Properties

Label 912.2.a.d.1.1
Level $912$
Weight $2$
Character 912.1
Self dual yes
Analytic conductor $7.282$
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] = [912,2,Mod(1,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("912.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.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: \(7.28235666434\)
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 \(\chi\) \(=\) 912.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -3.00000 q^{7} +1.00000 q^{9} +3.00000 q^{11} -6.00000 q^{13} -1.00000 q^{15} +3.00000 q^{17} +1.00000 q^{19} +3.00000 q^{21} -4.00000 q^{23} -4.00000 q^{25} -1.00000 q^{27} -10.0000 q^{29} -2.00000 q^{31} -3.00000 q^{33} -3.00000 q^{35} +8.00000 q^{37} +6.00000 q^{39} -8.00000 q^{41} +1.00000 q^{43} +1.00000 q^{45} -3.00000 q^{47} +2.00000 q^{49} -3.00000 q^{51} -6.00000 q^{53} +3.00000 q^{55} -1.00000 q^{57} +7.00000 q^{61} -3.00000 q^{63} -6.00000 q^{65} -8.00000 q^{67} +4.00000 q^{69} -12.0000 q^{71} -11.0000 q^{73} +4.00000 q^{75} -9.00000 q^{77} +1.00000 q^{81} -4.00000 q^{83} +3.00000 q^{85} +10.0000 q^{87} +10.0000 q^{89} +18.0000 q^{91} +2.00000 q^{93} +1.00000 q^{95} -2.00000 q^{97} +3.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) 0 0
\(7\) −3.00000 −1.13389 −0.566947 0.823754i \(-0.691875\pi\)
−0.566947 + 0.823754i \(0.691875\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −6.00000 −1.66410 −0.832050 0.554700i \(-0.812833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −8.00000 −1.24939 −0.624695 0.780869i \(-0.714777\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) 0 0
\(63\) −3.00000 −0.377964
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) −9.00000 −1.02565
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −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
\(86\) 0 0
\(87\) 10.0000 1.07211
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 18.0000 1.88691
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 3.00000 0.292770
\(106\) 0 0
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 0 0
\(109\) 20.0000 1.91565 0.957826 0.287348i \(-0.0927736\pi\)
0.957826 + 0.287348i \(0.0927736\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) −6.00000 −0.554700
\(118\) 0 0
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 8.00000 0.721336
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 13.0000 1.13582 0.567908 0.823092i \(-0.307753\pi\)
0.567908 + 0.823092i \(0.307753\pi\)
\(132\) 0 0
\(133\) −3.00000 −0.260133
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 3.00000 0.256307 0.128154 0.991754i \(-0.459095\pi\)
0.128154 + 0.991754i \(0.459095\pi\)
\(138\) 0 0
\(139\) 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
\(142\) 0 0
\(143\) −18.0000 −1.50524
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) −2.00000 −0.164957
\(148\) 0 0
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) −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
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 12.0000 0.907115
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) 0 0
\(185\) 8.00000 0.588172
\(186\) 0 0
\(187\) 9.00000 0.658145
\(188\) 0 0
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) 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
\(202\) 0 0
\(203\) 30.0000 2.10559
\(204\) 0 0
\(205\) −8.00000 −0.558744
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 0 0
\(213\) 12.0000 0.822226
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 11.0000 0.743311
\(220\) 0 0
\(221\) −18.0000 −1.21081
\(222\) 0 0
\(223\) −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
\(226\) 0 0
\(227\) −18.0000 −1.19470 −0.597351 0.801980i \(-0.703780\pi\)
