↑

Properties

Label 8550.2.a.n.1.1
Level $8550$
Weight $2$
Character 8550.1
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8550,2,Mod(1,8550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8550, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.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: \(68.2720937282\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 570)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 8550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{7} -1.00000 q^{8} -4.00000 q^{11} +6.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{19} +4.00000 q^{22} +4.00000 q^{23} -6.00000 q^{26} +2.00000 q^{28} -6.00000 q^{29} -6.00000 q^{31} -1.00000 q^{32} -4.00000 q^{34} -10.0000 q^{37} -1.00000 q^{38} -4.00000 q^{41} -12.0000 q^{43} -4.00000 q^{44} -4.00000 q^{46} +4.00000 q^{47} -3.00000 q^{49} +6.00000 q^{52} -10.0000 q^{53} -2.00000 q^{56} +6.00000 q^{58} -10.0000 q^{59} +2.00000 q^{61} +6.00000 q^{62} +1.00000 q^{64} -12.0000 q^{67} +4.00000 q^{68} -8.00000 q^{71} +2.00000 q^{73} +10.0000 q^{74} +1.00000 q^{76} -8.00000 q^{77} +10.0000 q^{79} +4.00000 q^{82} +2.00000 q^{83} +12.0000 q^{86} +4.00000 q^{88} +8.00000 q^{89} +12.0000 q^{91} +4.00000 q^{92} -4.00000 q^{94} -2.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 6.00000 1.66410 0.832050 − 0.554700i \(-0.187167\pi\)
0.832050 + 0.554700i \(0.187167\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 4.00000 0.852803
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 − 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 − 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) −1.00000 −0.162221
\(39\) 0 0
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 − 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 − 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 4.00000 0.583460 0.291730 − 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 6.00000 0.832050
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 6.00000 0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 − 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.00000 0.441726
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) 4.00000 0.426401
\(89\) 8.00000 0.847998 0.423999 − 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 12.0000 1.25794
\(92\) 4.00000 0.417029
\(93\) 0 0
\(94\) −4.00000 −0.412568
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 − 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 8.00000 0.773389 0.386695 − 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −16.0000 −1.53252 −0.766261 − 0.642529i \(-0.777885\pi\)
−0.766261 + 0.642529i \(0.777885\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) −14.0000 −1.31701 −0.658505 − 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) −6.00000 −0.538816
\(125\) 0 0
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 − 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 − 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000 0.671345
\(143\) −24.0000 −2.00698
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −10.0000 −0.821995
\(149\) −14.0000 −1.14692 −0.573462 − 0.819232i \(-0.694400\pi\)
−0.573462 + 0.819232i \(0.694400\pi\)
\(150\) 0 0
\(151\) −2.00000 −0.162758 −0.0813788 − 0.996683i \(-0.525932\pi\)
−0.0813788 + 0.996683i \(0.525932\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) −12.0000 −0.957704 −0.478852 − 0.877896i \(-0.658947\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) −10.0000 −0.795557
\(159\) 0 0
\(160\) 0 0
\(161\) 8.00000 0.630488
\(162\) 0 0
\(163\) 16.0000 1.25322 0.626608 − 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) 24.0000 1.85718 0.928588 − 0.371113i \(-0.121024\pi\)
0.928588 + 0.371113i \(0.121024\pi\)
\(168\) 0 0
\(169\) 23.0000 1.76923
\(170\) 0 0
\(171\) 0 0
\(172\) −12.0000 −0.914991
\(173\) −18.0000 −1.36851 −0.684257 − 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) −8.00000 −0.599625
\(179\) 2.00000 0.149487 0.0747435 − 0.997203i \(-0.476186\pi\)
0.0747435 + 0.997203i \(0.476186\pi\)
\(180\) 0 0
\(181\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) −12.0000 −0.889499
\(183\) 0 0
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) 0 0
\(187\) −16.0000 −1.17004
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 − 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 22.0000 1.56744 0.783718 − 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 − 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −10.0000 −0.703598
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) 0 0
\(208\) 6.00000 0.416025
