Properties

Label 5775.2.a.g.1.1
Level $5775$
Weight $2$
Character 5775.1
Self dual yes
Analytic conductor $46.114$
Analytic rank $0$
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] = [5775,2,Mod(1,5775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5775, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5775.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5775.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: \(46.1136071673\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5775.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{21} +1.00000 q^{22} +4.00000 q^{23} +3.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -4.00000 q^{31} -5.00000 q^{32} -1.00000 q^{33} -6.00000 q^{34} -1.00000 q^{36} +2.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -2.00000 q^{41} +1.00000 q^{42} +12.0000 q^{43} +1.00000 q^{44} -4.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +6.00000 q^{51} -2.00000 q^{52} +2.00000 q^{53} -1.00000 q^{54} -3.00000 q^{56} +4.00000 q^{57} +2.00000 q^{58} -14.0000 q^{61} +4.00000 q^{62} -1.00000 q^{63} +7.00000 q^{64} +1.00000 q^{66} +8.00000 q^{67} -6.00000 q^{68} +4.00000 q^{69} -16.0000 q^{71} +3.00000 q^{72} -10.0000 q^{73} -2.00000 q^{74} -4.00000 q^{76} +1.00000 q^{77} -2.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} +2.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} -12.0000 q^{86} -2.00000 q^{87} -3.00000 q^{88} -6.00000 q^{89} -2.00000 q^{91} -4.00000 q^{92} -4.00000 q^{93} -5.00000 q^{96} -2.00000 q^{97} -1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 1.00000 0.213201
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −5.00000 −0.883883
\(33\) −1.00000 −0.174078
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.00000 −0.648886
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 1.00000 0.154303
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −2.00000 −0.277350
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) 4.00000 0.529813
\(58\) 2.00000 0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 4.00000 0.508001
\(63\) −1.00000 −0.125988
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −6.00000 −0.727607
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 3.00000 0.353553
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 1.00000 0.113961
\(78\) −2.00000 −0.226455
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) −2.00000 −0.214423
\(88\) −3.00000 −0.319801
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −4.00000 −0.417029
\(93\) −4.00000 −0.414781
\(94\) 0 0
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) −1.00000 −0.101015
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) −6.00000 −0.594089
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 1.00000 0.0944911
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 14.0000 1.26750
\(123\) −2.00000 −0.180334
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 3.00000 0.265165
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 1.00000 0.0870388
\(133\) −4.00000 −0.346844
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) 18.0000 1.54349
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −4.00000 −0.340503
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 16.0000 1.34269
\(143\) −2.00000 −0.167248
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 10.0000 0.827606
\(147\) 1.00000 0.0824786
\(148\) −2.00000 −0.164399
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 12.0000 0.973329
\(153\) 6.00000 0.485071
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −16.0000 −1.27289
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) −1.00000 −0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −4.00000 −0.310460
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) −3.00000 −0.231455
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −12.0000 −0.914991
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 2.00000 0.148250
\(183\) −14.0000 −1.03491
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) −6.00000 −0.438763
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 7.00000 0.505181
\(193\) −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
\(196\) −1.00000 −0.0714286
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 1.00000 0.0710669
\(199\) 28.0000 1.98487 0.992434 0.122782i \(-0.0391815\pi\)
0.992434 + 0.122782i \(0.0391815\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) −10.0000 −0.703598
\(203\) 2.00000 0.140372
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000 0.278019
