Properties

Label 8281.2.a.i.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, 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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -3.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} -3.00000 q^{6} -3.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -3.00000 q^{3} -1.00000 q^{4} +3.00000 q^{5} -3.00000 q^{6} -3.00000 q^{8} +6.00000 q^{9} +3.00000 q^{10} -3.00000 q^{11} +3.00000 q^{12} -9.00000 q^{15} -1.00000 q^{16} -2.00000 q^{17} +6.00000 q^{18} -1.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} +9.00000 q^{24} +4.00000 q^{25} -9.00000 q^{27} +7.00000 q^{29} -9.00000 q^{30} +3.00000 q^{31} +5.00000 q^{32} +9.00000 q^{33} -2.00000 q^{34} -6.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} -9.00000 q^{40} +3.00000 q^{41} -7.00000 q^{43} +3.00000 q^{44} +18.0000 q^{45} +1.00000 q^{47} +3.00000 q^{48} +4.00000 q^{50} +6.00000 q^{51} +3.00000 q^{53} -9.00000 q^{54} -9.00000 q^{55} +3.00000 q^{57} +7.00000 q^{58} -4.00000 q^{59} +9.00000 q^{60} -13.0000 q^{61} +3.00000 q^{62} +7.00000 q^{64} +9.00000 q^{66} -3.00000 q^{67} +2.00000 q^{68} +13.0000 q^{71} -18.0000 q^{72} -13.0000 q^{73} +2.00000 q^{74} -12.0000 q^{75} +1.00000 q^{76} -3.00000 q^{79} -3.00000 q^{80} +9.00000 q^{81} +3.00000 q^{82} -6.00000 q^{85} -7.00000 q^{86} -21.0000 q^{87} +9.00000 q^{88} +6.00000 q^{89} +18.0000 q^{90} -9.00000 q^{93} +1.00000 q^{94} -3.00000 q^{95} -15.0000 q^{96} -5.00000 q^{97} -18.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) −3.00000 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(4\) −1.00000 −0.500000
\(5\) 3.00000 1.34164 0.670820 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) −3.00000 −1.22474
\(7\) 0 0
\(8\) −3.00000 −1.06066
\(9\) 6.00000 2.00000
\(10\) 3.00000 0.948683
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 3.00000 0.866025
\(13\) 0 0
\(14\) 0 0
\(15\) −9.00000 −2.32379
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 6.00000 1.41421
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 9.00000 1.83712
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) −9.00000 −1.73205
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) −9.00000 −1.64317
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 5.00000 0.883883
\(33\) 9.00000 1.56670
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) −9.00000 −1.42302
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) 3.00000 0.452267
\(45\) 18.0000 2.68328
\(46\) 0 0
\(47\) 1.00000 0.145865 0.0729325 0.997337i \(-0.476764\pi\)
0.0729325 + 0.997337i \(0.476764\pi\)
\(48\) 3.00000 0.433013
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) −9.00000 −1.22474
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) 3.00000 0.397360
\(58\) 7.00000 0.919145
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 9.00000 1.16190
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 9.00000 1.10782
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) 0 0
\(71\) 13.0000 1.54282 0.771408 0.636341i \(-0.219553\pi\)
0.771408 + 0.636341i \(0.219553\pi\)
\(72\) −18.0000 −2.12132
\(73\) −13.0000 −1.52153 −0.760767 0.649025i \(-0.775177\pi\)
−0.760767 + 0.649025i \(0.775177\pi\)
\(74\) 2.00000 0.232495
\(75\) −12.0000 −1.38564
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 0 0
\(79\) −3.00000 −0.337526 −0.168763 0.985657i \(-0.553977\pi\)
−0.168763 + 0.985657i \(0.553977\pi\)
\(80\) −3.00000 −0.335410
\(81\) 9.00000 1.00000
\(82\) 3.00000 0.331295
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) −7.00000 −0.754829
\(87\) −21.0000 −2.25144
\(88\) 9.00000 0.959403
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 18.0000 1.89737
\(91\) 0 0
\(92\) 0 0
\(93\) −9.00000 −0.933257
\(94\) 1.00000 0.103142
\(95\) −3.00000 −0.307794
\(96\) −15.0000 −1.53093
\(97\) −5.00000 −0.507673 −0.253837 0.967247i \(-0.581693\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 0 0
\(99\) −18.0000 −1.80907
\(100\) −4.00000 −0.400000
\(101\) −5.00000 −0.497519 −0.248759 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(102\) 6.00000 0.594089
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 9.00000 0.866025