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 − 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 16.0000 1.08366
\(219\) 0 0
\(220\) 0 0
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 − 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −26.0000 −1.71813 −0.859064 − 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −12.0000 −0.786146 −0.393073 − 0.919507i \(-0.628588\pi\)
−0.393073 + 0.919507i \(0.628588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 0 0
\(238\) −8.00000 −0.518563
\(239\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) 6.00000 0.381771
\(248\) 6.00000 0.381000
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000 0.741362
\(263\) 12.0000 0.739952 0.369976 − 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.00000 −0.122628
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) −14.0000 −0.853595 −0.426798 − 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 − 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −28.0000 −1.68236 −0.841178 − 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) −4.00000 −0.238620 −0.119310 − 0.992857i \(-0.538068\pi\)
−0.119310 + 0.992857i \(0.538068\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 − 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 2.00000 0.116841 0.0584206 − 0.998292i \(-0.481394\pi\)
0.0584206 + 0.998292i \(0.481394\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 2.00000 0.115087
\(303\) 0 0
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 − 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) 0 0
\(311\) −16.0000 −0.907277 −0.453638 − 0.891186i \(-0.649874\pi\)
−0.453638 + 0.891186i \(0.649874\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 − 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 12.0000 0.677199
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 0 0
\(326\) −16.0000 −0.886158
\(327\) 0 0
\(328\) 4.00000 0.220863
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 − 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 2.00000 0.109764
\(333\) 0 0
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 − 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −23.0000 −1.25104
\(339\) 0 0
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 18.0000 0.967686
\(347\) −2.00000 −0.107366 −0.0536828 − 0.998558i \(-0.517096\pi\)
−0.0536828 + 0.998558i \(0.517096\pi\)
\(348\) 0 0
\(349\) 22.0000 1.17763 0.588817 − 0.808267i \(-0.299594\pi\)
0.588817 + 0.808267i \(0.299594\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.00000 0.213201
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) −2.00000 −0.105703
\(359\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 12.0000 0.628971
\(365\) 0 0
\(366\) 0 0
\(367\) 22.0000 1.14839 0.574195 − 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) 4.00000 0.208514
\(369\) 0 0
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 16.0000 0.827340
\(375\) 0 0
\(376\) −4.00000 −0.206284
\(377\) −36.0000 −1.85409
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 − 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −8.00000 −0.409316
\(383\) −16.0000 −0.817562 −0.408781 − 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 0 0
\(388\) −2.00000 −0.101535
\(389\) 10.0000 0.507020 0.253510 − 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 3.00000 0.151523
\(393\) 0 0
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) 0 0
\(397\) 24.0000 1.20453 0.602263 − 0.798298i \(-0.294266\pi\)
0.602263 + 0.798298i \(0.294266\pi\)
\(398\) 4.00000 0.200502
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 − 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −36.0000 −1.79329
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 12.0000 0.591198
\(413\) −20.0000 −0.984136
\(414\) 0 0
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 0 0
\(418\) 4.00000 0.195646
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −8.00000 −0.389896 −0.194948 − 0.980814i \(-0.562454\pi\)
−0.194948 + 0.980814i \(0.562454\pi\)
\(422\) 8.00000 0.389434
\(423\) 0 0
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 0 0
\(427\) 4.00000 0.193574
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 0 0
\(431\) 40.0000 1.92673 0.963366 − 0.268190i \(-0.0864254\pi\)
0.963366 + 0.268190i \(0.0864254\pi\)
\(432\) 0 0
\(433\) −30.0000 −1.44171 −0.720854 − 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 12.0000 0.576018
\(435\) 0 0
\(436\) −16.0000 −0.766261
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) −38.0000 −1.81364 −0.906821 − 0.421517i \(-0.861498\pi\)
