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Properties

Label 4650.2.a.p.1.1
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
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] = [4650,2,Mod(1,4650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4650, 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("4650.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.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: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 4650.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +1.00000 q^{12} +4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{21} +3.00000 q^{22} -5.00000 q^{23} -1.00000 q^{24} -4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} +1.00000 q^{36} -4.00000 q^{37} +1.00000 q^{38} +4.00000 q^{39} -10.0000 q^{41} +1.00000 q^{42} +5.00000 q^{43} -3.00000 q^{44} +5.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} +4.00000 q^{52} -5.00000 q^{53} -1.00000 q^{54} +1.00000 q^{56} -1.00000 q^{57} -2.00000 q^{58} -6.00000 q^{59} -2.00000 q^{61} +1.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +3.00000 q^{66} -2.00000 q^{67} -5.00000 q^{69} -5.00000 q^{71} -1.00000 q^{72} +7.00000 q^{73} +4.00000 q^{74} -1.00000 q^{76} +3.00000 q^{77} -4.00000 q^{78} +3.00000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +2.00000 q^{83} -1.00000 q^{84} -5.00000 q^{86} +2.00000 q^{87} +3.00000 q^{88} +1.00000 q^{89} -4.00000 q^{91} -5.00000 q^{92} -1.00000 q^{93} -8.00000 q^{94} -1.00000 q^{96} +10.0000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964 −0.188982 − 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 3.00000 0.639602
\(23\) −5.00000 −1.04257 −0.521286 − 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) −3.00000 −0.522233
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.00000 −0.657596 −0.328798 − 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 1.00000 0.162221
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −10.0000 −1.56174 −0.780869 − 0.624695i \(-0.785223\pi\)
−0.780869 + 0.624695i \(0.785223\pi\)
\(42\) 1.00000 0.154303
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) 5.00000 0.737210
\(47\) 8.00000 1.16692 0.583460 − 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −5.00000 −0.686803 −0.343401 − 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −1.00000 −0.132453
\(58\) −2.00000 −0.262613
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 1.00000 0.127000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −2.00000 −0.244339 −0.122169 − 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) 0 0
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 − 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) −1.00000 −0.117851
\(73\) 7.00000 0.819288 0.409644 − 0.912245i \(-0.365653\pi\)
0.409644 + 0.912245i \(0.365653\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 3.00000 0.341882
\(78\) −4.00000 −0.452911
\(79\) 3.00000 0.337526 0.168763 − 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.0000 1.10432
\(83\) 2.00000 0.219529 0.109764 − 0.993958i \(-0.464990\pi\)
0.109764 + 0.993958i \(0.464990\pi\)
\(84\) −1.00000 −0.109109
\(85\) 0 0
\(86\) −5.00000 −0.539164
\(87\) 2.00000 0.214423
\(88\) 3.00000 0.319801
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −5.00000 −0.521286
\(93\) −1.00000 −0.103695
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 6.00000 0.606092
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 − 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 5.00000 0.485643
\(107\) −19.0000 −1.83680 −0.918400 − 0.395654i \(-0.870518\pi\)
−0.918400 + 0.395654i \(0.870518\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) −1.00000 −0.0944911
