Properties

Label 1225.2.a.d.1.1
Level $1225$
Weight $2$
Character 1225.1
Self dual yes
Analytic conductor $9.782$
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] = [1225,2,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.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: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{8} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{8} -2.00000 q^{9} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{16} -2.00000 q^{17} +2.00000 q^{18} +6.00000 q^{19} +3.00000 q^{23} +3.00000 q^{24} +2.00000 q^{26} -5.00000 q^{27} +7.00000 q^{29} +2.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} +2.00000 q^{36} +8.00000 q^{37} -6.00000 q^{38} -2.00000 q^{39} +5.00000 q^{41} -7.00000 q^{43} -3.00000 q^{46} -1.00000 q^{48} -2.00000 q^{51} +2.00000 q^{52} -6.00000 q^{53} +5.00000 q^{54} +6.00000 q^{57} -7.00000 q^{58} +10.0000 q^{59} +7.00000 q^{61} -2.00000 q^{62} +7.00000 q^{64} +5.00000 q^{67} +2.00000 q^{68} +3.00000 q^{69} -2.00000 q^{71} -6.00000 q^{72} +6.00000 q^{73} -8.00000 q^{74} -6.00000 q^{76} +2.00000 q^{78} -2.00000 q^{79} +1.00000 q^{81} -5.00000 q^{82} +11.0000 q^{83} +7.00000 q^{86} +7.00000 q^{87} +9.00000 q^{89} -3.00000 q^{92} +2.00000 q^{93} -5.00000 q^{96} +16.0000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350 0.288675 0.957427i \(-0.406785\pi\)
0.288675 + 0.957427i \(0.406785\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 3.00000 1.06066
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 2.00000 0.471405
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −6.00000 −0.973329
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.00000 −0.442326
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 5.00000 0.680414
\(55\) 0 0
\(56\) 0 0
\(57\) 6.00000 0.794719
\(58\) −7.00000 −0.919145
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 7.00000 0.896258 0.448129 0.893969i \(-0.352090\pi\)
0.448129 + 0.893969i \(0.352090\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 2.00000 0.242536
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −6.00000 −0.707107
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 2.00000 0.226455
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −5.00000 −0.552158
\(83\) 11.0000 1.20741 0.603703 0.797209i \(-0.293691\pi\)
0.603703 + 0.797209i \(0.293691\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.00000 0.754829
\(87\) 7.00000 0.750479
\(88\) 0 0
\(89\) 9.00000 0.953998 0.476999 0.878904i \(-0.341725\pi\)
0.476999 + 0.878904i \(0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.00000 −0.312772
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 9.00000 0.895533 0.447767 0.894150i \(-0.352219\pi\)
0.447767 + 0.894150i \(0.352219\pi\)
\(102\) 2.00000 0.198030
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −11.0000 −1.06341 −0.531705 0.846930i \(-0.678449\pi\)
−0.531705 + 0.846930i \(0.678449\pi\)
\(108\) 5.00000 0.481125
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 8.00000 0.759326
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) −7.00000 −0.649934
\(117\) 4.00000 0.369800
\(118\) −10.0000 −0.920575
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −7.00000 −0.633750
\(123\) 5.00000 0.450835
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 3.00000 0.265165
\(129\) −7.00000 −0.616316
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −5.00000 −0.431934
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) −3.00000 −0.255377