−0.597351 + 0.801980i \(0.703780\pi\)
\(228\) 0 0
\(229\) −15.0000 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(230\) 0 0
\(231\) 9.00000 0.592157
\(232\) 0 0
\(233\) −11.0000 −0.720634 −0.360317 0.932830i \(-0.617331\pi\)
−0.360317 + 0.932830i \(0.617331\pi\)
\(234\) 0 0
\(235\) −3.00000 −0.195698
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.0000 0.970269 0.485135 0.874439i \(-0.338771\pi\)
0.485135 + 0.874439i \(0.338771\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 2.00000 0.127775
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −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
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 0 0
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) −10.0000 −0.618984
\(262\) 0 0
\(263\) 21.0000 1.29492 0.647458 0.762101i \(-0.275832\pi\)
0.647458 + 0.762101i \(0.275832\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) −10.0000 −0.611990
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −12.0000 −0.728948 −0.364474 0.931214i \(-0.618751\pi\)
−0.364474 + 0.931214i \(0.618751\pi\)
\(272\) 0 0
\(273\) −18.0000 −1.08941
\(274\) 0 0
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) 13.0000 0.781094 0.390547 0.920583i \(-0.372286\pi\)
0.390547 + 0.920583i \(0.372286\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) −19.0000 −1.12943 −0.564716 0.825285i \(-0.691014\pi\)
−0.564716 + 0.825285i \(0.691014\pi\)
\(284\) 0 0
\(285\) −1.00000 −0.0592349
\(286\) 0 0
\(287\) 24.0000 1.41668
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) 4.00000 0.233682 0.116841 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −3.00000 −0.174078
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −3.00000 −0.172917
\(302\) 0 0
\(303\) −2.00000 −0.114897
\(304\) 0 0
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) 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
\(310\) 0 0
\(311\) −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 \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) −3.00000 −0.169031
\(316\) 0 0
\(317\) −12.0000 −0.673987 −0.336994 0.941507i \(-0.609410\pi\)
−0.336994 + 0.941507i \(0.609410\pi\)
\(318\) 0 0
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) −2.00000 −0.111629
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) 24.0000 1.33128
\(326\) 0 0
\(327\) −20.0000 −1.10600
\(328\) 0 0
\(329\) 9.00000 0.496186
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 0 0
\(333\) 8.00000 0.438397
\(334\) 0 0
\(335\) −8.00000 −0.437087
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −6.00000 −0.324918
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) −3.00000 −0.161048 −0.0805242 0.996753i \(-0.525659\pi\)
−0.0805242 + 0.996753i \(0.525659\pi\)
\(348\) 0 0
\(349\) 25.0000 1.33822 0.669110 0.743164i \(-0.266676\pi\)
0.669110 + 0.743164i \(0.266676\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −12.0000 −0.636894
\(356\) 0 0
\(357\) 9.00000 0.476331
\(358\) 0 0
\(359\) −25.0000 −1.31945 −0.659725 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −11.0000 −0.575766
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) −8.00000 −0.416463
\(370\) 0 0
\(371\) 18.0000 0.934513
\(372\) 0 0
\(373\) −16.0000 −0.828449 −0.414224 0.910175i \(-0.635947\pi\)
−0.414224 + 0.910175i \(0.635947\pi\)
\(374\) 0 0
\(375\) 9.00000 0.464758
\(376\) 0 0
\(377\) 60.0000 3.09016
\(378\) 0 0
\(379\) 30.0000 1.54100 0.770498 0.637442i \(-0.220007\pi\)
0.770498 + 0.637442i \(0.220007\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) 0 0
\(383\) −14.0000 −0.715367 −0.357683 0.933843i \(-0.616433\pi\)
−0.357683 + 0.933843i \(0.616433\pi\)
\(384\) 0 0
\(385\) −9.00000 −0.458682