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) −9.00000 −0.858116
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 3.00000 0.280976
\(115\) 0 0
\(116\) −7.00000 −0.649934
\(117\) 0 0
\(118\) −4.00000 −0.368230
\(119\) 0 0
\(120\) 27.0000 2.46475
\(121\) −2.00000 −0.181818
\(122\) −13.0000 −1.17696
\(123\) −9.00000 −0.811503
\(124\) −3.00000 −0.269408
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) −3.00000 −0.265165
\(129\) 21.0000 1.84895
\(130\) 0 0
\(131\) 5.00000 0.436852 0.218426 0.975854i \(-0.429908\pi\)
0.218426 + 0.975854i \(0.429908\pi\)
\(132\) −9.00000 −0.783349
\(133\) 0 0
\(134\) −3.00000 −0.259161
\(135\) −27.0000 −2.32379
\(136\) 6.00000 0.514496
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) −15.0000 −1.27228 −0.636142 0.771572i \(-0.719471\pi\)
−0.636142 + 0.771572i \(0.719471\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) 13.0000 1.09094
\(143\) 0 0
\(144\) −6.00000 −0.500000
\(145\) 21.0000 1.74396
\(146\) −13.0000 −1.07589
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) 15.0000 1.22885 0.614424 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) −12.0000 −0.979796
\(151\) −21.0000 −1.70896 −0.854478 0.519488i \(-0.826123\pi\)
−0.854478 + 0.519488i \(0.826123\pi\)
\(152\) 3.00000 0.243332
\(153\) −12.0000 −0.970143
\(154\) 0 0
\(155\) 9.00000 0.722897
\(156\) 0 0
\(157\) 19.0000 1.51637 0.758183 0.652042i \(-0.226088\pi\)
0.758183 + 0.652042i \(0.226088\pi\)
\(158\) −3.00000 −0.238667
\(159\) −9.00000 −0.713746
\(160\) 15.0000 1.18585
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −1.00000 −0.0783260 −0.0391630 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) −3.00000 −0.234261
\(165\) 27.0000 2.10195
\(166\) 0 0
\(167\) −13.0000 −1.00597 −0.502985 0.864295i \(-0.667765\pi\)
−0.502985 + 0.864295i \(0.667765\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −6.00000 −0.460179
\(171\) −6.00000 −0.458831
\(172\) 7.00000 0.533745
\(173\) 19.0000 1.44454 0.722272 0.691609i \(-0.243098\pi\)
0.722272 + 0.691609i \(0.243098\pi\)
\(174\) −21.0000 −1.59201
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 12.0000 0.901975
\(178\) 6.00000 0.449719
\(179\) 17.0000 1.27064 0.635320 0.772249i \(-0.280868\pi\)
0.635320 + 0.772249i \(0.280868\pi\)
\(180\) −18.0000 −1.34164
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 39.0000 2.88296
\(184\) 0 0
\(185\) 6.00000 0.441129
\(186\) −9.00000 −0.659912
\(187\) 6.00000 0.438763
\(188\) −1.00000 −0.0729325
\(189\) 0 0
\(190\) −3.00000 −0.217643
\(191\) −17.0000 −1.23008 −0.615038 0.788497i \(-0.710860\pi\)
−0.615038 + 0.788497i \(0.710860\pi\)
\(192\) −21.0000 −1.51554
\(193\) 7.00000 0.503871 0.251936 0.967744i \(-0.418933\pi\)
0.251936 + 0.967744i \(0.418933\pi\)
\(194\) −5.00000 −0.358979
\(195\) 0 0
\(196\) 0 0
\(197\) −1.00000 −0.0712470 −0.0356235 0.999365i \(-0.511342\pi\)
−0.0356235 + 0.999365i \(0.511342\pi\)
\(198\) −18.0000 −1.27920
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) −12.0000 −0.848528
\(201\) 9.00000 0.634811
\(202\) −5.00000 −0.351799
\(203\) 0 0
\(204\) −6.00000 −0.420084
\(205\) 9.00000 0.628587
\(206\) 5.00000 0.348367
\(207\) 0 0
\(208\) 0 0
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) −3.00000 −0.206041
\(213\) −39.0000 −2.67224
\(214\) 8.00000 0.546869
\(215\) −21.0000 −1.43219
\(216\) 27.0000 1.83712
\(217\) 0 0
\(218\) 7.00000 0.474100
\(219\) 39.0000 2.63538
\(220\) 9.00000 0.606780
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) −9.00000 −0.602685 −0.301342 0.953516i \(-0.597435\pi\)
−0.301342 + 0.953516i \(0.597435\pi\)
\(224\) 0 0
\(225\) 24.0000 1.60000
\(226\) 15.0000 0.997785
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) −3.00000 −0.198680
\(229\) −13.0000 −0.859064 −0.429532 0.903052i \(-0.641321\pi\)
−0.429532 + 0.903052i \(0.641321\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −21.0000 −1.37872
\(233\) −21.0000 −1.37576 −0.687878 0.725826i \(-0.741458\pi\)
−0.687878 + 0.725826i \(0.741458\pi\)
\(234\) 0 0
\(235\) 3.00000 0.195698
\(236\) 4.00000 0.260378
\(237\) 9.00000 0.584613
\(238\) 0 0