\(113\) 15.0000 1.41108 0.705541 − 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) 4.00000 0.369800
\(118\) 6.00000 0.552345
\(119\) 0 0
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 2.00000 0.181071
\(123\) −10.0000 −0.901670
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) −8.00000 −0.709885 −0.354943 − 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) −2.00000 −0.174741 −0.0873704 − 0.996176i \(-0.527846\pi\)
−0.0873704 + 0.996176i \(0.527846\pi\)
\(132\) −3.00000 −0.261116
\(133\) 1.00000 0.0867110
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 5.00000 0.425628
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 8.00000 0.673722
\(142\) 5.00000 0.419591
\(143\) −12.0000 −1.00349
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −7.00000 −0.579324
\(147\) −6.00000 −0.494872
\(148\) −4.00000 −0.328798
\(149\) 15.0000 1.22885 0.614424 − 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 − 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 1.00000 0.0811107
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) 4.00000 0.320256
\(157\) −9.00000 −0.718278 −0.359139 − 0.933284i \(-0.616930\pi\)
−0.359139 + 0.933284i \(0.616930\pi\)
\(158\) −3.00000 −0.238667
\(159\) −5.00000 −0.396526
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 − 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) −10.0000 −0.780869
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) −21.0000 −1.62503 −0.812514 − 0.582941i \(-0.801902\pi\)
−0.812514 + 0.582941i \(0.801902\pi\)
\(168\) 1.00000 0.0771517
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 5.00000 0.381246
\(173\) 2.00000 0.152057 0.0760286 − 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) −6.00000 −0.450988
\(178\) −1.00000 −0.0749532
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 − 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 4.00000 0.296500
\(183\) −2.00000 −0.147844
\(184\) 5.00000 0.368605
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 − 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) −10.0000 −0.717958
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −18.0000 −1.28245 −0.641223 − 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 3.00000 0.213201
\(199\) −21.0000 −1.48865 −0.744325 − 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 15.0000 1.05540
\(203\) −2.00000 −0.140372
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.00000 −0.347524
\(208\) 4.00000 0.277350
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 7.00000 0.481900 0.240950 − 0.970538i \(-0.422541\pi\)
0.240950 + 0.970538i \(0.422541\pi\)
\(212\) −5.00000 −0.343401
\(213\) −5.00000 −0.342594
\(214\) 19.0000 1.29881
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 1.00000 0.0678844
\(218\) −2.00000 −0.135457
\(219\) 7.00000 0.473016
\(220\) 0 0
\(221\) 0 0
\(222\) 4.00000 0.268462
\(223\) 20.0000 1.33930 0.669650 − 0.742677i \(-0.266444\pi\)
0.669650 + 0.742677i \(0.266444\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −15.0000 −0.997785
\(227\) 9.00000 0.597351 0.298675 − 0.954355i \(-0.403455\pi\)
0.298675 + 0.954355i \(0.403455\pi\)
\(228\) −1.00000 −0.0662266
\(229\) 15.0000 0.991228 0.495614 − 0.868543i \(-0.334943\pi\)
0.495614 + 0.868543i \(0.334943\pi\)
\(230\) 0 0
\(231\) 3.00000 0.197386
\(232\) −2.00000 −0.131306
\(233\) 21.0000 1.37576 0.687878 − 0.725826i \(-0.258542\pi\)
0.687878 + 0.725826i \(0.258542\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 3.00000 0.194871
\(238\) 0 0
\(239\) −14.0000 −0.905585 −0.452792 − 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 0 0
\(241\) −12.0000 −0.772988 −0.386494 − 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 2.00000 0.128565