\(208\) −2.00000 −0.138675
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) −2.00000 −0.137361
\(213\) −16.0000 −1.09630
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 4.00000 0.271538
\(218\) 10.0000 0.677285
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −2.00000 −0.134231
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 5.00000 0.334077
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) −6.00000 −0.393919
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 6.00000 0.388922
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 2.00000 0.127515
\(247\) 8.00000 0.509028
\(248\) −12.0000 −0.762001
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 1.00000 0.0629941
\(253\) −4.00000 −0.251478
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) −12.0000 −0.747087
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) −20.0000 −1.23560
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) 4.00000 0.245256
\(267\) −6.00000 −0.367194
\(268\) −8.00000 −0.488678
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) −6.00000 −0.363803
\(273\) −2.00000 −0.121046
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) −20.0000 −1.19952
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 2.00000 0.118056
\(288\) −5.00000 −0.294628
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 10.0000 0.585206
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 6.00000 0.348743
\(297\) −1.00000 −0.0580259
\(298\) −6.00000 −0.347571
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) 16.0000 0.920697
\(303\) 10.0000 0.574485
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) −1.00000 −0.0569803
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 6.00000 0.339683
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −2.00000 −0.112154
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 4.00000 0.222911
\(323\) 24.0000 1.33540
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) −10.0000 −0.553001
\(328\) −6.00000 −0.331295
\(329\) 0 0
\(330\) 0 0
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) −4.00000 −0.219529
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 9.00000 0.489535
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) −4.00000 −0.216295
\(343\) −1.00000 −0.0539949
\(344\) 36.0000 1.94099
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) 2.00000 0.107211
\(349\) 18.0000 0.963518 0.481759 0.876304i \(-0.339998\pi\)
0.481759 + 0.876304i \(0.339998\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 5.00000 0.266501
\(353\) 30.0000 1.59674 0.798369 0.602168i \(-0.205696\pi\)
0.798369 + 0.602168i \(0.205696\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) −6.00000 −0.317554
\(358\) −20.0000 −1.05703
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −14.0000 −0.735824
\(363\) 1.00000 0.0524864
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 14.0000 0.731792
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) −4.00000 −0.208514
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 4.00000 0.207390
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 6.00000 0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) −4.00000 −0.206010
\(378\) 1.00000 0.0514344
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 8.00000 0.409316
\(383\) 8.00000 0.408781 0.204390 0.978889i \(-0.434479\pi\)
0.204390 + 0.978889i \(0.434479\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 12.0000 0.609994
\(388\) 2.00000 0.101535
\(389\) −26.0000 −1.31825 −0.659126 0.752032i \(-0.729074\pi\)
−0.659126 + 0.752032i \(0.729074\pi\)
\(390\) 0 0
\(391\) 24.0000 1.21373
\(392\) 3.00000 0.151523
\(393\) 20.0000 1.00887
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) 1.00000 0.0502519
\(397\) 18.0000 0.903394 0.451697 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) −28.0000 −1.40351
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) 18.0000 0.898877 0.449439 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −8.00000 −0.399004
\(403\) −8.00000 −0.398508
\(404\) −10.0000 −0.497519
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −2.00000 −0.0991363
\(408\) 18.0000 0.891133
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 8.00000 0.394132
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −10.0000 −0.490290
\(417\) 20.0000 0.979404
\(418\) 4.00000 0.195646
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −20.0000 −0.973585
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 16.0000 0.775203
\(427\) 14.0000 0.677507
\(428\) 4.00000 0.193347
\(429\) −2.00000 −0.0965609