−0.906821 + 0.421517i \(0.861498\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) 6.00000 0.285069 0.142534 − 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −12.0000 −0.566315 −0.283158 − 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 16.0000 0.753411
\(452\) −14.0000 −0.658505
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 26.0000 1.21490
\(459\) 0 0
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 − 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 18.0000 0.836531 0.418265 − 0.908325i \(-0.362638\pi\)
0.418265 + 0.908325i \(0.362638\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) −22.0000 −1.01804 −0.509019 − 0.860755i \(-0.669992\pi\)
−0.509019 + 0.860755i \(0.669992\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) 0 0
\(472\) 10.0000 0.460287
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) 0 0
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 − 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −60.0000 −2.73576
\(482\) −10.0000 −0.455488
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 − 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) −2.00000 −0.0905357
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −24.0000 −1.08091
\(494\) −6.00000 −0.269953
\(495\) 0 0
\(496\) −6.00000 −0.269408
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 − 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8.00000 −0.356702 −0.178351 − 0.983967i \(-0.557076\pi\)
−0.178351 + 0.983967i \(0.557076\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) −38.0000 −1.68432 −0.842160 − 0.539227i \(-0.818716\pi\)
−0.842160 + 0.539227i \(0.818716\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 20.0000 0.878750
\(519\) 0 0
\(520\) 0 0
\(521\) −40.0000 −1.75243 −0.876216 − 0.481919i \(-0.839940\pi\)
−0.876216 + 0.481919i \(0.839940\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 − 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 2.00000 0.0867110
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) 12.0000 0.518321
\(537\) 0 0
\(538\) 14.0000 0.603583
\(539\) 12.0000 0.516877
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 − 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 10.0000 0.423714 0.211857 − 0.977301i \(-0.432049\pi\)
0.211857 + 0.977301i \(0.432049\pi\)
\(558\) 0 0
\(559\) −72.0000 −3.04528
\(560\) 0 0
\(561\) 0 0
\(562\) 4.00000 0.168730
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) 0 0
\(568\) 8.00000 0.335673
\(569\) −12.0000 −0.503066 −0.251533 − 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 − 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −24.0000 −1.00349
\(573\) 0 0
\(574\) 8.00000 0.333914
\(575\) 0 0
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 − 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 1.00000 0.0415945
\(579\) 0 0
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) 0 0
\(583\) 40.0000 1.65663
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −2.00000 −0.0826192
\(587\) 22.0000 0.908037 0.454019 − 0.890992i \(-0.349990\pi\)
0.454019 + 0.890992i \(0.349990\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) 0 0
\(592\) −10.0000 −0.410997
\(593\) −8.00000 −0.328521 −0.164260 − 0.986417i \(-0.552524\pi\)
−0.164260 + 0.986417i \(0.552524\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) −24.0000 −0.981433
\(599\) 28.0000 1.14405 0.572024 − 0.820237i \(-0.306158\pi\)
0.572024 + 0.820237i \(0.306158\pi\)
\(600\) 0 0
\(601\) −6.00000 −0.244745 −0.122373 − 0.992484i \(-0.539050\pi\)
−0.122373 + 0.992484i \(0.539050\pi\)
\(602\) 24.0000 0.978167
\(603\) 0 0
\(604\) −2.00000 −0.0813788
\(605\) 0 0
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 − 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 − 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 8.00000 0.322329
\(617\) −48.0000 −1.93241 −0.966204 − 0.257780i \(-0.917009\pi\)
−0.966204 + 0.257780i \(0.917009\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.0000 0.641542
\(623\) 16.0000 0.641026
\(624\) 0 0
\(625\) 0 0
\(626\) 6.00000 0.239808
\(627\) 0 0
\(628\) −12.0000 −0.478852
\(629\) −40.0000 −1.59490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 − 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) −10.0000 −0.397779
\(633\) 0 0
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) −24.0000 −0.950169
\(639\) 0 0
\(640\) 0 0
\(641\) 20.0000 0.789953 0.394976 − 0.918691i \(-0.370753\pi\)
0.394976 + 0.918691i \(0.370753\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 − 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 8.00000 0.315244
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 40.0000 1.57014
\(650\) 0 0
\(651\) 0 0
\(652\) 16.0000 0.626608