\(243\) 1.00000 0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 10.0000 0.637577
\(247\) −4.00000 −0.254514
\(248\) 1.00000 0.0635001
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 − 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 15.0000 0.943042
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −5.00000 −0.311891 −0.155946 − 0.987766i \(-0.549842\pi\)
−0.155946 + 0.987766i \(0.549842\pi\)
\(258\) −5.00000 −0.311286
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 2.00000 0.123560
\(263\) −24.0000 −1.47990 −0.739952 − 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −1.00000 −0.0613139
\(267\) 1.00000 0.0611990
\(268\) −2.00000 −0.122169
\(269\) 2.00000 0.121942 0.0609711 − 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 9.00000 0.546711 0.273356 − 0.961913i \(-0.411866\pi\)
0.273356 + 0.961913i \(0.411866\pi\)
\(272\) 0 0
\(273\) −4.00000 −0.242091
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) −5.00000 −0.300965
\(277\) −2.00000 −0.120168 −0.0600842 − 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 20.0000 1.19952
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −32.0000 −1.90896 −0.954480 − 0.298275i \(-0.903589\pi\)
−0.954480 + 0.298275i \(0.903589\pi\)
\(282\) −8.00000 −0.476393
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −5.00000 −0.296695
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 10.0000 0.590281
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 10.0000 0.586210
\(292\) 7.00000 0.409644
\(293\) 18.0000 1.05157 0.525786 − 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) −3.00000 −0.174078
\(298\) −15.0000 −0.868927
\(299\) −20.0000 −1.15663
\(300\) 0 0
\(301\) −5.00000 −0.288195
\(302\) 16.0000 0.920697
\(303\) −15.0000 −0.861727
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) 4.00000 0.228292 0.114146 − 0.993464i \(-0.463587\pi\)
0.114146 + 0.993464i \(0.463587\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 0 0
\(311\) 16.0000 0.907277 0.453638 − 0.891186i \(-0.350126\pi\)
0.453638 + 0.891186i \(0.350126\pi\)
\(312\) −4.00000 −0.226455
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 9.00000 0.507899
\(315\) 0 0
\(316\) 3.00000 0.168763
\(317\) −18.0000 −1.01098 −0.505490 − 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 5.00000 0.280386
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) −19.0000 −1.06048
\(322\) −5.00000 −0.278639
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 2.00000 0.110600
\(328\) 10.0000 0.552158
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 − 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 2.00000 0.109764
\(333\) −4.00000 −0.219199
\(334\) 21.0000 1.14907
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 34.0000 1.85210 0.926049 − 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −3.00000 −0.163178
\(339\) 15.0000 0.814688
\(340\) 0 0
\(341\) 3.00000 0.162459
\(342\) 1.00000 0.0540738
\(343\) 13.0000 0.701934
\(344\) −5.00000 −0.269582
\(345\) 0 0
\(346\) −2.00000 −0.107521
\(347\) 22.0000 1.18102 0.590511 − 0.807030i \(-0.298926\pi\)
0.590511 + 0.807030i \(0.298926\pi\)
\(348\) 2.00000 0.107211
\(349\) 20.0000 1.07058 0.535288 − 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 0 0
\(351\) 4.00000 0.213504
\(352\) 3.00000 0.159901
\(353\) −24.0000 −1.27739 −0.638696 − 0.769460i \(-0.720526\pi\)
−0.638696 + 0.769460i \(0.720526\pi\)
\(354\) 6.00000 0.318896
\(355\) 0 0
\(356\) 1.00000 0.0529999
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 0.158334 0.0791670 − 0.996861i \(-0.474774\pi\)
0.0791670 + 0.996861i \(0.474774\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 5.00000 0.262794
\(363\) −2.00000 −0.104973