\(139\) 8.00000 0.678551 0.339276 0.940687i \(-0.389818\pi\)
0.339276 + 0.940687i \(0.389818\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 2.00000 0.167836
\(143\) 0 0
\(144\) 2.00000 0.166667
\(145\) 0 0
\(146\) −6.00000 −0.496564
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 1.00000 0.0819232 0.0409616 0.999161i \(-0.486958\pi\)
0.0409616 + 0.999161i \(0.486958\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 18.0000 1.45999
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −12.0000 −0.957704 −0.478852 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 2.00000 0.159111
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) −11.0000 −0.853766
\(167\) 3.00000 0.232147 0.116073 0.993241i \(-0.462969\pi\)
0.116073 + 0.993241i \(0.462969\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −12.0000 −0.917663
\(172\) 7.00000 0.533745
\(173\) 12.0000 0.912343 0.456172 0.889892i \(-0.349220\pi\)
0.456172 + 0.889892i \(0.349220\pi\)
\(174\) −7.00000 −0.530669
\(175\) 0 0
\(176\) 0 0
\(177\) 10.0000 0.751646
\(178\) −9.00000 −0.674579
\(179\) −2.00000 −0.149487 −0.0747435 0.997203i \(-0.523814\pi\)
−0.0747435 + 0.997203i \(0.523814\pi\)
\(180\) 0 0
\(181\) −3.00000 −0.222988 −0.111494 0.993765i \(-0.535564\pi\)
−0.111494 + 0.993765i \(0.535564\pi\)
\(182\) 0 0
\(183\) 7.00000 0.517455
\(184\) 9.00000 0.663489
\(185\) 0 0
\(186\) −2.00000 −0.146647
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 7.00000 0.505181
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 0 0
\(197\) 8.00000 0.569976 0.284988 0.958531i \(-0.408010\pi\)
0.284988 + 0.958531i \(0.408010\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) −9.00000 −0.633238
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 7.00000 0.487713
\(207\) −6.00000 −0.417029
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 6.00000 0.412082
\(213\) −2.00000 −0.137038
\(214\) 11.0000 0.751945
\(215\) 0 0
\(216\) −15.0000 −1.02062
\(217\) 0 0
\(218\) −5.00000 −0.338643
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 4.00000 0.269069
\(222\) −8.00000 −0.536925
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) −6.00000 −0.397360
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 21.0000 1.37872
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) −2.00000 −0.129914
\(238\) 0 0
\(239\) −18.0000 −1.16432 −0.582162 0.813073i \(-0.697793\pi\)
−0.582162 + 0.813073i \(0.697793\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 11.0000 0.707107
\(243\) 16.0000 1.02640
\(244\) −7.00000 −0.448129
\(245\) 0 0
\(246\) −5.00000 −0.318788
\(247\) −12.0000 −0.763542
\(248\) 6.00000 0.381000
\(249\) 11.0000 0.697097
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) −20.0000 −1.24757 −0.623783 0.781598i \(-0.714405\pi\)
−0.623783 + 0.781598i \(0.714405\pi\)
\(258\) 7.00000 0.435801
\(259\) 0 0
\(260\) 0 0
\(261\) −14.0000 −0.866578
\(262\) −4.00000 −0.247121
\(263\) 9.00000 0.554964 0.277482 0.960731i \(-0.410500\pi\)
0.277482 + 0.960731i \(0.410500\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 9.00000 0.550791
\(268\) −5.00000 −0.305424
\(269\) −11.0000 −0.670682 −0.335341 0.942097i \(-0.608852\pi\)
−0.335341 + 0.942097i \(0.608852\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −3.00000 −0.180579
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −8.00000 −0.479808
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) 2.00000 0.118678