\(386\) 0 0
\(387\) 1.00000 0.0508329
\(388\) 0 0
\(389\) −15.0000 −0.760530 −0.380265 0.924878i \(-0.624167\pi\)
−0.380265 + 0.924878i \(0.624167\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) −13.0000 −0.655763
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) 3.00000 0.150188
\(400\) 0 0
\(401\) −28.0000 −1.39825 −0.699127 0.714998i \(-0.746428\pi\)
−0.699127 + 0.714998i \(0.746428\pi\)
\(402\) 0 0
\(403\) 12.0000 0.597763
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 24.0000 1.18964
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4.00000 −0.196352
\(416\) 0 0
\(417\) −5.00000 −0.244851
\(418\) 0 0
\(419\) −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 \(0.484480\pi\)
\(422\) 0 0
\(423\) −3.00000 −0.145865
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) −21.0000 −1.01626
\(428\) 0 0
\(429\) 18.0000 0.869048
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) −26.0000 −1.24948 −0.624740 0.780833i \(-0.714795\pi\)
−0.624740 + 0.780833i \(0.714795\pi\)
\(434\) 0 0
\(435\) 10.0000 0.479463
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −10.0000 −0.477274 −0.238637 0.971109i \(-0.576701\pi\)
−0.238637 + 0.971109i \(0.576701\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) 0 0
\(443\) −39.0000 −1.85295 −0.926473 0.376361i \(-0.877175\pi\)
−0.926473 + 0.376361i \(0.877175\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) −15.0000 −0.709476
\(448\) 0 0
\(449\) −20.0000 −0.943858 −0.471929 0.881636i \(-0.656442\pi\)
−0.471929 + 0.881636i \(0.656442\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 0 0
\(453\) −8.00000 −0.375873
\(454\) 0 0
\(455\) 18.0000 0.843853
\(456\) 0 0
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) −33.0000 −1.53696 −0.768482 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) 31.0000 1.44069 0.720346 0.693615i \(-0.243983\pi\)
0.720346 + 0.693615i \(0.243983\pi\)
\(464\) 0 0
\(465\) 2.00000 0.0927478
\(466\) 0 0
\(467\) 17.0000 0.786666 0.393333 0.919396i \(-0.371322\pi\)
0.393333 + 0.919396i \(0.371322\pi\)
\(468\) 0 0
\(469\) 24.0000 1.10822
\(470\) 0 0
\(471\) 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
\(478\) 0 0
\(479\) 40.0000 1.82765 0.913823 0.406112i \(-0.133116\pi\)
0.913823 + 0.406112i \(0.133116\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 0 0
\(483\) −12.0000 −0.546019
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) −30.0000 −1.35113
\(494\) 0 0
\(495\) 3.00000 0.134840
\(496\) 0 0
\(497\) 36.0000 1.61482
\(498\) 0 0
\(499\) 35.0000 1.56682 0.783408 0.621508i \(-0.213480\pi\)
0.783408 + 0.621508i \(0.213480\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 0 0
\(503\) −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
\(506\) 0 0
\(507\) −23.0000 −1.02147
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 33.0000 1.45983
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −14.0000 −0.616914
\(516\) 0 0
\(517\) −9.00000 −0.395820
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) −14.0000 −0.612177 −0.306089 0.952003i \(-0.599020\pi\)
−0.306089 + 0.952003i \(0.599020\pi\)
\(524\) 0 0
\(525\) −12.0000 −0.523723
\(526\) 0 0
\(527\) −6.00000 −0.261364
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 48.0000 2.07911
\(534\) 0 0
\(535\) 2.00000 0.0864675
\(536\) 0 0
\(537\) −10.0000 −0.431532
\(538\) 0 0
\(539\) 6.00000 0.258438
\(540\) 0 0
\(541\) −13.0000 −0.558914 −0.279457 0.960158i \(-0.590154\pi\)
−0.279457 + 0.960158i \(0.590154\pi\)
\(542\) 0 0
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 0 0
\(549\) 7.00000 0.298753
\(550\) 0 0
\(551\) −10.0000 −0.426014
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 0 0
