i \(-0.223441\pi\)
0.763577 + 0.645717i \(0.223441\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) −32.0000 −1.31519
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) 30.0000 1.22885
\(597\) −5.00000 −0.204636
\(598\) −48.0000 −1.96287
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 8.00000 0.326327 0.163163 0.986599i \(-0.447830\pi\)
0.163163 + 0.986599i \(0.447830\pi\)
\(602\) 0 0
\(603\) 8.00000 0.325785
\(604\) −16.0000 −0.651031
\(605\) 2.00000 0.0813116
\(606\) −4.00000 −0.162489
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) −18.0000 −0.728202
\(612\) −6.00000 −0.242536
\(613\) 9.00000 0.363507 0.181753 0.983344i \(-0.441823\pi\)
0.181753 + 0.983344i \(0.441823\pi\)
\(614\) −24.0000 −0.968561
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) 23.0000 0.925945 0.462973 0.886373i \(-0.346783\pi\)
0.462973 + 0.886373i \(0.346783\pi\)
\(618\) −28.0000 −1.12633
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 4.00000 0.160644
\(621\) −4.00000 −0.160514
\(622\) 14.0000 0.561349
\(623\) 0 0
\(624\) 24.0000 0.960769
\(625\) 11.0000 0.440000
\(626\) 28.0000 1.11911
\(627\) 3.00000 0.119808
\(628\) 4.00000 0.159617
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 7.00000 0.278666 0.139333 0.990246i \(-0.455504\pi\)
0.139333 + 0.990246i \(0.455504\pi\)
\(632\) 0 0
\(633\) 28.0000 1.11290
\(634\) 24.0000 0.953162
\(635\) 2.00000 0.0793676
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) −60.0000 −2.37542
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −4.00000 −0.157867
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 6.00000 0.236067
\(647\) 27.0000 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 48.0000 1.88271
\(651\) 0 0
\(652\) −32.0000 −1.25322
\(653\) −1.00000 −0.0391330 −0.0195665 0.999809i \(-0.506229\pi\)
−0.0195665 + 0.999809i \(0.506229\pi\)
\(654\) 40.0000 1.56412
\(655\) −13.0000 −0.507952
\(656\) −32.0000 −1.24939
\(657\) 11.0000 0.429151
\(658\) 0 0
\(659\) 10.0000 0.389545 0.194772 0.980848i \(-0.437603\pi\)
0.194772 + 0.980848i \(0.437603\pi\)
\(660\) −6.00000 −0.233550
\(661\) −12.0000 −0.466746 −0.233373 0.972387i \(-0.574976\pi\)
−0.233373 + 0.972387i \(0.574976\pi\)
\(662\) −24.0000 −0.932786
\(663\) 18.0000 0.699062
\(664\) 0 0
\(665\) 0 0
\(666\) −16.0000 −0.619987
\(667\) −40.0000 −1.54881
\(668\) −36.0000 −1.39288
\(669\) 4.00000 0.154649
\(670\) 16.0000 0.618134
\(671\) 21.0000 0.810696
\(672\) 0 0
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) 44.0000 1.69482
\(675\) 4.00000 0.153960
\(676\) 46.0000 1.76923
\(677\) 22.0000 0.845529 0.422764 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) −12.0000 −0.460857
\(679\) 0 0
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) −12.0000 −0.459504
\(683\) −6.00000 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(684\) 2.00000 0.0764719
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) −15.0000 −0.572286
\(688\) 4.00000 0.152499
\(689\) −36.0000 −1.37149
\(690\) −8.00000 −0.304555
\(691\) −17.0000 −0.646710 −0.323355 0.946278i \(-0.604811\pi\)
−0.323355 + 0.946278i \(0.604811\pi\)
\(692\) −28.0000 −1.06440
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) −5.00000 −0.189661