\(653\) −6.00000 −0.234798 −0.117399 − 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −4.00000 −0.156174
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) −22.0000 −0.856998 −0.428499 − 0.903542i \(-0.640958\pi\)
−0.428499 + 0.903542i \(0.640958\pi\)
\(660\) 0 0
\(661\) −8.00000 −0.311164 −0.155582 − 0.987823i \(-0.549725\pi\)
−0.155582 + 0.987823i \(0.549725\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) 24.0000 0.928588
\(669\) 0 0
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) −26.0000 −1.00223 −0.501113 − 0.865382i \(-0.667076\pi\)
−0.501113 + 0.865382i \(0.667076\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) −6.00000 −0.230599 −0.115299 − 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 0 0
\(682\) −24.0000 −0.919007
\(683\) −12.0000 −0.459167 −0.229584 − 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 0 0
\(688\) −12.0000 −0.457496
\(689\) −60.0000 −2.28582
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 − 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) −18.0000 −0.684257
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) 0 0
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) −22.0000 −0.832712
\(699\) 0 0
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 − 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) −4.00000 −0.150756
\(705\) 0 0
\(706\) 0 0
\(707\) 20.0000 0.752177
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 − 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −8.00000 −0.299813
\(713\) −24.0000 −0.898807
\(714\) 0 0
\(715\) 0 0
\(716\) 2.00000 0.0747435
\(717\) 0 0
\(718\) 0 0
\(719\) −40.0000 −1.49175 −0.745874 − 0.666087i \(-0.767968\pi\)
−0.745874 + 0.666087i \(0.767968\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) −1.00000 −0.0372161
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −26.0000 −0.964287 −0.482143 − 0.876092i \(-0.660142\pi\)
−0.482143 + 0.876092i \(0.660142\pi\)
\(728\) −12.0000 −0.444750
\(729\) 0 0
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) 8.00000 0.295487 0.147743 − 0.989026i \(-0.452799\pi\)
0.147743 + 0.989026i \(0.452799\pi\)
\(734\) −22.0000 −0.812035
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 − 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 20.0000 0.734223
\(743\) 48.0000 1.76095 0.880475 − 0.474093i \(-0.157224\pi\)
0.880475 + 0.474093i \(0.157224\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.0000 0.512576
\(747\) 0 0
\(748\) −16.0000 −0.585018
\(749\) 16.0000 0.584627
\(750\) 0 0
\(751\) −38.0000 −1.38664 −0.693320 − 0.720630i \(-0.743853\pi\)
−0.693320 + 0.720630i \(0.743853\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 − 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) −8.00000 −0.290573
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 − 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 0 0
\(763\) −32.0000 −1.15848
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) −60.0000 −2.16647
\(768\) 0 0
\(769\) −18.0000 −0.649097 −0.324548 − 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) 6.00000 0.215805 0.107903 − 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) −16.0000 −0.572159
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) −4.00000 −0.142585 −0.0712923 − 0.997455i \(-0.522712\pi\)
−0.0712923 + 0.997455i \(0.522712\pi\)
\(788\) 22.0000 0.783718
\(789\) 0 0
\(790\) 0 0
\(791\) −28.0000 −0.995565
\(792\) 0 0
\(793\) 12.0000 0.426132
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) 2.00000 0.0708436 0.0354218 − 0.999372i \(-0.488723\pi\)
0.0354218 + 0.999372i \(0.488723\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 0 0
\(802\) −24.0000 −0.847469
\(803\) −8.00000 −0.282314
\(804\) 0 0
\(805\) 0 0
\(806\) 36.0000 1.26805
\(807\) 0 0
\(808\) −10.0000 −0.351799
\(809\) 42.0000 1.47664 0.738321 − 0.674450i \(-0.235619\pi\)
0.738321 + 0.674450i \(0.235619\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 − 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) −12.0000 −0.421117
\(813\) 0 0
\(814\) −40.0000 −1.40200
\(815\) 0 0
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 6.00000 0.209785
\(819\) 0 0
\(820\) 0 0
\(821\) 42.0000 1.46581 0.732905 − 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) 26.0000 0.906303 0.453152 − 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) −12.0000 −0.418040
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) 48.0000 1.66912 0.834562 − 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) −40.0000 −1.38926 −0.694629 − 0.719368i \(-0.744431\pi\)
−0.694629 + 0.719368i \(0.744431\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 6.00000 0.208013