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 10.0000 0.589256
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) −6.00000 −0.351123
\(293\) −24.0000 −1.40209 −0.701047 0.713115i \(-0.747284\pi\)
−0.701047 + 0.713115i \(0.747284\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 24.0000 1.39497
\(297\) 0 0
\(298\) −1.00000 −0.0579284
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) 0 0
\(302\) −14.0000 −0.805609
\(303\) 9.00000 0.517036
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −23.0000 −1.31268 −0.656340 0.754466i \(-0.727896\pi\)
−0.656340 + 0.754466i \(0.727896\pi\)
\(308\) 0 0
\(309\) −7.00000 −0.398216
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) −6.00000 −0.339683
\(313\) −24.0000 −1.35656 −0.678280 0.734803i \(-0.737274\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) 6.00000 0.336994 0.168497 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 6.00000 0.336463
\(319\) 0 0
\(320\) 0 0
\(321\) −11.0000 −0.613960
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 5.00000 0.276501
\(328\) 15.0000 0.828236
\(329\) 0 0
\(330\) 0 0
\(331\) 6.00000 0.329790 0.164895 0.986311i \(-0.447272\pi\)
0.164895 + 0.986311i \(0.447272\pi\)
\(332\) −11.0000 −0.603703
\(333\) −16.0000 −0.876795
\(334\) −3.00000 −0.164153
\(335\) 0 0
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 12.0000 0.648886
\(343\) 0 0
\(344\) −21.0000 −1.13224
\(345\) 0 0
\(346\) −12.0000 −0.645124
\(347\) −15.0000 −0.805242 −0.402621 0.915367i \(-0.631901\pi\)
−0.402621 + 0.915367i \(0.631901\pi\)
\(348\) −7.00000 −0.375239
\(349\) −17.0000 −0.909989 −0.454995 0.890494i \(-0.650359\pi\)
−0.454995 + 0.890494i \(0.650359\pi\)
\(350\) 0 0
\(351\) 10.0000 0.533761
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) −10.0000 −0.531494
\(355\) 0 0
\(356\) −9.00000 −0.476999
\(357\) 0 0
\(358\) 2.00000 0.105703
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 3.00000 0.157676
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 0 0
\(366\) −7.00000 −0.365896
\(367\) −27.0000 −1.40939 −0.704694 0.709511i \(-0.748916\pi\)
−0.704694 + 0.709511i \(0.748916\pi\)
\(368\) −3.00000 −0.156386
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) 24.0000 1.24267 0.621336 0.783544i \(-0.286590\pi\)
0.621336 + 0.783544i \(0.286590\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −14.0000 −0.721037
\(378\) 0 0
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) −12.0000 −0.613973
\(383\) −5.00000 −0.255488 −0.127744 0.991807i \(-0.540774\pi\)
−0.127744 + 0.991807i \(0.540774\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 14.0000 0.711660
\(388\) −16.0000 −0.812277
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) −6.00000 −0.303433
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) −8.00000 −0.403034
\(395\) 0 0
\(396\) 0 0
\(397\) 16.0000 0.803017 0.401508 0.915855i \(-0.368486\pi\)
0.401508 + 0.915855i \(0.368486\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 0 0
\(401\) 3.00000 0.149813 0.0749064 0.997191i \(-0.476134\pi\)
0.0749064 + 0.997191i \(0.476134\pi\)
\(402\) −5.00000 −0.249377
\(403\) −4.00000 −0.199254
\(404\) −9.00000 −0.447767
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) −6.00000 −0.297044
\(409\) −25.0000 −1.23617 −0.618085 0.786111i \(-0.712091\pi\)
−0.618085 + 0.786111i \(0.712091\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7.00000 0.344865
\(413\) 0 0
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) 10.0000 0.490290
\(417\) 8.00000 0.391762