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 50.0000 1.89253
\(699\) 11.0000 0.416058
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 12.0000 0.452911
\(703\) 8.00000 0.301726
\(704\) 24.0000 0.904534
\(705\) −3.00000 −0.112987
\(706\) 28.0000 1.05379
\(707\) 0 0
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 24.0000 0.900704
\(711\) 0 0
\(712\) 0 0
\(713\) −8.00000 −0.299602
\(714\) 0 0
\(715\) 18.0000 0.673162
\(716\) −20.0000 −0.747435
\(717\) 15.0000 0.560185
\(718\) −50.0000 −1.86598
\(719\) 35.0000 1.30528 0.652640 0.757668i \(-0.273661\pi\)
0.652640 + 0.757668i \(0.273661\pi\)
\(720\) 4.00000 0.149071
\(721\) 0 0
\(722\) −2.00000 −0.0744323
\(723\) 12.0000 0.446285
\(724\) −4.00000 −0.148659
\(725\) 40.0000 1.48556
\(726\) −4.00000 −0.148454
\(727\) 7.00000 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 22.0000 0.814257
\(731\) 3.00000 0.110959
\(732\) 14.0000 0.517455
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 32.0000 1.17954
\(737\) −24.0000 −0.884051
\(738\) −16.0000 −0.588968
\(739\) −45.0000 −1.65535 −0.827676 0.561206i \(-0.810337\pi\)
−0.827676 + 0.561206i \(0.810337\pi\)
\(740\) −16.0000 −0.588172
\(741\) −6.00000 −0.220416
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 0 0
\(745\) −15.0000 −0.549557
\(746\) 32.0000 1.17160
\(747\) −4.00000 −0.146352
\(748\) 18.0000 0.658145
\(749\) 0 0
\(750\) 18.0000 0.657267
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 12.0000 0.437595
\(753\) 27.0000 0.983935
\(754\) 120.000 4.37014
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) 60.0000 2.17930
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) 13.0000 0.471250 0.235625 0.971844i \(-0.424286\pi\)
0.235625 + 0.971844i \(0.424286\pi\)
\(762\) −4.00000 −0.144905
\(763\) 0 0
\(764\) −6.00000 −0.217072
\(765\) 3.00000 0.108465
\(766\) 28.0000 1.01168
\(767\) 0 0
\(768\) −16.0000 −0.577350
\(769\) 45.0000 1.62274 0.811371 0.584532i \(-0.198722\pi\)
0.811371 + 0.584532i \(0.198722\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) 8.00000 0.287926
\(773\) 36.0000 1.29483 0.647415 0.762138i \(-0.275850\pi\)
0.647415 + 0.762138i \(0.275850\pi\)
\(774\) 2.00000 0.0718885
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 30.0000 1.07555
\(779\) 8.00000 0.286630
\(780\) 12.0000 0.429669
\(781\) −36.0000 −1.28818
\(782\) 24.0000 0.858238
\(783\) 10.0000 0.357371
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 26.0000 0.927389
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) −4.00000 −0.142494
\(789\) 21.0000 0.747620
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −42.0000 −1.49146
\(794\) −14.0000 −0.496841
\(795\) −6.00000 −0.212798
\(796\) 10.0000 0.354441
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 9.00000 0.318397
\(800\) −32.0000 −1.13137
\(801\) −10.0000 −0.353333
\(802\) 56.0000 1.97743
\(803\) −33.0000 −1.16454
\(804\) −16.0000 −0.564276
\(805\) 0 0
\(806\) 24.0000 0.845364
\(807\) −30.0000 −1.05605
\(808\) 0 0
\(809\) 5.00000 0.175791 0.0878953 0.996130i \(-0.471986\pi\)
0.0878953 + 0.996130i \(0.471986\pi\)
\(810\) 2.00000 0.0702728
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 12.0000 0.420858
\(814\) 48.0000 1.68240
\(815\) 16.0000 0.560456