\(430\) 0 0
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) 16.0000 0.765384
\(438\) 10.0000 0.477818
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −12.0000 −0.570782
\(443\) 8.00000 0.380091 0.190046 0.981775i \(-0.439136\pi\)
0.190046 + 0.981775i \(0.439136\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 6.00000 0.283790
\(448\) −7.00000 −0.330719
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −6.00000 −0.282216
\(453\) −16.0000 −0.751746
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) −30.0000 −1.40334 −0.701670 0.712502i \(-0.747562\pi\)
−0.701670 + 0.712502i \(0.747562\pi\)
\(458\) −14.0000 −0.654177
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) −1.00000 −0.0465242
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 28.0000 1.29569 0.647843 0.761774i \(-0.275671\pi\)
0.647843 + 0.761774i \(0.275671\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) −16.0000 −0.734904
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 2.00000 0.0915737
\(478\) 24.0000 1.09773
\(479\) −32.0000 −1.46212 −0.731059 0.682315i \(-0.760973\pi\)
−0.731059 + 0.682315i \(0.760973\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 10.0000 0.455488
\(483\) −4.00000 −0.182006
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −42.0000 −1.90125
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 2.00000 0.0901670
\(493\) −12.0000 −0.540453
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) 16.0000 0.717698
\(498\) −4.00000 −0.179244
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) −3.00000 −0.133631
\(505\) 0 0
\(506\) 4.00000 0.177822
\(507\) −9.00000 −0.399704
\(508\) 8.00000 0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 10.0000 0.442374
\(512\) 11.0000 0.486136
\(513\) 4.00000 0.176604
\(514\) −14.0000 −0.617514
\(515\) 0 0
\(516\) −12.0000 −0.528271
\(517\) 0 0
\(518\) 2.00000 0.0878750
\(519\) 10.0000 0.438951
\(520\) 0 0
\(521\) 34.0000 1.48957 0.744784 0.667306i \(-0.232553\pi\)
0.744784 + 0.667306i \(0.232553\pi\)
\(522\) 2.00000 0.0875376
\(523\) −20.0000 −0.874539 −0.437269 0.899331i \(-0.644054\pi\)
−0.437269 + 0.899331i \(0.644054\pi\)
\(524\) −20.0000 −0.873704
\(525\) 0 0
\(526\) 0 0
\(527\) −24.0000 −1.04546
\(528\) 1.00000 0.0435194
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) −4.00000 −0.173259
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) 24.0000 1.03664
\(537\) 20.0000 0.863064
\(538\) 26.0000 1.12094
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) 8.00000 0.343629
\(543\) 14.0000 0.600798
\(544\) −30.0000 −1.28624
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 2.00000 0.0854358
\(549\) −14.0000 −0.597505
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 12.0000 0.510754
\(553\) −16.0000 −0.680389
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 4.00000 0.169334
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) −6.00000 −0.253320
\(562\) 6.00000 0.253095
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) −1.00000 −0.0419961
\(568\) −48.0000 −2.01404
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 2.00000 0.0836242
\(573\) −8.00000 −0.334205
\(574\) −2.00000 −0.0834784
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) −19.0000 −0.790296
\(579\) −14.0000 −0.581820
\(580\) 0 0
\(581\) −4.00000 −0.165948
\(582\) 2.00000 0.0829027
\(583\) −2.00000 −0.0828315
\(584\) −30.0000 −1.24141
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 36.0000 1.48588 0.742940 0.669359i \(-0.233431\pi\)
0.742940 + 0.669359i \(0.233431\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) −2.00000 −0.0821995
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 28.0000 1.14596
\(598\) −8.00000 −0.327144
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 0 0
\(601\) −18.0000 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(602\) 12.0000 0.489083
\(603\) 8.00000 0.325785
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −20.0000 −0.811107
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) 0 0
\(612\) −6.00000 −0.242536
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 8.00000 0.321807
\(619\) −48.0000 −1.92928 −0.964641 0.263566i \(-0.915101\pi\)
−0.964641 + 0.263566i \(0.915101\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) −20.0000 −0.801927
\(623\) 6.00000 0.240385
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) −4.00000 −0.159745
\(628\) −2.00000 −0.0798087
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 48.0000 1.90934
\(633\) 20.0000 0.794929