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) −14.0000 −0.730794 −0.365397 − 0.930852i \(-0.619067\pi\)
−0.365397 + 0.930852i \(0.619067\pi\)
\(368\) −5.00000 −0.260643
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) 5.00000 0.259587
\(372\) −1.00000 −0.0518476
\(373\) 11.0000 0.569558 0.284779 − 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 8.00000 0.412021
\(378\) 1.00000 0.0514344
\(379\) −15.0000 −0.770498 −0.385249 − 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) 16.0000 0.818631
\(383\) −28.0000 −1.43073 −0.715367 − 0.698749i \(-0.753740\pi\)
−0.715367 + 0.698749i \(0.753740\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 5.00000 0.254164
\(388\) 10.0000 0.507673
\(389\) −28.0000 −1.41966 −0.709828 − 0.704375i \(-0.751227\pi\)
−0.709828 + 0.704375i \(0.751227\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 6.00000 0.303046
\(393\) −2.00000 −0.100887
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) 31.0000 1.55585 0.777923 − 0.628360i \(-0.216273\pi\)
0.777923 + 0.628360i \(0.216273\pi\)
\(398\) 21.0000 1.05263
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) 5.00000 0.249688 0.124844 − 0.992176i \(-0.460157\pi\)
0.124844 + 0.992176i \(0.460157\pi\)
\(402\) 2.00000 0.0997509
\(403\) −4.00000 −0.199254
\(404\) −15.0000 −0.746278
\(405\) 0 0
\(406\) 2.00000 0.0992583
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 − 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) 6.00000 0.295241
\(414\) 5.00000 0.245737
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) −20.0000 −0.979404
\(418\) −3.00000 −0.146735
\(419\) −18.0000 −0.879358 −0.439679 − 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 − 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) −7.00000 −0.340755
\(423\) 8.00000 0.388973
\(424\) 5.00000 0.242821
\(425\) 0 0
\(426\) 5.00000 0.242251
\(427\) 2.00000 0.0967868
\(428\) −19.0000 −0.918400
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −19.0000 −0.913082 −0.456541 − 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) −1.00000 −0.0480015
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 5.00000 0.239182
\(438\) −7.00000 −0.334473
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) −29.0000 −1.37783 −0.688916 − 0.724841i \(-0.741913\pi\)
−0.688916 + 0.724841i \(0.741913\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −20.0000 −0.947027
\(447\) 15.0000 0.709476
\(448\) −1.00000 −0.0472456
\(449\) 2.00000 0.0943858 0.0471929 − 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 30.0000 1.41264
\(452\) 15.0000 0.705541
\(453\) −16.0000 −0.751746
\(454\) −9.00000 −0.422391
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) −30.0000 −1.40334 −0.701670 − 0.712502i \(-0.747562\pi\)
−0.701670 + 0.712502i \(0.747562\pi\)
\(458\) −15.0000 −0.700904
\(459\) 0 0
\(460\) 0 0
\(461\) 16.0000 0.745194 0.372597 − 0.927993i \(-0.378467\pi\)
0.372597 + 0.927993i \(0.378467\pi\)
\(462\) −3.00000 −0.139573
\(463\) 14.0000 0.650635 0.325318 − 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −21.0000 −0.972806
\(467\) −28.0000 −1.29569 −0.647843 − 0.761774i \(-0.724329\pi\)
−0.647843 + 0.761774i \(0.724329\pi\)
\(468\) 4.00000 0.184900
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) −9.00000 −0.414698
\(472\) 6.00000 0.276172
\(473\) −15.0000 −0.689701
\(474\) −3.00000 −0.137795
\(475\) 0 0
\(476\) 0 0
\(477\) −5.00000 −0.228934
\(478\) 14.0000 0.640345
\(479\) −25.0000 −1.14228 −0.571140 − 0.820853i \(-0.693499\pi\)
−0.571140 + 0.820853i \(0.693499\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 12.0000 0.546585
\(483\) 5.00000 0.227508
\(484\) −2.00000 −0.0909091