\(816\) −12.0000 −0.420084
\(817\) −1.00000 −0.0349856
\(818\) 20.0000 0.699284
\(819\) 0 0
\(820\) −16.0000 −0.558744
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 6.00000 0.209274
\(823\) 19.0000 0.662298 0.331149 0.943578i \(-0.392564\pi\)
0.331149 + 0.943578i \(0.392564\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) −52.0000 −1.80822 −0.904109 0.427303i \(-0.859464\pi\)
−0.904109 + 0.427303i \(0.859464\pi\)
\(828\) 8.00000 0.278019
\(829\) −20.0000 −0.694629 −0.347314 0.937749i \(-0.612906\pi\)
−0.347314 + 0.937749i \(0.612906\pi\)
\(830\) −8.00000 −0.277684
\(831\) −13.0000 −0.450965
\(832\) −48.0000 −1.66410
\(833\) 0 0
\(834\) 10.0000 0.346272
\(835\) 18.0000 0.622916
\(836\) −6.00000 −0.207514
\(837\) 2.00000 0.0691301
\(838\) 40.0000 1.38178
\(839\) −30.0000 −1.03572 −0.517858 0.855467i \(-0.673270\pi\)
−0.517858 + 0.855467i \(0.673270\pi\)
\(840\) 0 0
\(841\) 71.0000 2.44828
\(842\) −4.00000 −0.137849
\(843\) −2.00000 −0.0688837
\(844\) −56.0000 −1.92760
\(845\) −23.0000 −0.791224
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) 24.0000 0.824163
\(849\) 19.0000 0.652078
\(850\) −24.0000 −0.823193
\(851\) 32.0000 1.09695
\(852\) −24.0000 −0.822226
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) −48.0000 −1.63965 −0.819824 0.572615i \(-0.805929\pi\)
−0.819824 + 0.572615i \(0.805929\pi\)
\(858\) −36.0000 −1.22902
\(859\) −35.0000 −1.19418 −0.597092 0.802173i \(-0.703677\pi\)
−0.597092 + 0.802173i \(0.703677\pi\)
\(860\) 2.00000 0.0681994
\(861\) 0 0
\(862\) 36.0000 1.22616
\(863\) 44.0000 1.49778 0.748889 0.662696i \(-0.230588\pi\)
0.748889 + 0.662696i \(0.230588\pi\)
\(864\) −8.00000 −0.272166
\(865\) 14.0000 0.476014
\(866\) −52.0000 −1.76703
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 0 0
\(870\) 20.0000 0.678064
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) −22.0000 −0.743311
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 20.0000 0.674967
\(879\) 4.00000 0.134917
\(880\) −12.0000 −0.404520
\(881\) −7.00000 −0.235836 −0.117918 0.993023i \(-0.537622\pi\)
−0.117918 + 0.993023i \(0.537622\pi\)
\(882\) 0 0
\(883\) −21.0000 −0.706706 −0.353353 0.935490i \(-0.614959\pi\)
−0.353353 + 0.935490i \(0.614959\pi\)
\(884\) −36.0000 −1.21081
\(885\) 0 0
\(886\) −78.0000 −2.62046
\(887\) 52.0000 1.74599 0.872995 0.487730i \(-0.162175\pi\)
0.872995 + 0.487730i \(0.162175\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −20.0000 −0.670402
\(891\) −3.00000 −0.100504
\(892\) −8.00000 −0.267860
\(893\) −3.00000 −0.100391
\(894\) 30.0000 1.00335
\(895\) 10.0000 0.334263
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 40.0000 1.33482
\(899\) 20.0000 0.667037
\(900\) −8.00000 −0.266667
\(901\) 18.0000 0.599667
\(902\) 48.0000 1.59823
\(903\) 0 0
\(904\) 0 0
\(905\) 2.00000 0.0664822
\(906\) −16.0000 −0.531564
\(907\) −2.00000 −0.0664089 −0.0332045 0.999449i \(-0.510571\pi\)
−0.0332045 + 0.999449i \(0.510571\pi\)
\(908\) −36.0000 −1.19470
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 2.00000 0.0662630 0.0331315 0.999451i \(-0.489452\pi\)
0.0331315 + 0.999451i \(0.489452\pi\)
\(912\) 4.00000 0.132453
\(913\) 12.0000 0.397142
\(914\) −6.00000 −0.198462
\(915\) −7.00000 −0.231413
\(916\) 30.0000 0.991228