\(418\) 0 0
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) 0 0
\(421\) 15.0000 0.731055 0.365528 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) 10.0000 0.486792
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) 2.00000 0.0969003
\(427\) 0 0
\(428\) 11.0000 0.531705
\(429\) 0 0
\(430\) 0 0
\(431\) −32.0000 −1.54139 −0.770693 0.637207i \(-0.780090\pi\)
−0.770693 + 0.637207i \(0.780090\pi\)
\(432\) 5.00000 0.240563
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.00000 −0.239457
\(437\) 18.0000 0.861057
\(438\) −6.00000 −0.286691
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −4.00000 −0.190261
\(443\) −31.0000 −1.47285 −0.736427 0.676517i \(-0.763489\pi\)
−0.736427 + 0.676517i \(0.763489\pi\)
\(444\) −8.00000 −0.379663
\(445\) 0 0
\(446\) 0 0
\(447\) 1.00000 0.0472984
\(448\) 0 0
\(449\) 31.0000 1.46298 0.731490 0.681852i \(-0.238825\pi\)
0.731490 + 0.681852i \(0.238825\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.00000 −0.282216
\(453\) 14.0000 0.657777
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 18.0000 0.842927
\(457\) −32.0000 −1.49690 −0.748448 0.663193i \(-0.769201\pi\)
−0.748448 + 0.663193i \(0.769201\pi\)
\(458\) 22.0000 1.02799
\(459\) 10.0000 0.466760
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) −3.00000 −0.139422 −0.0697109 0.997567i \(-0.522208\pi\)
−0.0697109 + 0.997567i \(0.522208\pi\)
\(464\) −7.00000 −0.324967
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) 30.0000 1.38086
\(473\) 0 0
\(474\) 2.00000 0.0918630
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 18.0000 0.823301
\(479\) 42.0000 1.91903 0.959514 0.281659i \(-0.0908848\pi\)
0.959514 + 0.281659i \(0.0908848\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 14.0000 0.637683
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) −16.0000 −0.725775
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 21.0000 0.950625
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −26.0000 −1.17336 −0.586682 0.809818i \(-0.699566\pi\)
−0.586682 + 0.809818i \(0.699566\pi\)
\(492\) −5.00000 −0.225417
\(493\) −14.0000 −0.630528
\(494\) 12.0000 0.539906
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) 0 0
\(498\) −11.0000 −0.492922
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 3.00000 0.134030
\(502\) −30.0000 −1.33897
\(503\) 15.0000 0.668817 0.334408 0.942428i \(-0.391463\pi\)
0.334408 + 0.942428i \(0.391463\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 16.0000 0.709885
\(509\) 27.0000 1.19675 0.598377 0.801215i \(-0.295813\pi\)
0.598377 + 0.801215i \(0.295813\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.0000 0.486136
\(513\) −30.0000 −1.32453
\(514\) 20.0000 0.882162
\(515\) 0 0
\(516\) 7.00000 0.308158
\(517\) 0 0
\(518\) 0 0
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 14.0000 0.612763
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −9.00000 −0.392419
\(527\) −4.00000 −0.174243
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −20.0000 −0.867926
\(532\) 0 0
\(533\) −10.0000 −0.433148
\(534\) −9.00000 −0.389468
\(535\) 0 0
\(536\) 15.0000 0.647901
\(537\) −2.00000 −0.0863064
\(538\) 11.0000 0.474244
\(539\) 0 0
\(540\) 0 0
\(541\) −3.00000 −0.128980 −0.0644900 0.997918i \(-0.520542\pi\)
−0.0644900 + 0.997918i \(0.520542\pi\)
\(542\) 2.00000 0.0859074
\(543\) −3.00000 −0.128742
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) 0 0
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) 42.0000 1.78926