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 2.00000 0.0792429
\(638\) −2.00000 −0.0791808
\(639\) −16.0000 −0.632950
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 4.00000 0.157867
\(643\) 12.0000 0.473234 0.236617 0.971603i \(-0.423961\pi\)
0.236617 + 0.971603i \(0.423961\pi\)
\(644\) 4.00000 0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 3.00000 0.117851
\(649\) 0 0
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) −8.00000 −0.313304
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 10.0000 0.391031
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 12.0000 0.466393
\(663\) 12.0000 0.466041
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) −8.00000 −0.309761
\(668\) 0 0
\(669\) −24.0000 −0.927894
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 5.00000 0.192879
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) −6.00000 −0.230429
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) −4.00000 −0.153168
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) 14.0000 0.534133
\(688\) −12.0000 −0.457496
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 16.0000 0.608669 0.304334 0.952565i \(-0.401566\pi\)
0.304334 + 0.952565i \(0.401566\pi\)
\(692\) −10.0000 −0.380143
\(693\) 1.00000 0.0379869
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) −12.0000 −0.454532
\(698\) −18.0000 −0.681310
\(699\) 26.0000 0.983410
\(700\) 0 0
\(701\) −2.00000 −0.0755390 −0.0377695 0.999286i \(-0.512025\pi\)
−0.0377695 + 0.999286i \(0.512025\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 8.00000 0.301726
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) −30.0000 −1.12906
\(707\) −10.0000 −0.376089
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) −18.0000 −0.674579
\(713\) −16.0000 −0.599205
\(714\) 6.00000 0.224544
\(715\) 0 0
\(716\) −20.0000 −0.747435
\(717\) −24.0000 −0.896296
\(718\) −24.0000 −0.895672
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) −10.0000 −0.371904
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) −1.00000 −0.0371135
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 72.0000 2.66302
\(732\) 14.0000 0.517455
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) −8.00000 −0.294684
\(738\) 2.00000 0.0736210
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) 8.00000 0.293887
\(742\) 2.00000 0.0734223
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) −12.0000 −0.439941
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 4.00000 0.146352
\(748\) 6.00000 0.219382
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) 42.0000 1.52652 0.763258 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 20.0000 0.726433
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 8.00000 0.289809
\(763\) 10.0000 0.362024
\(764\) 8.00000 0.289430
\(765\) 0 0
\(766\) −8.00000 −0.289052
\(767\) 0 0
\(768\) −17.0000 −0.613435
\(769\) −18.0000 −0.649097 −0.324548 0.945869i \(-0.605212\pi\)
−0.324548 + 0.945869i \(0.605212\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 14.0000 0.503871
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) −12.0000 −0.431331
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) −2.00000 −0.0717496
\(778\) 26.0000 0.932145
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 16.0000 0.572525
\(782\) −24.0000 −0.858238
\(783\) −2.00000 −0.0714742
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −20.0000 −0.713376
\(787\) −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
\(791\) −6.00000 −0.213335
\(792\) −3.00000 −0.106600
\(793\) −28.0000 −0.994309
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −28.0000 −0.992434
\(797\) 34.0000 1.20434 0.602171 0.798367i \(-0.294303\pi\)
0.602171 + 0.798367i \(0.294303\pi\)
\(798\) 4.00000 0.141598
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) −18.0000 −0.635602
\(803\) 10.0000 0.352892
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) −26.0000 −0.915243
\(808\) 30.0000 1.05540
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) −2.00000 −0.0701862
\(813\) −8.00000 −0.280572
\(814\) 2.00000 0.0701000
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 48.0000 1.67931
\(818\) −14.0000 −0.489499
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 2.00000 0.0697580
\(823\) 20.0000 0.697156 0.348578 0.937280i \(-0.386665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(824\) −24.0000 −0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) −4.00000 −0.139010
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 14.0000 0.485363
\(833\) 6.00000 0.207888