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 4.00000 0.181257 0.0906287 − 0.995885i \(-0.471112\pi\)
0.0906287 + 0.995885i \(0.471112\pi\)
\(488\) 2.00000 0.0905357
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 33.0000 1.48927 0.744635 − 0.667472i \(-0.232624\pi\)
0.744635 + 0.667472i \(0.232624\pi\)
\(492\) −10.0000 −0.450835
\(493\) 0 0
\(494\) 4.00000 0.179969
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 5.00000 0.224281
\(498\) −2.00000 −0.0896221
\(499\) 22.0000 0.984855 0.492428 − 0.870353i \(-0.336110\pi\)
0.492428 + 0.870353i \(0.336110\pi\)
\(500\) 0 0
\(501\) −21.0000 −0.938211
\(502\) −12.0000 −0.535586
\(503\) −14.0000 −0.624229 −0.312115 − 0.950044i \(-0.601037\pi\)
−0.312115 + 0.950044i \(0.601037\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) −15.0000 −0.666831
\(507\) 3.00000 0.133235
\(508\) −8.00000 −0.354943
\(509\) 32.0000 1.41838 0.709188 − 0.705020i \(-0.249062\pi\)
0.709188 + 0.705020i \(0.249062\pi\)
\(510\) 0 0
\(511\) −7.00000 −0.309662
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) 5.00000 0.220541
\(515\) 0 0
\(516\) 5.00000 0.220113
\(517\) −24.0000 −1.05552
\(518\) −4.00000 −0.175750
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −19.0000 −0.830812 −0.415406 − 0.909636i \(-0.636360\pi\)
−0.415406 + 0.909636i \(0.636360\pi\)
\(524\) −2.00000 −0.0873704
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 0 0
\(528\) −3.00000 −0.130558
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 1.00000 0.0433555
\(533\) −40.0000 −1.73259
\(534\) −1.00000 −0.0432742
\(535\) 0 0
\(536\) 2.00000 0.0863868
\(537\) 0 0
\(538\) −2.00000 −0.0862261
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 24.0000 1.03184 0.515920 − 0.856637i \(-0.327450\pi\)
0.515920 + 0.856637i \(0.327450\pi\)
\(542\) −9.00000 −0.386583
\(543\) −5.00000 −0.214571
\(544\) 0 0
\(545\) 0 0
\(546\) 4.00000 0.171184
\(547\) −28.0000 −1.19719 −0.598597 − 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −2.00000 −0.0854358
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −2.00000 −0.0852029
\(552\) 5.00000 0.212814
\(553\) −3.00000 −0.127573
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) −9.00000 −0.381342 −0.190671 − 0.981654i \(-0.561066\pi\)
−0.190671 + 0.981654i \(0.561066\pi\)
\(558\) 1.00000 0.0423334
\(559\) 20.0000 0.845910
\(560\) 0 0
\(561\) 0 0
\(562\) 32.0000 1.34984
\(563\) −36.0000 −1.51722 −0.758610 − 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 8.00000 0.336861
\(565\) 0 0
\(566\) 14.0000 0.588464
\(567\) −1.00000 −0.0419961
\(568\) 5.00000 0.209795
\(569\) −39.0000 −1.63497 −0.817483 − 0.575953i \(-0.804631\pi\)
−0.817483 + 0.575953i \(0.804631\pi\)
\(570\) 0 0
\(571\) 26.0000 1.08807 0.544033 − 0.839064i \(-0.316897\pi\)
0.544033 + 0.839064i \(0.316897\pi\)
\(572\) −12.0000 −0.501745
\(573\) −16.0000 −0.668410
\(574\) −10.0000 −0.417392
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 17.0000 0.707107
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) −2.00000 −0.0829740
\(582\) −10.0000 −0.414513
\(583\) 15.0000 0.621237
\(584\) −7.00000 −0.289662
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) −6.00000 −0.247647 −0.123823 − 0.992304i \(-0.539516\pi\)
−0.123823 + 0.992304i \(0.539516\pi\)
\(588\) −6.00000 −0.247436
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) −18.0000 −0.740421
\(592\) −4.00000 −0.164399
\(593\) −6.00000 −0.246390 −0.123195 − 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) −21.0000 −0.859473
\(598\) 20.0000 0.817861
\(599\) 13.0000 0.531166 0.265583 − 0.964088i \(-0.414436\pi\)
0.265583 + 0.964088i \(0.414436\pi\)
\(600\) 0 0
\(601\) −44.0000 −1.79480 −0.897399 − 0.441221i \(-0.854546\pi\)
−0.897399 + 0.441221i \(0.854546\pi\)