\(552\) 9.00000 0.383065
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −8.00000 −0.339276
\(557\) 44.0000 1.86434 0.932170 0.362021i \(-0.117913\pi\)
0.932170 + 0.362021i \(0.117913\pi\)
\(558\) 4.00000 0.169334
\(559\) 14.0000 0.592137
\(560\) 0 0
\(561\) 0 0
\(562\) 14.0000 0.590554
\(563\) −33.0000 −1.39078 −0.695392 0.718631i \(-0.744769\pi\)
−0.695392 + 0.718631i \(0.744769\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) −18.0000 −0.753277 −0.376638 0.926360i \(-0.622920\pi\)
−0.376638 + 0.926360i \(0.622920\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 0 0
\(576\) −14.0000 −0.583333
\(577\) 32.0000 1.33218 0.666089 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) 13.0000 0.540729
\(579\) 4.00000 0.166234
\(580\) 0 0
\(581\) 0 0
\(582\) −16.0000 −0.663221
\(583\) 0 0
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) 0 0
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 8.00000 0.329076
\(592\) −8.00000 −0.328798
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.00000 −0.0409616
\(597\) −4.00000 −0.163709
\(598\) 6.00000 0.245358
\(599\) −44.0000 −1.79779 −0.898896 0.438163i \(-0.855629\pi\)
−0.898896 + 0.438163i \(0.855629\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) −9.00000 −0.365600
\(607\) 33.0000 1.33943 0.669714 0.742619i \(-0.266417\pi\)
0.669714 + 0.742619i \(0.266417\pi\)
\(608\) −30.0000 −1.21666
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −4.00000 −0.161690
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 23.0000 0.928204
\(615\) 0 0
\(616\) 0 0
\(617\) 4.00000 0.161034 0.0805170 0.996753i \(-0.474343\pi\)
0.0805170 + 0.996753i \(0.474343\pi\)
\(618\) 7.00000 0.281581
\(619\) 28.0000 1.12542 0.562708 0.826656i \(-0.309760\pi\)
0.562708 + 0.826656i \(0.309760\pi\)
\(620\) 0 0
\(621\) −15.0000 −0.601929
\(622\) −30.0000 −1.20289
\(623\) 0 0
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 24.0000 0.959233
\(627\) 0 0
\(628\) 12.0000 0.478852
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −6.00000 −0.238667
\(633\) −10.0000 −0.397464
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) 4.00000 0.158238
\(640\) 0 0
\(641\) −11.0000 −0.434474 −0.217237 0.976119i \(-0.569704\pi\)
−0.217237 + 0.976119i \(0.569704\pi\)
\(642\) 11.0000 0.434135
\(643\) 4.00000 0.157745 0.0788723 0.996885i \(-0.474868\pi\)
0.0788723 + 0.996885i \(0.474868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) −47.0000 −1.84776 −0.923880 0.382682i \(-0.875001\pi\)
−0.923880 + 0.382682i \(0.875001\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 26.0000 1.01746 0.508729 0.860927i \(-0.330115\pi\)
0.508729 + 0.860927i \(0.330115\pi\)
\(654\) −5.00000 −0.195515
\(655\) 0 0
\(656\) −5.00000 −0.195217
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) 6.00000 0.233727 0.116863 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) −9.00000 −0.350059 −0.175030 0.984563i \(-0.556002\pi\)
−0.175030 + 0.984563i \(0.556002\pi\)
\(662\) −6.00000 −0.233197
\(663\) 4.00000 0.155347
\(664\) 33.0000 1.28065
\(665\) 0 0
\(666\) 16.0000 0.619987
\(667\) 21.0000 0.813123
\(668\) −3.00000 −0.116073
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 18.0000 0.693334
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 14.0000 0.538064 0.269032 0.963131i \(-0.413296\pi\)
0.269032 + 0.963131i \(0.413296\pi\)
\(678\) −6.00000 −0.230429
\(679\) 0 0
\(680\) 0 0
\(681\) 20.0000 0.766402
\(682\) 0 0