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 9.00000 0.580948
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) 13.0000 0.832240
\(245\) 0 0
\(246\) −9.00000 −0.573819
\(247\) 0 0
\(248\) −9.00000 −0.571501
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) −23.0000 −1.45175 −0.725874 0.687828i \(-0.758564\pi\)
−0.725874 + 0.687828i \(0.758564\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 11.0000 0.690201
\(255\) 18.0000 1.12720
\(256\) −17.0000 −1.06250
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 21.0000 1.30740
\(259\) 0 0
\(260\) 0 0
\(261\) 42.0000 2.59973
\(262\) 5.00000 0.308901
\(263\) −27.0000 −1.66489 −0.832446 0.554107i \(-0.813060\pi\)
−0.832446 + 0.554107i \(0.813060\pi\)
\(264\) −27.0000 −1.66174
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) −18.0000 −1.10158
\(268\) 3.00000 0.183254
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) −27.0000 −1.64317
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 10.0000 0.604122
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −15.0000 −0.899640
\(279\) 18.0000 1.07763
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) −3.00000 −0.178647
\(283\) 1.00000 0.0594438 0.0297219 0.999558i \(-0.490538\pi\)
0.0297219 + 0.999558i \(0.490538\pi\)
\(284\) −13.0000 −0.771408
\(285\) 9.00000 0.533114
\(286\) 0 0
\(287\) 0 0
\(288\) 30.0000 1.76777
\(289\) −13.0000 −0.764706
\(290\) 21.0000 1.23316
\(291\) 15.0000 0.879316
\(292\) 13.0000 0.760767
\(293\) 11.0000 0.642627 0.321313 0.946973i \(-0.395876\pi\)
0.321313 + 0.946973i \(0.395876\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) −6.00000 −0.348743
\(297\) 27.0000 1.56670
\(298\) 15.0000 0.868927
\(299\) 0 0
\(300\) 12.0000 0.692820
\(301\) 0 0
\(302\) −21.0000 −1.20841
\(303\) 15.0000 0.861727
\(304\) 1.00000 0.0573539
\(305\) −39.0000 −2.23313
\(306\) −12.0000 −0.685994
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) −15.0000 −0.853320
\(310\) 9.00000 0.511166
\(311\) −9.00000 −0.510343 −0.255172 0.966896i \(-0.582132\pi\)
−0.255172 + 0.966896i \(0.582132\pi\)
\(312\) 0 0
\(313\) 19.0000 1.07394 0.536972 0.843600i \(-0.319568\pi\)
0.536972 + 0.843600i \(0.319568\pi\)
\(314\) 19.0000 1.07223
\(315\) 0 0
\(316\) 3.00000 0.168763
\(317\) −9.00000 −0.505490 −0.252745 0.967533i \(-0.581333\pi\)
−0.252745 + 0.967533i \(0.581333\pi\)
\(318\) −9.00000 −0.504695
\(319\) −21.0000 −1.17577
\(320\) 21.0000 1.17394
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −1.00000 −0.0553849
\(327\) −21.0000 −1.16130
\(328\) −9.00000 −0.496942
\(329\) 0 0
\(330\) 27.0000 1.48630
\(331\) −29.0000 −1.59398 −0.796992 0.603990i \(-0.793577\pi\)
−0.796992 + 0.603990i \(0.793577\pi\)
\(332\) 0 0
\(333\) 12.0000 0.657596
\(334\) −13.0000 −0.711328
\(335\) −9.00000 −0.491723
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −45.0000 −2.44406
\(340\) 6.00000 0.325396
\(341\) −9.00000 −0.487377
\(342\) −6.00000 −0.324443
\(343\) 0 0
\(344\) 21.0000 1.13224
\(345\) 0 0
\(346\) 19.0000 1.02145
\(347\) −8.00000 −0.429463 −0.214731 0.976673i \(-0.568888\pi\)
−0.214731 + 0.976673i \(0.568888\pi\)
\(348\) 21.0000 1.12572
\(349\) 23.0000 1.23116 0.615581 0.788074i \(-0.288921\pi\)
0.615581 + 0.788074i \(0.288921\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −15.0000 −0.799503
\(353\) −25.0000 −1.33062 −0.665308 0.746569i \(-0.731700\pi\)
−0.665308 + 0.746569i \(0.731700\pi\)
\(354\) 12.0000 0.637793
\(355\) 39.0000 2.06991
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) 17.0000 0.898478
\(359\) 17.0000 0.897226 0.448613 0.893726i \(-0.351918\pi\)
0.448613 + 0.893726i \(0.351918\pi\)
\(360\) −54.0000 −2.84605
\(361\) −18.0000 −0.947368
\(362\) −22.0000 −1.15629
\(363\) 6.00000 0.314918
\(364\) 0 0
\(365\) −39.0000 −2.04135
\(366\) 39.0000 2.03856
\(367\) −31.0000 −1.61819 −0.809093 0.587680i \(-0.800041\pi\)
−0.809093 + 0.587680i \(0.800041\pi\)
\(368\) 0 0
\(369\) 18.0000 0.937043
\(370\) 6.00000 0.311925
\(371\) 0 0
\(372\) 9.00000 0.466628
\(373\) −9.00000 −0.466002 −0.233001 0.972476i \(-0.574855\pi\)
−0.233001 + 0.972476i \(0.574855\pi\)
\(374\) 6.00000 0.310253
\(375\) 9.00000 0.464758