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) −4.00000 −0.138260
\(838\) −16.0000 −0.552711
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −6.00000 −0.206774
\(843\) −6.00000 −0.206651
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −2.00000 −0.0686803
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 16.0000 0.548151
\(853\) −14.0000 −0.479351 −0.239675 0.970853i \(-0.577041\pi\)
−0.239675 + 0.970853i \(0.577041\pi\)
\(854\) −14.0000 −0.479070
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −10.0000 −0.341593 −0.170797 0.985306i \(-0.554634\pi\)
−0.170797 + 0.985306i \(0.554634\pi\)
\(858\) 2.00000 0.0682789
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 2.00000 0.0681598
\(862\) −16.0000 −0.544962
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −14.0000 −0.475739
\(867\) 19.0000 0.645274
\(868\) −4.00000 −0.135769
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −30.0000 −1.01593
\(873\) −2.00000 −0.0676897
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 10.0000 0.337869
\(877\) 6.00000 0.202606 0.101303 0.994856i \(-0.467699\pi\)
0.101303 + 0.994856i \(0.467699\pi\)
\(878\) 32.0000 1.07995
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 6.00000 0.201347
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 24.0000 0.803579
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) −3.00000 −0.100223
\(897\) 8.00000 0.267112
\(898\) −18.0000 −0.600668
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −2.00000 −0.0665927
\(903\) −12.0000 −0.399335
\(904\) 18.0000 0.598671
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 20.0000 0.663723
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −4.00000 −0.132453
\(913\) −4.00000 −0.132381
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) −20.0000 −0.660458
\(918\) −6.00000 −0.198030
\(919\) 56.0000 1.84727 0.923635 0.383274i \(-0.125203\pi\)
0.923635 + 0.383274i \(0.125203\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 6.00000 0.197599
\(923\) −32.0000 −1.05329
\(924\) −1.00000 −0.0328976
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) −8.00000 −0.262754
\(928\) 10.0000 0.328266
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) 4.00000 0.131095
\(932\) −26.0000 −0.851658
\(933\) 20.0000 0.654771
\(934\) −28.0000 −0.916188
\(935\) 0 0
\(936\) 6.00000 0.196116
\(937\) −34.0000 −1.11073 −0.555366 0.831606i \(-0.687422\pi\)
−0.555366 + 0.831606i \(0.687422\pi\)
\(938\) 8.00000 0.261209
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 12.0000 0.390154
\(947\) 48.0000 1.55979 0.779895 0.625910i \(-0.215272\pi\)
0.779895 + 0.625910i \(0.215272\pi\)
\(948\) −16.0000 −0.519656
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) −18.0000 −0.583383
\(953\) 34.0000 1.10137 0.550684 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) −2.00000 −0.0647524
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 2.00000 0.0646508
\(958\) 32.0000 1.03387
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) −4.00000 −0.128965
\(963\) −4.00000 −0.128898
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 4.00000 0.128698
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 3.00000 0.0964237
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −8.00000 −0.256732 −0.128366 0.991727i \(-0.540973\pi\)
−0.128366 + 0.991727i \(0.540973\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −20.0000 −0.641171
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) −58.0000 −1.85558 −0.927792 0.373097i \(-0.878296\pi\)
−0.927792 + 0.373097i \(0.878296\pi\)
\(978\) −8.00000 −0.255812
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 36.0000 1.14881
\(983\) 56.0000 1.78612 0.893061 0.449935i \(-0.148553\pi\)
0.893061 + 0.449935i \(0.148553\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 12.0000 0.382158
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) 48.0000 1.52631
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 20.0000 0.635001
\(993\) −12.0000 −0.380808
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) −4.00000 −0.126745
\(997\) −46.0000 −1.45683 −0.728417 0.685134i \(-0.759744\pi\)
−0.728417 + 0.685134i \(0.759744\pi\)
\(998\) −4.00000 −0.126618
\(999\) 2.00000 0.0632772
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 5775.2.a.g.1.1 1
5.4 even 2 1155.2.a.k.1.1 1
15.14 odd 2 3465.2.a.b.1.1 1
35.34 odd 2 8085.2.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.k.1.1 1 5.4 even 2
3465.2.a.b.1.1 1 15.14 odd 2
5775.2.a.g.1.1 1 1.1 even 1 trivial
8085.2.a.u.1.1 1 35.34 odd 2