\(683\) −35.0000 −1.33924 −0.669619 0.742705i \(-0.733543\pi\)
−0.669619 + 0.742705i \(0.733543\pi\)
\(684\) 12.0000 0.458831
\(685\) 0 0
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) 7.00000 0.266872
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −12.0000 −0.456172
\(693\) 0 0
\(694\) 15.0000 0.569392
\(695\) 0 0
\(696\) 21.0000 0.796003
\(697\) −10.0000 −0.378777
\(698\) 17.0000 0.643459
\(699\) −14.0000 −0.529529
\(700\) 0 0
\(701\) −1.00000 −0.0377695 −0.0188847 0.999822i \(-0.506012\pi\)
−0.0188847 + 0.999822i \(0.506012\pi\)
\(702\) −10.0000 −0.377426
\(703\) 48.0000 1.81035
\(704\) 0 0
\(705\) 0 0
\(706\) −26.0000 −0.978523
\(707\) 0 0
\(708\) −10.0000 −0.375823
\(709\) −49.0000 −1.84023 −0.920117 0.391644i \(-0.871906\pi\)
−0.920117 + 0.391644i \(0.871906\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 27.0000 1.01187
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) −18.0000 −0.672222
\(718\) −10.0000 −0.373197
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) −14.0000 −0.520666
\(724\) 3.00000 0.111494
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) 47.0000 1.74313 0.871567 0.490277i \(-0.163104\pi\)
0.871567 + 0.490277i \(0.163104\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 14.0000 0.517809
\(732\) −7.00000 −0.258727
\(733\) 16.0000 0.590973 0.295487 0.955347i \(-0.404518\pi\)
0.295487 + 0.955347i \(0.404518\pi\)
\(734\) 27.0000 0.996588
\(735\) 0 0
\(736\) −15.0000 −0.552907
\(737\) 0 0
\(738\) 10.0000 0.368105
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) 0 0
\(743\) 49.0000 1.79764 0.898818 0.438322i \(-0.144427\pi\)
0.898818 + 0.438322i \(0.144427\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) −22.0000 −0.804938
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −12.0000 −0.437886 −0.218943 0.975738i \(-0.570261\pi\)
−0.218943 + 0.975738i \(0.570261\pi\)
\(752\) 0 0
\(753\) 30.0000 1.09326
\(754\) 14.0000 0.509850
\(755\) 0 0
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 10.0000 0.363216
\(759\) 0 0
\(760\) 0 0
\(761\) 14.0000 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(762\) 16.0000 0.579619
\(763\) 0 0
\(764\) −12.0000 −0.434145
\(765\) 0 0
\(766\) 5.00000 0.180657
\(767\) −20.0000 −0.722158
\(768\) −17.0000 −0.613435
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) −20.0000 −0.720282
\(772\) −4.00000 −0.143963
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) −14.0000 −0.503220
\(775\) 0 0
\(776\) 48.0000 1.72310
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) 30.0000 1.07486
\(780\) 0 0
\(781\) 0 0
\(782\) 6.00000 0.214560
\(783\) −35.0000 −1.25080
\(784\) 0 0
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 17.0000 0.605985 0.302992 0.952993i \(-0.402014\pi\)
0.302992 + 0.952993i \(0.402014\pi\)
\(788\) −8.00000 −0.284988
\(789\) 9.00000 0.320408
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −14.0000 −0.497155
\(794\) −16.0000 −0.567819
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) −40.0000 −1.41687 −0.708436 0.705775i \(-0.750599\pi\)
−0.708436 + 0.705775i \(0.750599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) −3.00000 −0.105934
\(803\) 0 0
\(804\) −5.00000 −0.176336
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) −11.0000 −0.387218
\(808\) 27.0000 0.949857
\(809\) 9.00000 0.316423 0.158212 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 0 0
\(815\) 0 0
\(816\) 2.00000 0.0700140
\(817\) −42.0000 −1.46939
\(818\) 25.0000 0.874105