\(376\) −3.00000 −0.154713
\(377\) 0 0
\(378\) 0 0
\(379\) −33.0000 −1.69510 −0.847548 0.530719i \(-0.821922\pi\)
−0.847548 + 0.530719i \(0.821922\pi\)
\(380\) 3.00000 0.153897
\(381\) −33.0000 −1.69064
\(382\) −17.0000 −0.869796
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 9.00000 0.459279
\(385\) 0 0
\(386\) 7.00000 0.356291
\(387\) −42.0000 −2.13498
\(388\) 5.00000 0.253837
\(389\) −33.0000 −1.67317 −0.836583 0.547840i \(-0.815450\pi\)
−0.836583 + 0.547840i \(0.815450\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −15.0000 −0.756650
\(394\) −1.00000 −0.0503793
\(395\) −9.00000 −0.452839
\(396\) 18.0000 0.904534
\(397\) −1.00000 −0.0501886 −0.0250943 0.999685i \(-0.507989\pi\)
−0.0250943 + 0.999685i \(0.507989\pi\)
\(398\) −20.0000 −1.00251
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −2.00000 −0.0998752 −0.0499376 0.998752i \(-0.515902\pi\)
−0.0499376 + 0.998752i \(0.515902\pi\)
\(402\) 9.00000 0.448879
\(403\) 0 0
\(404\) 5.00000 0.248759
\(405\) 27.0000 1.34164
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) −18.0000 −0.891133
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 9.00000 0.444478
\(411\) −30.0000 −1.47979
\(412\) −5.00000 −0.246332
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 45.0000 2.20366
\(418\) 3.00000 0.146735
\(419\) −25.0000 −1.22133 −0.610665 0.791889i \(-0.709098\pi\)
−0.610665 + 0.791889i \(0.709098\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) 7.00000 0.340755
\(423\) 6.00000 0.291730
\(424\) −9.00000 −0.437079
\(425\) −8.00000 −0.388057
\(426\) −39.0000 −1.88956
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 0 0
\(430\) −21.0000 −1.01271
\(431\) 9.00000 0.433515 0.216757 0.976226i \(-0.430452\pi\)
0.216757 + 0.976226i \(0.430452\pi\)
\(432\) 9.00000 0.433013
\(433\) 27.0000 1.29754 0.648769 0.760986i \(-0.275284\pi\)
0.648769 + 0.760986i \(0.275284\pi\)
\(434\) 0 0
\(435\) −63.0000 −3.02062
\(436\) −7.00000 −0.335239
\(437\) 0 0
\(438\) 39.0000 1.86349
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 27.0000 1.28717
\(441\) 0 0
\(442\) 0 0
\(443\) −11.0000 −0.522626 −0.261313 0.965254i \(-0.584155\pi\)
−0.261313 + 0.965254i \(0.584155\pi\)
\(444\) 6.00000 0.284747
\(445\) 18.0000 0.853282
\(446\) −9.00000 −0.426162
\(447\) −45.0000 −2.12843
\(448\) 0 0
\(449\) 15.0000 0.707894 0.353947 0.935266i \(-0.384839\pi\)
0.353947 + 0.935266i \(0.384839\pi\)
\(450\) 24.0000 1.13137
\(451\) −9.00000 −0.423793
\(452\) −15.0000 −0.705541
\(453\) 63.0000 2.96000
\(454\) −4.00000 −0.187729
\(455\) 0 0
\(456\) −9.00000 −0.421464
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) −13.0000 −0.607450
\(459\) 18.0000 0.840168
\(460\) 0 0
\(461\) 35.0000 1.63011 0.815056 0.579382i \(-0.196706\pi\)
0.815056 + 0.579382i \(0.196706\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) −7.00000 −0.324967
\(465\) −27.0000 −1.25210
\(466\) −21.0000 −0.972806
\(467\) −7.00000 −0.323921 −0.161961 0.986797i \(-0.551782\pi\)
−0.161961 + 0.986797i \(0.551782\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3.00000 0.138380
\(471\) −57.0000 −2.62642
\(472\) 12.0000 0.552345
\(473\) 21.0000 0.965581
\(474\) 9.00000 0.413384
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 18.0000 0.824163
\(478\) −4.00000 −0.182956
\(479\) −35.0000 −1.59919 −0.799595 0.600539i \(-0.794953\pi\)
−0.799595 + 0.600539i \(0.794953\pi\)
\(480\) −45.0000 −2.05396
\(481\) 0 0
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) −15.0000 −0.681115
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 39.0000 1.76545
\(489\) 3.00000 0.135665
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 9.00000 0.405751
\(493\) −14.0000 −0.630528
\(494\) 0 0
\(495\) −54.0000 −2.42712
\(496\) −3.00000 −0.134704
\(497\) 0 0
\(498\) 0 0
\(499\) −31.0000 −1.38775 −0.693875 0.720095i \(-0.744098\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) 3.00000 0.134164
\(501\) 39.0000 1.74239
\(502\) −23.0000 −1.02654
\(503\) −31.0000 −1.38222 −0.691111 0.722749i \(-0.742878\pi\)
−0.691111 + 0.722749i \(0.742878\pi\)
\(504\) 0 0
\(505\) −15.0000 −0.667491
\(506\) 0 0
\(507\) 0 0
\(508\) −11.0000 −0.488046