\(602\) 5.00000 0.203785
\(603\) −2.00000 −0.0814463
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 15.0000 0.609333
\(607\) 27.0000 1.09590 0.547948 − 0.836512i \(-0.315409\pi\)
0.547948 + 0.836512i \(0.315409\pi\)
\(608\) 1.00000 0.0405554
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) 32.0000 1.29458
\(612\) 0 0
\(613\) −24.0000 −0.969351 −0.484675 − 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) −5.00000 −0.201292 −0.100646 − 0.994922i \(-0.532091\pi\)
−0.100646 + 0.994922i \(0.532091\pi\)
\(618\) 0 0
\(619\) −12.0000 −0.482321 −0.241160 − 0.970485i \(-0.577528\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(620\) 0 0
\(621\) −5.00000 −0.200643
\(622\) −16.0000 −0.641542
\(623\) −1.00000 −0.0400642
\(624\) 4.00000 0.160128
\(625\) 0 0
\(626\) −6.00000 −0.239808
\(627\) 3.00000 0.119808
\(628\) −9.00000 −0.359139
\(629\) 0 0
\(630\) 0 0
\(631\) 25.0000 0.995234 0.497617 − 0.867397i \(-0.334208\pi\)
0.497617 + 0.867397i \(0.334208\pi\)
\(632\) −3.00000 −0.119334
\(633\) 7.00000 0.278225
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −5.00000 −0.198263
\(637\) −24.0000 −0.950915
\(638\) 6.00000 0.237542
\(639\) −5.00000 −0.197797
\(640\) 0 0
\(641\) 50.0000 1.97488 0.987441 − 0.157991i \(-0.0505015\pi\)
0.987441 + 0.157991i \(0.0505015\pi\)
\(642\) 19.0000 0.749870
\(643\) 7.00000 0.276053 0.138027 − 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 5.00000 0.197028
\(645\) 0 0
\(646\) 0 0
\(647\) −7.00000 −0.275198 −0.137599 − 0.990488i \(-0.543939\pi\)
−0.137599 + 0.990488i \(0.543939\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 18.0000 0.706562
\(650\) 0 0
\(651\) 1.00000 0.0391931
\(652\) 4.00000 0.156652
\(653\) 42.0000 1.64359 0.821794 − 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 0 0
\(656\) −10.0000 −0.390434
\(657\) 7.00000 0.273096
\(658\) 8.00000 0.311872
\(659\) 6.00000 0.233727 0.116863 − 0.993148i \(-0.462716\pi\)
0.116863 + 0.993148i \(0.462716\pi\)
\(660\) 0 0
\(661\) 36.0000 1.40024 0.700119 − 0.714026i \(-0.253130\pi\)
0.700119 + 0.714026i \(0.253130\pi\)
\(662\) −8.00000 −0.310929
\(663\) 0 0
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) −10.0000 −0.387202
\(668\) −21.0000 −0.812514
\(669\) 20.0000 0.773245
\(670\) 0 0
\(671\) 6.00000 0.231627
\(672\) 1.00000 0.0385758
\(673\) 30.0000 1.15642 0.578208 − 0.815890i \(-0.303752\pi\)
0.578208 + 0.815890i \(0.303752\pi\)
\(674\) −34.0000 −1.30963
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) −13.0000 −0.499631 −0.249815 − 0.968294i \(-0.580370\pi\)
−0.249815 + 0.968294i \(0.580370\pi\)
\(678\) −15.0000 −0.576072
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 9.00000 0.344881
\(682\) −3.00000 −0.114876
\(683\) −35.0000 −1.33924 −0.669619 − 0.742705i \(-0.733543\pi\)
−0.669619 + 0.742705i \(0.733543\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) −13.0000 −0.496342
\(687\) 15.0000 0.572286
\(688\) 5.00000 0.190623
\(689\) −20.0000 −0.761939
\(690\) 0 0
\(691\) −41.0000 −1.55971 −0.779857 − 0.625958i \(-0.784708\pi\)
−0.779857 + 0.625958i \(0.784708\pi\)
\(692\) 2.00000 0.0760286
\(693\) 3.00000 0.113961
\(694\) −22.0000 −0.835109
\(695\) 0 0
\(696\) −2.00000 −0.0758098
\(697\) 0 0
\(698\) −20.0000 −0.757011
\(699\) 21.0000 0.794293
\(700\) 0 0
\(701\) −35.0000 −1.32193 −0.660966 − 0.750416i \(-0.729853\pi\)
−0.660966 + 0.750416i \(0.729853\pi\)
\(702\) −4.00000 −0.150970
\(703\) 4.00000 0.150863
\(704\) −3.00000 −0.113067
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 15.0000 0.564133
\(708\) −6.00000 −0.225494
\(709\) 5.00000 0.187779 0.0938895 − 0.995583i \(-0.470070\pi\)
0.0938895 + 0.995583i \(0.470070\pi\)
\(710\) 0 0
\(711\) 3.00000 0.112509