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) −25.0000 −0.871445 −0.435723 0.900081i \(-0.643507\pi\)
−0.435723 + 0.900081i \(0.643507\pi\)
\(824\) −21.0000 −0.731570
\(825\) 0 0
\(826\) 0 0
\(827\) 39.0000 1.35616 0.678081 0.734987i \(-0.262812\pi\)
0.678081 + 0.734987i \(0.262812\pi\)
\(828\) 6.00000 0.208514
\(829\) −30.0000 −1.04194 −0.520972 0.853574i \(-0.674430\pi\)
−0.520972 + 0.853574i \(0.674430\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) −14.0000 −0.485363
\(833\) 0 0
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) −24.0000 −0.829066
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) −15.0000 −0.516934
\(843\) −14.0000 −0.482186
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 2.00000 0.0685189
\(853\) 6.00000 0.205436 0.102718 0.994711i \(-0.467246\pi\)
0.102718 + 0.994711i \(0.467246\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −33.0000 −1.12792
\(857\) 12.0000 0.409912 0.204956 0.978771i \(-0.434295\pi\)
0.204956 + 0.978771i \(0.434295\pi\)
\(858\) 0 0
\(859\) −32.0000 −1.09183 −0.545913 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 32.0000 1.08992
\(863\) 9.00000 0.306364 0.153182 0.988198i \(-0.451048\pi\)
0.153182 + 0.988198i \(0.451048\pi\)
\(864\) 25.0000 0.850517
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −10.0000 −0.338837
\(872\) 15.0000 0.507964
\(873\) −32.0000 −1.08304
\(874\) −18.0000 −0.608859
\(875\) 0 0
\(876\) −6.00000 −0.202721
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −24.0000 −0.809961
\(879\) −24.0000 −0.809500
\(880\) 0 0
\(881\) −43.0000 −1.44871 −0.724353 0.689429i \(-0.757862\pi\)
−0.724353 + 0.689429i \(0.757862\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 31.0000 1.04147
\(887\) −43.0000 −1.44380 −0.721899 0.691998i \(-0.756731\pi\)
−0.721899 + 0.691998i \(0.756731\pi\)
\(888\) 24.0000 0.805387
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) −1.00000 −0.0334450
\(895\) 0 0
\(896\) 0 0
\(897\) −6.00000 −0.200334
\(898\) −31.0000 −1.03448
\(899\) 14.0000 0.466926
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) −14.0000 −0.465119
\(907\) 19.0000 0.630885 0.315442 0.948945i \(-0.397847\pi\)
0.315442 + 0.948945i \(0.397847\pi\)
\(908\) −20.0000 −0.663723
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) −6.00000 −0.198680
\(913\) 0 0
\(914\) 32.0000 1.05847
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 0 0
\(918\) −10.0000 −0.330049
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) −23.0000 −0.757876
\(922\) −18.0000 −0.592798
\(923\) 4.00000 0.131662
\(924\) 0 0
\(925\) 0 0
\(926\) 3.00000 0.0985861
\(927\) 14.0000 0.459820
\(928\) −35.0000 −1.14893
\(929\) −29.0000 −0.951459 −0.475730 0.879592i \(-0.657816\pi\)
−0.475730 + 0.879592i \(0.657816\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 14.0000 0.458585
\(933\) 30.0000 0.982156
\(934\) 3.00000 0.0981630
\(935\) 0 0
\(936\) 12.0000 0.392232
\(937\) 8.00000 0.261349 0.130674 0.991425i \(-0.458286\pi\)
0.130674 + 0.991425i \(0.458286\pi\)
\(938\) 0 0
\(939\) −24.0000 −0.783210
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) 12.0000 0.390981
\(943\) 15.0000 0.488467
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 0 0
\(947\) 57.0000 1.85225 0.926126 0.377215i \(-0.123118\pi\)
0.926126 + 0.377215i \(0.123118\pi\)
\(948\) 2.00000 0.0649570
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −60.0000 −1.94359 −0.971795 0.235826i \(-0.924220\pi\)