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 18.0000 0.797053
\(511\) 0 0
\(512\) −11.0000 −0.486136
\(513\) 9.00000 0.397360
\(514\) −2.00000 −0.0882162
\(515\) 15.0000 0.660979
\(516\) −21.0000 −0.924473
\(517\) −3.00000 −0.131940
\(518\) 0 0
\(519\) −57.0000 −2.50202
\(520\) 0 0
\(521\) −17.0000 −0.744784 −0.372392 0.928076i \(-0.621462\pi\)
−0.372392 + 0.928076i \(0.621462\pi\)
\(522\) 42.0000 1.83829
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) −5.00000 −0.218426
\(525\) 0 0
\(526\) −27.0000 −1.17726
\(527\) −6.00000 −0.261364
\(528\) −9.00000 −0.391675
\(529\) −23.0000 −1.00000
\(530\) 9.00000 0.390935
\(531\) −24.0000 −1.04151
\(532\) 0 0
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) 24.0000 1.03761
\(536\) 9.00000 0.388741
\(537\) −51.0000 −2.20081
\(538\) 18.0000 0.776035
\(539\) 0 0
\(540\) 27.0000 1.16190
\(541\) −37.0000 −1.59075 −0.795377 0.606115i \(-0.792727\pi\)
−0.795377 + 0.606115i \(0.792727\pi\)
\(542\) −16.0000 −0.687259
\(543\) 66.0000 2.83233
\(544\) −10.0000 −0.428746
\(545\) 21.0000 0.899541
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −10.0000 −0.427179
\(549\) −78.0000 −3.32896
\(550\) −12.0000 −0.511682
\(551\) −7.00000 −0.298210
\(552\) 0 0
\(553\) 0 0
\(554\) 22.0000 0.934690
\(555\) −18.0000 −0.764057
\(556\) 15.0000 0.636142
\(557\) 3.00000 0.127114 0.0635570 0.997978i \(-0.479756\pi\)
0.0635570 + 0.997978i \(0.479756\pi\)
\(558\) 18.0000 0.762001
\(559\) 0 0
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) −18.0000 −0.759284
\(563\) 4.00000 0.168580 0.0842900 0.996441i \(-0.473138\pi\)
0.0842900 + 0.996441i \(0.473138\pi\)
\(564\) 3.00000 0.126323
\(565\) 45.0000 1.89316
\(566\) 1.00000 0.0420331
\(567\) 0 0
\(568\) −39.0000 −1.63640
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 9.00000 0.376969
\(571\) 43.0000 1.79949 0.899747 0.436412i \(-0.143751\pi\)
0.899747 + 0.436412i \(0.143751\pi\)
\(572\) 0 0
\(573\) 51.0000 2.13056
\(574\) 0 0
\(575\) 0 0
\(576\) 42.0000 1.75000
\(577\) −1.00000 −0.0416305 −0.0208153 0.999783i \(-0.506626\pi\)
−0.0208153 + 0.999783i \(0.506626\pi\)
\(578\) −13.0000 −0.540729
\(579\) −21.0000 −0.872730
\(580\) −21.0000 −0.871978
\(581\) 0 0
\(582\) 15.0000 0.621770
\(583\) −9.00000 −0.372742
\(584\) 39.0000 1.61383
\(585\) 0 0
\(586\) 11.0000 0.454406
\(587\) −33.0000 −1.36206 −0.681028 0.732257i \(-0.738467\pi\)
−0.681028 + 0.732257i \(0.738467\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) −12.0000 −0.494032
\(591\) 3.00000 0.123404
\(592\) −2.00000 −0.0821995
\(593\) 27.0000 1.10876 0.554379 0.832265i \(-0.312956\pi\)
0.554379 + 0.832265i \(0.312956\pi\)
\(594\) 27.0000 1.10782
\(595\) 0 0
\(596\) −15.0000 −0.614424
\(597\) 60.0000 2.45564
\(598\) 0 0
\(599\) −25.0000 −1.02147 −0.510736 0.859738i \(-0.670627\pi\)
−0.510736 + 0.859738i \(0.670627\pi\)
\(600\) 36.0000 1.46969
\(601\) 35.0000 1.42768 0.713840 0.700309i \(-0.246954\pi\)
0.713840 + 0.700309i \(0.246954\pi\)
\(602\) 0 0
\(603\) −18.0000 −0.733017
\(604\) 21.0000 0.854478
\(605\) −6.00000 −0.243935
\(606\) 15.0000 0.609333
\(607\) 11.0000 0.446476 0.223238 0.974764i \(-0.428337\pi\)
0.223238 + 0.974764i \(0.428337\pi\)
\(608\) −5.00000 −0.202777
\(609\) 0 0
\(610\) −39.0000 −1.57906
\(611\) 0 0
\(612\) 12.0000 0.485071
\(613\) −25.0000 −1.00974 −0.504870 0.863195i \(-0.668460\pi\)
−0.504870 + 0.863195i \(0.668460\pi\)
\(614\) 12.0000 0.484281
\(615\) −27.0000 −1.08875
\(616\) 0 0
\(617\) −33.0000 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(618\) −15.0000 −0.603388
\(619\) 11.0000 0.442127 0.221064 0.975259i \(-0.429047\pi\)
0.221064 + 0.975259i \(0.429047\pi\)
\(620\) −9.00000 −0.361449
\(621\) 0 0
\(622\) −9.00000 −0.360867
\(623\) 0 0
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 19.0000 0.759393
\(627\) −9.00000 −0.359425
\(628\) −19.0000 −0.758183
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) 9.00000 0.358001
\(633\) −21.0000 −0.834675
\(634\) −9.00000 −0.357436
\(635\) 33.0000 1.30957
\(636\) 9.00000 0.356873
\(637\) 0 0
\(638\) −21.0000 −0.831398
\(639\) 78.0000 3.08563
\(640\) −9.00000 −0.355756