\(917\) 0 0
\(918\) −6.00000 −0.198030
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) −66.0000 −2.17359
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 62.0000 2.03745
\(927\) −14.0000 −0.459820
\(928\) −80.0000 −2.62613
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 4.00000 0.131165
\(931\) 0 0
\(932\) −22.0000 −0.720634
\(933\) 7.00000 0.229170
\(934\) −34.0000 −1.11251
\(935\) −9.00000 −0.294331
\(936\) 0 0
\(937\) −53.0000 −1.73143 −0.865717 0.500533i \(-0.833137\pi\)
−0.865717 + 0.500533i \(0.833137\pi\)
\(938\) 0 0
\(939\) 14.0000 0.456873
\(940\) 6.00000 0.195698
\(941\) −2.00000 −0.0651981 −0.0325991 0.999469i \(-0.510378\pi\)
−0.0325991 + 0.999469i \(0.510378\pi\)
\(942\) 4.00000 0.130327
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 0 0
\(946\) −6.00000 −0.195077
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) 66.0000 2.14245
\(950\) 8.00000 0.259554
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −16.0000 −0.518291 −0.259145 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) 12.0000 0.388514
\(955\) 3.00000 0.0970777
\(956\) −30.0000 −0.970269
\(957\) −30.0000 −0.969762
\(958\) −80.0000 −2.58468
\(959\) 0 0
\(960\) −8.00000 −0.258199
\(961\) −27.0000 −0.870968
\(962\) −96.0000 −3.09516
\(963\) −2.00000 −0.0644491
\(964\) −24.0000 −0.772988
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 4.00000 0.128432
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −2.00000 −0.0641500
\(973\) 0 0
\(974\) −16.0000 −0.512673
\(975\) 24.0000 0.768615
\(976\) 28.0000 0.896258
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −32.0000 −1.02325
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 16.0000 0.510581
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) −60.0000 −1.91079
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −4.00000 −0.127193
\(990\) −6.00000 −0.190693
\(991\) −8.00000 −0.254128 −0.127064 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −16.0000 −0.508001
\(993\) −12.0000 −0.380808
\(994\) 0 0
\(995\) −5.00000 −0.158511
\(996\) 8.00000 0.253490
\(997\) 7.00000 0.221692 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(998\) 70.0000 2.21581
\(999\) −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 2793.2.a.a.1.1 1
3.2 odd 2 8379.2.a.q.1.1 1
7.6 odd 2 57.2.a.b.1.1 1
21.20 even 2 171.2.a.c.1.1 1
28.27 even 2 912.2.a.d.1.1 1
35.13 even 4 1425.2.c.a.799.2 2
35.27 even 4 1425.2.c.a.799.1 2
35.34 odd 2 1425.2.a.i.1.1 1
56.13 odd 2 3648.2.a.h.1.1 1
56.27 even 2 3648.2.a.y.1.1 1
77.76 even 2 6897.2.a.g.1.1 1
84.83 odd 2 2736.2.a.h.1.1 1
91.90 odd 2 9633.2.a.p.1.1 1
105.104 even 2 4275.2.a.a.1.1 1
133.132 even 2 1083.2.a.d.1.1 1
399.398 odd 2 3249.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 1 7.6 odd 2
171.2.a.c.1.1 1 21.20 even 2
912.2.a.d.1.1 1 28.27 even 2
1083.2.a.d.1.1 1 133.132 even 2
1425.2.a.i.1.1 1 35.34 odd 2
1425.2.c.a.799.1 2 35.27 even 4
1425.2.c.a.799.2 2 35.13 even 4
2736.2.a.h.1.1 1 84.83 odd 2
2793.2.a.a.1.1 1 1.1 even 1 trivial
3249.2.a.a.1.1 1 399.398 odd 2
3648.2.a.h.1.1 1 56.13 odd 2
3648.2.a.y.1.1 1 56.27 even 2
4275.2.a.a.1.1 1 105.104 even 2
6897.2.a.g.1.1 1 77.76 even 2
8379.2.a.q.1.1 1 3.2 odd 2
9633.2.a.p.1.1 1 91.90 odd 2