−0.971795 + 0.235826i \(0.924220\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) 18.0000 0.582162
\(957\) 0 0
\(958\) −42.0000 −1.35696
\(959\) 0 0
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 16.0000 0.515861
\(963\) 22.0000 0.708940
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 13.0000 0.418052 0.209026 0.977910i \(-0.432971\pi\)
0.209026 + 0.977910i \(0.432971\pi\)
\(968\) −33.0000 −1.06066
\(969\) −12.0000 −0.385496
\(970\) 0 0
\(971\) −32.0000 −1.02693 −0.513464 0.858111i \(-0.671638\pi\)
−0.513464 + 0.858111i \(0.671638\pi\)
\(972\) −16.0000 −0.513200
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) −60.0000 −1.91957 −0.959785 0.280736i \(-0.909421\pi\)
−0.959785 + 0.280736i \(0.909421\pi\)
\(978\) −4.00000 −0.127906
\(979\) 0 0
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 26.0000 0.829693
\(983\) −3.00000 −0.0956851 −0.0478426 0.998855i \(-0.515235\pi\)
−0.0478426 + 0.998855i \(0.515235\pi\)
\(984\) 15.0000 0.478183
\(985\) 0 0
\(986\) 14.0000 0.445851
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −21.0000 −0.667761
\(990\) 0 0
\(991\) −50.0000 −1.58830 −0.794151 0.607720i \(-0.792084\pi\)
−0.794151 + 0.607720i \(0.792084\pi\)
\(992\) −10.0000 −0.317500
\(993\) 6.00000 0.190404
\(994\) 0 0
\(995\) 0 0
\(996\) −11.0000 −0.348548
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −16.0000 −0.506471
\(999\) −40.0000 −1.26554
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 1225.2.a.d.1.1 1
5.2 odd 4 245.2.b.c.99.1 2
5.3 odd 4 245.2.b.c.99.2 2
5.4 even 2 1225.2.a.f.1.1 1
7.2 even 3 175.2.e.b.151.1 2
7.4 even 3 175.2.e.b.51.1 2
7.6 odd 2 1225.2.a.b.1.1 1
15.2 even 4 2205.2.d.d.1324.2 2
15.8 even 4 2205.2.d.d.1324.1 2
35.2 odd 12 35.2.j.a.4.1 4
35.3 even 12 245.2.j.c.79.1 4
35.4 even 6 175.2.e.a.51.1 2
35.9 even 6 175.2.e.a.151.1 2
35.12 even 12 245.2.j.c.214.1 4
35.13 even 4 245.2.b.b.99.2 2
35.17 even 12 245.2.j.c.79.2 4
35.18 odd 12 35.2.j.a.9.1 yes 4
35.23 odd 12 35.2.j.a.4.2 yes 4
35.27 even 4 245.2.b.b.99.1 2
35.32 odd 12 35.2.j.a.9.2 yes 4
35.33 even 12 245.2.j.c.214.2 4
35.34 odd 2 1225.2.a.g.1.1 1
105.2 even 12 315.2.bf.a.109.2 4
105.23 even 12 315.2.bf.a.109.1 4
105.32 even 12 315.2.bf.a.289.1 4
105.53 even 12 315.2.bf.a.289.2 4
105.62 odd 4 2205.2.d.e.1324.2 2
105.83 odd 4 2205.2.d.e.1324.1 2
140.23 even 12 560.2.bw.b.529.2 4
140.67 even 12 560.2.bw.b.289.2 4
140.107 even 12 560.2.bw.b.529.1 4
140.123 even 12 560.2.bw.b.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.2.j.a.4.1 4 35.2 odd 12
35.2.j.a.4.2 yes 4 35.23 odd 12
35.2.j.a.9.1 yes 4 35.18 odd 12
35.2.j.a.9.2 yes 4 35.32 odd 12
175.2.e.a.51.1 2 35.4 even 6
175.2.e.a.151.1 2 35.9 even 6
175.2.e.b.51.1 2 7.4 even 3
175.2.e.b.151.1 2 7.2 even 3
245.2.b.b.99.1 2 35.27 even 4
245.2.b.b.99.2 2 35.13 even 4
245.2.b.c.99.1 2 5.2 odd 4
245.2.b.c.99.2 2 5.3 odd 4
245.2.j.c.79.1 4 35.3 even 12
245.2.j.c.79.2 4 35.17 even 12
245.2.j.c.214.1 4 35.12 even 12
245.2.j.c.214.2 4 35.33 even 12
315.2.bf.a.109.1 4 105.23 even 12
315.2.bf.a.109.2 4 105.2 even 12
315.2.bf.a.289.1 4 105.32 even 12
315.2.bf.a.289.2 4 105.53 even 12
560.2.bw.b.289.1 4 140.123 even 12
560.2.bw.b.289.2 4 140.67 even 12
560.2.bw.b.529.1 4 140.107 even 12
560.2.bw.b.529.2 4 140.23 even 12
1225.2.a.b.1.1 1 7.6 odd 2
1225.2.a.d.1.1 1 1.1 even 1 trivial
1225.2.a.f.1.1 1 5.4 even 2
1225.2.a.g.1.1 1 35.34 odd 2
2205.2.d.d.1324.1 2 15.8 even 4
2205.2.d.d.1324.2 2 15.2 even 4
2205.2.d.e.1324.1 2 105.83 odd 4
2205.2.d.e.1324.2 2 105.62 odd 4