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) −24.0000 −0.947204
\(643\) −19.0000 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(644\) 0 0
\(645\) 63.0000 2.48062
\(646\) 2.00000 0.0786889
\(647\) 9.00000 0.353827 0.176913 0.984226i \(-0.443389\pi\)
0.176913 + 0.984226i \(0.443389\pi\)
\(648\) −27.0000 −1.06066
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 1.00000 0.0391630
\(653\) 18.0000 0.704394 0.352197 0.935926i \(-0.385435\pi\)
0.352197 + 0.935926i \(0.385435\pi\)
\(654\) −21.0000 −0.821165
\(655\) 15.0000 0.586098
\(656\) −3.00000 −0.117130
\(657\) −78.0000 −3.04307
\(658\) 0 0
\(659\) 29.0000 1.12968 0.564840 0.825201i \(-0.308938\pi\)
0.564840 + 0.825201i \(0.308938\pi\)
\(660\) −27.0000 −1.05097
\(661\) −9.00000 −0.350059 −0.175030 0.984563i \(-0.556002\pi\)
−0.175030 + 0.984563i \(0.556002\pi\)
\(662\) −29.0000 −1.12712
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) 0 0
\(668\) 13.0000 0.502985
\(669\) 27.0000 1.04388
\(670\) −9.00000 −0.347700
\(671\) 39.0000 1.50558
\(672\) 0 0
\(673\) −41.0000 −1.58043 −0.790217 0.612827i \(-0.790032\pi\)
−0.790217 + 0.612827i \(0.790032\pi\)
\(674\) 14.0000 0.539260
\(675\) −36.0000 −1.38564
\(676\) 0 0
\(677\) 7.00000 0.269032 0.134516 0.990911i \(-0.457052\pi\)
0.134516 + 0.990911i \(0.457052\pi\)
\(678\) −45.0000 −1.72821
\(679\) 0 0
\(680\) 18.0000 0.690268
\(681\) 12.0000 0.459841
\(682\) −9.00000 −0.344628
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 6.00000 0.229416
\(685\) 30.0000 1.14624
\(686\) 0 0
\(687\) 39.0000 1.48794
\(688\) 7.00000 0.266872
\(689\) 0 0
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −19.0000 −0.722272
\(693\) 0 0
\(694\) −8.00000 −0.303676
\(695\) −45.0000 −1.70695
\(696\) 63.0000 2.38801
\(697\) −6.00000 −0.227266
\(698\) 23.0000 0.870563
\(699\) 63.0000 2.38288
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) −21.0000 −0.791467
\(705\) −9.00000 −0.338960
\(706\) −25.0000 −0.940887
\(707\) 0 0
\(708\) −12.0000 −0.450988
\(709\) 11.0000 0.413114 0.206557 0.978435i \(-0.433774\pi\)
0.206557 + 0.978435i \(0.433774\pi\)
\(710\) 39.0000 1.46364
\(711\) −18.0000 −0.675053
\(712\) −18.0000 −0.674579
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −17.0000 −0.635320
\(717\) 12.0000 0.448148
\(718\) 17.0000 0.634434
\(719\) −9.00000 −0.335643 −0.167822 0.985817i \(-0.553673\pi\)
−0.167822 + 0.985817i \(0.553673\pi\)
\(720\) −18.0000 −0.670820
\(721\) 0 0
\(722\) −18.0000 −0.669891
\(723\) 78.0000 2.90085
\(724\) 22.0000 0.817624
\(725\) 28.0000 1.03989
\(726\) 6.00000 0.222681
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) −39.0000 −1.44345
\(731\) 14.0000 0.517809
\(732\) −39.0000 −1.44148
\(733\) −9.00000 −0.332423 −0.166211 0.986090i \(-0.553153\pi\)
−0.166211 + 0.986090i \(0.553153\pi\)
\(734\) −31.0000 −1.14423
\(735\) 0 0
\(736\) 0 0
\(737\) 9.00000 0.331519
\(738\) 18.0000 0.662589
\(739\) 1.00000 0.0367856 0.0183928 0.999831i \(-0.494145\pi\)
0.0183928 + 0.999831i \(0.494145\pi\)
\(740\) −6.00000 −0.220564
\(741\) 0 0
\(742\) 0 0
\(743\) 51.0000 1.87101 0.935504 0.353315i \(-0.114946\pi\)
0.935504 + 0.353315i \(0.114946\pi\)
\(744\) 27.0000 0.989868
\(745\) 45.0000 1.64867
\(746\) −9.00000 −0.329513
\(747\) 0 0
\(748\) −6.00000 −0.219382
\(749\) 0 0
\(750\) 9.00000 0.328634
\(751\) 28.0000 1.02173 0.510867 0.859660i \(-0.329324\pi\)
0.510867 + 0.859660i \(0.329324\pi\)
\(752\) −1.00000 −0.0364662
\(753\) 69.0000 2.51450
\(754\) 0 0
\(755\) −63.0000 −2.29280
\(756\) 0 0
\(757\) 3.00000 0.109037 0.0545184 0.998513i \(-0.482638\pi\)
0.0545184 + 0.998513i \(0.482638\pi\)
\(758\) −33.0000 −1.19861
\(759\) 0 0
\(760\) 9.00000 0.326464
\(761\) −9.00000 −0.326250 −0.163125 0.986605i \(-0.552157\pi\)
−0.163125 + 0.986605i \(0.552157\pi\)
\(762\) −33.0000 −1.19546
\(763\) 0 0
\(764\) 17.0000 0.615038
\(765\) −36.0000 −1.30158
\(766\) −21.0000 −0.758761
\(767\) 0 0
\(768\) 51.0000 1.84030
\(769\) 19.0000 0.685158 0.342579 0.939489i \(-0.388700\pi\)
0.342579 + 0.939489i \(0.388700\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −7.00000 −0.251936
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −42.0000 −1.50966