\(833\) −12.0000 −0.415775
\(834\) 0 0
\(835\) 0 0
\(836\) −4.00000 −0.138343
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 8.00000 0.275698
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) −10.0000 −0.343401
\(849\) 0 0
\(850\) 0 0
\(851\) −40.0000 −1.37118
\(852\) 0 0
\(853\) 36.0000 1.23262 0.616308 − 0.787505i \(-0.288628\pi\)
0.616308 + 0.787505i \(0.288628\pi\)
\(854\) −4.00000 −0.136877
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 30.0000 1.02478 0.512390 − 0.858753i \(-0.328760\pi\)
0.512390 + 0.858753i \(0.328760\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 − 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −40.0000 −1.36241
\(863\) 32.0000 1.08929 0.544646 − 0.838666i \(-0.316664\pi\)
0.544646 + 0.838666i \(0.316664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 30.0000 1.01944
\(867\) 0 0
\(868\) −12.0000 −0.407307
\(869\) −40.0000 −1.35691
\(870\) 0 0
\(871\) −72.0000 −2.43963
\(872\) 16.0000 0.541828
\(873\) 0 0
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 − 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 38.0000 1.28244
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 − 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −36.0000 −1.21150 −0.605748 − 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) 48.0000 1.61168 0.805841 − 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 40.0000 1.34156
\(890\) 0 0
\(891\) 0 0
\(892\) −24.0000 −0.803579
\(893\) 4.00000 0.133855
\(894\) 0 0
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 36.0000 1.20067
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) −16.0000 −0.532742
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 − 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 20.0000 0.663723
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 − 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −26.0000 −0.859064
\(917\) −24.0000 −0.792550
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 18.0000 0.592798
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) 0 0
\(926\) −18.0000 −0.591517
\(927\) 0 0
\(928\) 6.00000 0.196960
\(929\) −42.0000 −1.37798 −0.688988 − 0.724773i \(-0.741945\pi\)
−0.688988 + 0.724773i \(0.741945\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) 22.0000 0.719862
\(935\) 0 0
\(936\) 0 0
\(937\) −26.0000 −0.849383 −0.424691 − 0.905338i \(-0.639617\pi\)
−0.424691 + 0.905338i \(0.639617\pi\)
\(938\) 24.0000 0.783628
\(939\) 0 0
\(940\) 0 0
\(941\) −10.0000 −0.325991 −0.162995 − 0.986627i \(-0.552116\pi\)
−0.162995 + 0.986627i \(0.552116\pi\)
\(942\) 0 0
\(943\) −16.0000 −0.521032
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) −38.0000 −1.23483 −0.617417 − 0.786636i \(-0.711821\pi\)
−0.617417 + 0.786636i \(0.711821\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) −8.00000 −0.259281
\(953\) 6.00000 0.194359 0.0971795 − 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −24.0000 −0.775405
\(959\) 0 0
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 60.0000 1.93448
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 18.0000 0.578841 0.289420 − 0.957202i \(-0.406537\pi\)
0.289420 + 0.957202i \(0.406537\pi\)
\(968\) −5.00000 −0.160706
\(969\) 0 0
\(970\) 0 0
\(971\) 42.0000 1.34784 0.673922 − 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) −8.00000 −0.256468
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −42.0000 −1.34370 −0.671850 − 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) 0 0
\(979\) −32.0000 −1.02272
\(980\) 0 0
\(981\) 0 0
\(982\) 12.0000 0.382935
\(983\) 24.0000 0.765481 0.382741 − 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 24.0000 0.764316
\(987\) 0 0
\(988\) 6.00000 0.190885
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) −2.00000 −0.0635321 −0.0317660 − 0.999495i \(-0.510113\pi\)
−0.0317660 + 0.999495i \(0.510113\pi\)
\(992\) 6.00000 0.190500
\(993\) 0 0
\(994\) 16.0000 0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) 52.0000 1.64686 0.823428 − 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) 4.00000 0.126618
\(999\) 0 0
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 8550.2.a.n.1.1 1
3.2 odd 2 2850.2.a.bb.1.1 1
5.4 even 2 1710.2.a.q.1.1 1
15.2 even 4 2850.2.d.r.799.2 2
15.8 even 4 2850.2.d.r.799.1 2
15.14 odd 2 570.2.a.a.1.1 ✓ 1
60.59 even 2 4560.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.2.a.a.1.1 ✓ 1 15.14 odd 2
1710.2.a.q.1.1 1 5.4 even 2
2850.2.a.bb.1.1 1 3.2 odd 2
2850.2.d.r.799.1 2 15.8 even 4
2850.2.d.r.799.2 2 15.2 even 4
4560.2.a.t.1.1 1 60.59 even 2
8550.2.a.n.1.1 1 1.1 even 1 trivial