\(557\) 3.00000 0.127114 0.0635570 0.997978i \(-0.479756\pi\)
0.0635570 + 0.997978i \(0.479756\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) −9.00000 −0.379980
\(562\) 0 0
\(563\) −44.0000 −1.85438 −0.927189 0.374593i \(-0.877783\pi\)
−0.927189 + 0.374593i \(0.877783\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) 0 0
\(567\) −3.00000 −0.125988
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) −3.00000 −0.125327
\(574\) 0 0
\(575\) 16.0000 0.667246
\(576\) 0 0
\(577\) 3.00000 0.124892 0.0624458 0.998048i \(-0.480110\pi\)
0.0624458 + 0.998048i \(0.480110\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) 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
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) −9.00000 −0.368964
\(596\) 0 0
\(597\) −5.00000 −0.204636
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −8.00000 −0.326327 −0.163163 0.986599i \(-0.552170\pi\)
−0.163163 + 0.986599i \(0.552170\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 0 0
\(605\) −2.00000 −0.0813116
\(606\) 0 0
\(607\) −18.0000 −0.730597 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(608\) 0 0
\(609\) −30.0000 −1.21566
\(610\) 0 0
\(611\) 18.0000 0.728202
\(612\) 0 0
\(613\) 9.00000 0.363507 0.181753 0.983344i \(-0.441823\pi\)
0.181753 + 0.983344i \(0.441823\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) 23.0000 0.925945 0.462973 0.886373i \(-0.346783\pi\)
0.462973 + 0.886373i \(0.346783\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −30.0000 −1.20192
\(624\) 0 0
\(625\) 11.0000 0.440000
\(626\) 0 0
\(627\) −3.00000 −0.119808
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) −7.00000 −0.278666 −0.139333 0.990246i \(-0.544496\pi\)
−0.139333 + 0.990246i \(0.544496\pi\)
\(632\) 0 0
\(633\) −28.0000 −1.11290
\(634\) 0 0
\(635\) 2.00000 0.0793676
\(636\) 0 0
\(637\) −12.0000 −0.475457
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) 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
\(646\) 0 0
\(647\) 27.0000 1.06148 0.530740 0.847535i \(-0.321914\pi\)
0.530740 + 0.847535i \(0.321914\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) −1.00000 −0.0391330 −0.0195665 0.999809i \(-0.506229\pi\)
−0.0195665 + 0.999809i \(0.506229\pi\)
\(654\) 0 0
\(655\) 13.0000 0.507952
\(656\) 0 0
\(657\) −11.0000 −0.429151
\(658\) 0 0
\(659\) −10.0000 −0.389545 −0.194772 0.980848i \(-0.562397\pi\)
−0.194772 + 0.980848i \(0.562397\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 0 0
\(663\) 18.0000 0.699062
\(664\) 0 0
\(665\) −3.00000 −0.116335
\(666\) 0 0
\(667\) 40.0000 1.54881
\(668\) 0 0
\(669\) 4.00000 0.154649
\(670\) 0 0
\(671\) 21.0000 0.810696
\(672\) 0 0
\(673\) −16.0000 −0.616755 −0.308377 0.951264i \(-0.599786\pi\)
−0.308377 + 0.951264i \(0.599786\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 0 0
\(677\) −22.0000 −0.845529 −0.422764 0.906240i \(-0.638940\pi\)
−0.422764 + 0.906240i \(0.638940\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 18.0000 0.689761
\(682\) 0 0
\(683\) 6.00000 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(684\) 0 0
\(685\) 3.00000 0.114624
\(686\) 0 0
\(687\) 15.0000 0.572286
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −17.0000 −0.646710 −0.323355 0.946278i \(-0.604811\pi\)
−0.323355 + 0.946278i \(0.604811\pi\)
\(692\) 0 0
\(693\) −9.00000 −0.341882
\(694\) 0 0
\(695\) 5.00000 0.189661
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 11.0000 0.416058
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) 3.00000 0.112987
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) −18.0000 −0.673162
\(716\) 0 0
\(717\) −15.0000 −0.560185
\(718\) 0 0
\(719\) 35.0000 1.30528 0.652640 0.757668i \(-0.273661\pi\)