\(775\) 12.0000 0.431053
\(776\) 15.0000 0.538469
\(777\) 0 0
\(778\) −33.0000 −1.18311
\(779\) −3.00000 −0.107486
\(780\) 0 0
\(781\) −39.0000 −1.39553
\(782\) 0 0
\(783\) −63.0000 −2.25144
\(784\) 0 0
\(785\) 57.0000 2.03442
\(786\) −15.0000 −0.535032
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 1.00000 0.0356235
\(789\) 81.0000 2.88368
\(790\) −9.00000 −0.320206
\(791\) 0 0
\(792\) 54.0000 1.91881
\(793\) 0 0
\(794\) −1.00000 −0.0354887
\(795\) −27.0000 −0.957591
\(796\) 20.0000 0.708881
\(797\) 3.00000 0.106265 0.0531327 0.998587i \(-0.483079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) 0 0
\(799\) −2.00000 −0.0707549
\(800\) 20.0000 0.707107
\(801\) 36.0000 1.27200
\(802\) −2.00000 −0.0706225
\(803\) 39.0000 1.37628
\(804\) −9.00000 −0.317406
\(805\) 0 0
\(806\) 0 0
\(807\) −54.0000 −1.90089
\(808\) 15.0000 0.527698
\(809\) 11.0000 0.386739 0.193370 0.981126i \(-0.438058\pi\)
0.193370 + 0.981126i \(0.438058\pi\)
\(810\) 27.0000 0.948683
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) 48.0000 1.68343
\(814\) −6.00000 −0.210300
\(815\) −3.00000 −0.105085
\(816\) −6.00000 −0.210042
\(817\) 7.00000 0.244899
\(818\) 14.0000 0.489499
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) −54.0000 −1.88461 −0.942306 0.334751i \(-0.891348\pi\)
−0.942306 + 0.334751i \(0.891348\pi\)
\(822\) −30.0000 −1.04637
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −15.0000 −0.522550
\(825\) 36.0000 1.25336
\(826\) 0 0
\(827\) 4.00000 0.139094 0.0695468 0.997579i \(-0.477845\pi\)
0.0695468 + 0.997579i \(0.477845\pi\)
\(828\) 0 0
\(829\) 11.0000 0.382046 0.191023 0.981586i \(-0.438820\pi\)
0.191023 + 0.981586i \(0.438820\pi\)
\(830\) 0 0
\(831\) −66.0000 −2.28951
\(832\) 0 0
\(833\) 0 0
\(834\) 45.0000 1.55822
\(835\) −39.0000 −1.34965
\(836\) −3.00000 −0.103757
\(837\) −27.0000 −0.933257
\(838\) −25.0000 −0.863611
\(839\) 37.0000 1.27738 0.638691 0.769463i \(-0.279476\pi\)
0.638691 + 0.769463i \(0.279476\pi\)
\(840\) 0 0
\(841\) 20.0000 0.689655
\(842\) 18.0000 0.620321
\(843\) 54.0000 1.85986
\(844\) −7.00000 −0.240950
\(845\) 0 0
\(846\) 6.00000 0.206284
\(847\) 0 0
\(848\) −3.00000 −0.103020
\(849\) −3.00000 −0.102960
\(850\) −8.00000 −0.274398
\(851\) 0 0
\(852\) 39.0000 1.33612
\(853\) −6.00000 −0.205436 −0.102718 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) −18.0000 −0.615587
\(856\) −24.0000 −0.820303
\(857\) −33.0000 −1.12726 −0.563629 0.826028i \(-0.690595\pi\)
−0.563629 + 0.826028i \(0.690595\pi\)
\(858\) 0 0
\(859\) 25.0000 0.852989 0.426494 0.904490i \(-0.359748\pi\)
0.426494 + 0.904490i \(0.359748\pi\)
\(860\) 21.0000 0.716094
\(861\) 0 0
\(862\) 9.00000 0.306541
\(863\) 37.0000 1.25949 0.629747 0.776800i \(-0.283158\pi\)
0.629747 + 0.776800i \(0.283158\pi\)
\(864\) −45.0000 −1.53093
\(865\) 57.0000 1.93806
\(866\) 27.0000 0.917497
\(867\) 39.0000 1.32451
\(868\) 0 0
\(869\) 9.00000 0.305304
\(870\) −63.0000 −2.13590
\(871\) 0 0
\(872\) −21.0000 −0.711150
\(873\) −30.0000 −1.01535
\(874\) 0 0
\(875\) 0 0
\(876\) −39.0000 −1.31769
\(877\) −45.0000 −1.51954 −0.759771 0.650191i \(-0.774689\pi\)
−0.759771 + 0.650191i \(0.774689\pi\)
\(878\) −16.0000 −0.539974
\(879\) −33.0000 −1.11306
\(880\) 9.00000 0.303390
\(881\) 15.0000 0.505363 0.252681 0.967550i \(-0.418688\pi\)
0.252681 + 0.967550i \(0.418688\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 36.0000 1.21013
\(886\) −11.0000 −0.369552
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 18.0000 0.604040
\(889\) 0 0
\(890\) 18.0000 0.603361
\(891\) −27.0000 −0.904534
\(892\) 9.00000 0.301342
\(893\) −1.00000 −0.0334637
\(894\) −45.0000 −1.50503
\(895\) 51.0000 1.70474
\(896\) 0 0
\(897\) 0 0
\(898\) 15.0000 0.500556
\(899\) 21.0000 0.700389
\(900\) −24.0000 −0.800000
\(901\) −6.00000 −0.199889
\(902\) −9.00000 −0.299667
\(903\) 0 0
\(904\) −45.0000 −1.49668
\(905\) −66.0000 −2.19391
\(906\) 63.0000 2.09303
\(907\) −47.0000 −1.56061 −0.780305 0.625400i \(-0.784936\pi\)
−0.780305 + 0.625400i \(0.784936\pi\)
\(908\) 4.00000 0.132745
\(909\) −30.0000 −0.995037
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) −3.00000 −0.0993399
\(913\) 0 0
\(914\) −18.0000 −0.595387
\(915\) 117.000 3.86790
\(916\) 13.0000 0.429532
\(917\) 0 0
\(918\) 18.0000 0.594089
\(919\) −25.0000 −0.824674 −0.412337 0.911031i \(-0.635287\pi\)
−0.412337 + 0.911031i \(0.635287\pi\)
\(920\) 0 0
\(921\) −36.0000 −1.18624
\(922\) 35.0000 1.15266
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −8.00000 −0.262896
\(927\) 30.0000 0.985329
\(928\) 35.0000 1.14893
\(929\) −13.0000 −0.426516 −0.213258 0.976996i \(-0.568408\pi\)
−0.213258 + 0.976996i \(0.568408\pi\)
\(930\) −27.0000 −0.885365
\(931\) 0 0
\(932\) 21.0000 0.687878
\(933\) 27.0000 0.883940
\(934\) −7.00000 −0.229047
\(935\) 18.0000 0.588663
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −57.0000 −1.86012
\(940\) −3.00000 −0.0978492
\(941\) −17.0000 −0.554184 −0.277092 0.960843i \(-0.589371\pi\)
−0.277092 + 0.960843i \(0.589371\pi\)
\(942\) −57.0000 −1.85716
\(943\) 0 0
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) 21.0000 0.682769
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) −9.00000 −0.292306
\(949\) 0 0
\(950\) −4.00000 −0.129777
\(951\) 27.0000 0.875535
\(952\) 0 0
\(953\) −33.0000 −1.06897 −0.534487 0.845176i \(-0.679495\pi\)
−0.534487 + 0.845176i \(0.679495\pi\)
\(954\) 18.0000 0.582772
\(955\) −51.0000 −1.65032
\(956\) 4.00000 0.129369
\(957\) 63.0000 2.03650
\(958\) −35.0000 −1.13080
\(959\) 0 0
\(960\) −63.0000 −2.03332
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 48.0000 1.54678
\(964\) 26.0000 0.837404
\(965\) 21.0000 0.676014
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 6.00000 0.192847
\(969\) −6.00000 −0.192748
\(970\) −15.0000 −0.481621
\(971\) −1.00000 −0.0320915 −0.0160458 0.999871i \(-0.505108\pi\)
−0.0160458 + 0.999871i \(0.505108\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 13.0000 0.416120
\(977\) −5.00000 −0.159964 −0.0799821 0.996796i \(-0.525486\pi\)
−0.0799821 + 0.996796i \(0.525486\pi\)
\(978\) 3.00000 0.0959294
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 42.0000 1.34096
\(982\) 15.0000 0.478669
\(983\) 47.0000 1.49907 0.749534 0.661966i \(-0.230278\pi\)
0.749534 + 0.661966i \(0.230278\pi\)
\(984\) 27.0000 0.860729
\(985\) −3.00000 −0.0955879
\(986\) −14.0000 −0.445851
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) −54.0000 −1.71623
\(991\) 13.0000 0.412959 0.206479 0.978451i \(-0.433799\pi\)
0.206479 + 0.978451i \(0.433799\pi\)
\(992\) 15.0000 0.476250
\(993\) 87.0000 2.76086
\(994\) 0 0
\(995\) −60.0000 −1.90213
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) −31.0000 −0.981288
\(999\) −18.0000 −0.569495
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 8281.2.a.i.1.1 1
7.2 even 3 1183.2.e.a.508.1 2
7.4 even 3 1183.2.e.a.170.1 2
7.6 odd 2 8281.2.a.j.1.1 1
13.3 even 3 637.2.f.b.295.1 2
13.9 even 3 637.2.f.b.393.1 2
13.12 even 2 8281.2.a.c.1.1 1
91.3 odd 6 637.2.g.a.373.1 2
91.9 even 3 91.2.g.a.81.1 yes 2
91.16 even 3 91.2.h.a.74.1 yes 2
91.25 even 6 1183.2.e.c.170.1 2
91.48 odd 6 637.2.f.a.393.1 2
91.51 even 6 1183.2.e.c.508.1 2
91.55 odd 6 637.2.f.a.295.1 2
91.61 odd 6 637.2.g.a.263.1 2
91.68 odd 6 637.2.h.a.165.1 2
91.74 even 3 91.2.h.a.16.1 yes 2
91.81 even 3 91.2.g.a.9.1 2
91.87 odd 6 637.2.h.a.471.1 2
91.90 odd 2 8281.2.a.g.1.1 1
273.74 odd 6 819.2.s.a.289.1 2
273.107 odd 6 819.2.s.a.802.1 2
273.191 odd 6 819.2.n.c.172.1 2
273.263 odd 6 819.2.n.c.100.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.g.a.9.1 2 91.81 even 3
91.2.g.a.81.1 yes 2 91.9 even 3
91.2.h.a.16.1 yes 2 91.74 even 3
91.2.h.a.74.1 yes 2 91.16 even 3
637.2.f.a.295.1 2 91.55 odd 6
637.2.f.a.393.1 2 91.48 odd 6
637.2.f.b.295.1 2 13.3 even 3
637.2.f.b.393.1 2 13.9 even 3
637.2.g.a.263.1 2 91.61 odd 6
637.2.g.a.373.1 2 91.3 odd 6
637.2.h.a.165.1 2 91.68 odd 6
637.2.h.a.471.1 2 91.87 odd 6
819.2.n.c.100.1 2 273.263 odd 6
819.2.n.c.172.1 2 273.191 odd 6
819.2.s.a.289.1 2 273.74 odd 6
819.2.s.a.802.1 2 273.107 odd 6
1183.2.e.a.170.1 2 7.4 even 3
1183.2.e.a.508.1 2 7.2 even 3
1183.2.e.c.170.1 2 91.25 even 6
1183.2.e.c.508.1 2 91.51 even 6
8281.2.a.c.1.1 1 13.12 even 2
8281.2.a.g.1.1 1 91.90 odd 2
8281.2.a.i.1.1 1 1.1 even 1 trivial
8281.2.a.j.1.1 1 7.6 odd 2