0.652640 + 0.757668i \(0.273661\pi\)
\(720\) 0 0
\(721\) 42.0000 1.56416
\(722\) 0 0
\(723\) −12.0000 −0.446285
\(724\) 0 0
\(725\) 40.0000 1.48556
\(726\) 0 0
\(727\) 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
\(730\) 0 0
\(731\) 3.00000 0.110959
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) −2.00000 −0.0737711
\(736\) 0 0
\(737\) −24.0000 −0.884051
\(738\) 0 0
\(739\) 45.0000 1.65535 0.827676 0.561206i \(-0.189663\pi\)
0.827676 + 0.561206i \(0.189663\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 15.0000 0.549557
\(746\) 0 0
\(747\) −4.00000 −0.146352
\(748\) 0 0
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) 0 0
\(753\) 27.0000 0.983935
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) 0 0
\(759\) 12.0000 0.435572
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) 0 0
\(763\) −60.0000 −2.17215
\(764\) 0 0
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −45.0000 −1.62274 −0.811371 0.584532i \(-0.801278\pi\)
−0.811371 + 0.584532i \(0.801278\pi\)
\(770\) 0 0
\(771\) −8.00000 −0.288113
\(772\) 0 0
\(773\) −36.0000 −1.29483 −0.647415 0.762138i \(-0.724150\pi\)
−0.647415 + 0.762138i \(0.724150\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 24.0000 0.860995
\(778\) 0 0
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 10.0000 0.357371
\(784\) 0 0
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) −8.00000 −0.285169 −0.142585 0.989783i \(-0.545541\pi\)
−0.142585 + 0.989783i \(0.545541\pi\)
\(788\) 0 0
\(789\) −21.0000 −0.747620
\(790\) 0 0
\(791\) 18.0000 0.640006
\(792\) 0 0
\(793\) −42.0000 −1.49146
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) −33.0000 −1.16454
\(804\) 0 0
\(805\) 12.0000 0.422944
\(806\) 0 0
\(807\) 30.0000 1.05605
\(808\) 0 0
\(809\) 5.00000 0.175791 0.0878953 0.996130i \(-0.471986\pi\)
0.0878953 + 0.996130i \(0.471986\pi\)
\(810\) 0 0
\(811\) −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
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 0 0
\(817\) 1.00000 0.0349856
\(818\) 0 0
\(819\) 18.0000 0.628971
\(820\) 0 0
\(821\) −33.0000 −1.15171 −0.575854 0.817553i \(-0.695330\pi\)
−0.575854 + 0.817553i \(0.695330\pi\)
\(822\) 0 0
\(823\) −19.0000 −0.662298 −0.331149 0.943578i \(-0.607436\pi\)
−0.331149 + 0.943578i \(0.607436\pi\)
\(824\) 0 0
\(825\) 12.0000 0.417786
\(826\) 0 0
\(827\) 52.0000 1.80822 0.904109 0.427303i \(-0.140536\pi\)
0.904109 + 0.427303i \(0.140536\pi\)
\(828\) 0 0
\(829\) 20.0000 0.694629 0.347314 0.937749i \(-0.387094\pi\)
0.347314 + 0.937749i \(0.387094\pi\)
\(830\) 0 0
\(831\) −13.0000 −0.450965
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) −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
\(842\) 0 0
\(843\) −2.00000 −0.0688837
\(844\) 0 0
\(845\) 23.0000 0.791224
\(846\) 0 0
\(847\) 6.00000 0.206162
\(848\) 0 0
\(849\) 19.0000 0.652078
\(850\) 0 0
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 34.0000 1.16414 0.582069 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 48.0000 1.63965 0.819824 0.572615i \(-0.194071\pi\)
0.819824 + 0.572615i \(0.194071\pi\)
\(858\) 0 0
\(859\) −35.0000 −1.19418 −0.597092 0.802173i \(-0.703677\pi\)
−0.597092 + 0.802173i \(0.703677\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) −44.0000 −1.49778 −0.748889 0.662696i \(-0.769412\pi\)
−0.748889 + 0.662696i \(0.769412\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 27.0000 0.912767
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −4.00000 −0.134917
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 0 0
\(883\) 21.0000 0.706706 0.353353 0.935490i \(-0.385041\pi\)
0.353353 + 0.935490i \(0.385041\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 52.0000 1.74599 0.872995 0.487730i \(-0.162175\pi\)
0.872995 + 0.487730i \(0.162175\pi\)
\(888\) 0 0
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 0 0
\(893\) −3.00000 −0.100391
\(894\) 0 0
\(895\) 10.0000 0.334263
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) −18.0000 −0.599667
\(902\) 0 0
\(903\) 3.00000 0.0998337
\(904\) 0 0
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) 2.00000 0.0664089 0.0332045 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) 0 0
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −2.00000 −0.0662630 −0.0331315 0.999451i \(-0.510548\pi\)
−0.0331315 + 0.999451i \(0.510548\pi\)
\(912\) 0 0
\(913\) −12.0000 −0.397142
\(914\) 0 0
\(915\) −7.00000 −0.231413
\(916\) 0 0
\(917\) −39.0000 −1.28789
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 0 0
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 0 0
\(927\) −14.0000 −0.459820
\(928\) 0 0
\(929\) −50.0000 −1.64045 −0.820223 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) 2.00000 0.0655474
\(932\) 0 0
\(933\) 7.00000 0.229170
\(934\) 0 0
\(935\) 9.00000 0.294331
\(936\) 0 0
\(937\) 53.0000 1.73143 0.865717 0.500533i \(-0.166863\pi\)
0.865717 + 0.500533i \(0.166863\pi\)
\(938\) 0 0
\(939\) −14.0000 −0.456873
\(940\) 0 0
\(941\) 2.00000 0.0651981 0.0325991 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) 0 0
\(945\) 3.00000 0.0975900
\(946\) 0 0
\(947\) −28.0000 −0.909878 −0.454939 0.890523i \(-0.650339\pi\)
−0.454939 + 0.890523i \(0.650339\pi\)
\(948\) 0 0
\(949\) 66.0000 2.14245
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −16.0000 −0.518291 −0.259145 0.965838i \(-0.583441\pi\)
−0.259145 + 0.965838i \(0.583441\pi\)
\(954\) 0 0
\(955\) 3.00000 0.0970777
\(956\) 0 0
\(957\) 30.0000 0.969762
\(958\) 0 0
\(959\) −9.00000 −0.290625
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 2.00000 0.0644491
\(964\) 0 0
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 32.0000 1.02905 0.514525 0.857475i \(-0.327968\pi\)
0.514525 + 0.857475i \(0.327968\pi\)
\(968\) 0 0
\(969\) −3.00000 −0.0963739
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) −15.0000 −0.480878
\(974\) 0 0
\(975\) −24.0000 −0.768615
\(976\) 0 0
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) 30.0000 0.958804
\(980\) 0 0
\(981\) 20.0000 0.638551
\(982\) 0 0
\(983\) −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
\(986\) 0 0
\(987\) −9.00000 −0.286473
\(988\) 0 0
\(989\) −4.00000 −0.127193
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 12.0000 0.380808
\(994\) 0 0
\(995\) 5.00000 0.158511
\(996\) 0 0
\(997\) −7.00000 −0.221692 −0.110846 0.993838i \(-0.535356\pi\)
−0.110846 + 0.993838i \(0.535356\pi\)
\(998\) 0 0
\(999\) −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 912.2.a.d.1.1 1
3.2 odd 2 2736.2.a.h.1.1 1
4.3 odd 2 57.2.a.b.1.1 1
8.3 odd 2 3648.2.a.h.1.1 1
8.5 even 2 3648.2.a.y.1.1 1
12.11 even 2 171.2.a.c.1.1 1
20.3 even 4 1425.2.c.a.799.2 2
20.7 even 4 1425.2.c.a.799.1 2
20.19 odd 2 1425.2.a.i.1.1 1
28.27 even 2 2793.2.a.a.1.1 1
44.43 even 2 6897.2.a.g.1.1 1
52.51 odd 2 9633.2.a.p.1.1 1
60.59 even 2 4275.2.a.a.1.1 1
76.75 even 2 1083.2.a.d.1.1 1
84.83 odd 2 8379.2.a.q.1.1 1
228.227 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 4.3 odd 2
171.2.a.c.1.1 1 12.11 even 2
912.2.a.d.1.1 1 1.1 even 1 trivial
1083.2.a.d.1.1 1 76.75 even 2
1425.2.a.i.1.1 1 20.19 odd 2
1425.2.c.a.799.1 2 20.7 even 4
1425.2.c.a.799.2 2 20.3 even 4
2736.2.a.h.1.1 1 3.2 odd 2
2793.2.a.a.1.1 1 28.27 even 2
3249.2.a.a.1.1 1 228.227 odd 2
3648.2.a.h.1.1 1 8.3 odd 2
3648.2.a.y.1.1 1 8.5 even 2
4275.2.a.a.1.1 1 60.59 even 2
6897.2.a.g.1.1 1 44.43 even 2
8379.2.a.q.1.1 1 84.83 odd 2
9633.2.a.p.1.1 1 52.51 odd 2