\(712\) −1.00000 −0.0374766
\(713\) 5.00000 0.187251
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −14.0000 −0.522840
\(718\) −3.00000 −0.111959
\(719\) 10.0000 0.372937 0.186469 − 0.982461i \(-0.440296\pi\)
0.186469 + 0.982461i \(0.440296\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 18.0000 0.669891
\(723\) −12.0000 −0.446285
\(724\) −5.00000 −0.185824
\(725\) 0 0
\(726\) 2.00000 0.0742270
\(727\) −49.0000 −1.81731 −0.908655 − 0.417548i \(-0.862889\pi\)
−0.908655 + 0.417548i \(0.862889\pi\)
\(728\) 4.00000 0.148250
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −2.00000 −0.0739221
\(733\) 22.0000 0.812589 0.406294 − 0.913742i \(-0.366821\pi\)
0.406294 + 0.913742i \(0.366821\pi\)
\(734\) 14.0000 0.516749
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) 6.00000 0.221013
\(738\) 10.0000 0.368105
\(739\) −18.0000 −0.662141 −0.331070 − 0.943606i \(-0.607410\pi\)
−0.331070 + 0.943606i \(0.607410\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) −5.00000 −0.183556
\(743\) −15.0000 −0.550297 −0.275148 − 0.961402i \(-0.588727\pi\)
−0.275148 + 0.961402i \(0.588727\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) −11.0000 −0.402739
\(747\) 2.00000 0.0731762
\(748\) 0 0
\(749\) 19.0000 0.694245
\(750\) 0 0
\(751\) 22.0000 0.802791 0.401396 − 0.915905i \(-0.368525\pi\)
0.401396 + 0.915905i \(0.368525\pi\)
\(752\) 8.00000 0.291730
\(753\) 12.0000 0.437304
\(754\) −8.00000 −0.291343
\(755\) 0 0
\(756\) −1.00000 −0.0363696
\(757\) −20.0000 −0.726912 −0.363456 − 0.931611i \(-0.618403\pi\)
−0.363456 + 0.931611i \(0.618403\pi\)
\(758\) 15.0000 0.544825
\(759\) 15.0000 0.544466
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 − 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 8.00000 0.289809
\(763\) −2.00000 −0.0724049
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 28.0000 1.01168
\(767\) −24.0000 −0.866590
\(768\) 1.00000 0.0360844
\(769\) 45.0000 1.62274 0.811371 − 0.584532i \(-0.198722\pi\)
0.811371 + 0.584532i \(0.198722\pi\)
\(770\) 0 0
\(771\) −5.00000 −0.180071
\(772\) 10.0000 0.359908
\(773\) 29.0000 1.04306 0.521529 − 0.853234i \(-0.325362\pi\)
0.521529 + 0.853234i \(0.325362\pi\)
\(774\) −5.00000 −0.179721
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) 4.00000 0.143499
\(778\) 28.0000 1.00385
\(779\) 10.0000 0.358287
\(780\) 0 0
\(781\) 15.0000 0.536742
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) 2.00000 0.0713376
\(787\) −27.0000 −0.962446 −0.481223 − 0.876598i \(-0.659807\pi\)
−0.481223 + 0.876598i \(0.659807\pi\)
\(788\) −18.0000 −0.641223
\(789\) −24.0000 −0.854423
\(790\) 0 0
\(791\) −15.0000 −0.533339
\(792\) 3.00000 0.106600
\(793\) −8.00000 −0.284088
\(794\) −31.0000 −1.10015
\(795\) 0 0
\(796\) −21.0000 −0.744325
\(797\) 22.0000 0.779280 0.389640 − 0.920967i \(-0.372599\pi\)
0.389640 + 0.920967i \(0.372599\pi\)
\(798\) −1.00000 −0.0353996
\(799\) 0 0
\(800\) 0 0
\(801\) 1.00000 0.0353333
\(802\) −5.00000 −0.176556
\(803\) −21.0000 −0.741074
\(804\) −2.00000 −0.0705346
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 2.00000 0.0704033
\(808\) 15.0000 0.527698
\(809\) 9.00000 0.316423 0.158212 − 0.987405i \(-0.449427\pi\)
0.158212 + 0.987405i \(0.449427\pi\)
\(810\) 0 0
\(811\) 35.0000 1.22902 0.614508 − 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 9.00000 0.315644
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) −5.00000 −0.174928
\(818\) −4.00000 −0.139857
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 − 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) 2.00000 0.0697580
\(823\) 6.00000 0.209147 0.104573 − 0.994517i \(-0.466652\pi\)
0.104573 + 0.994517i \(0.466652\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −14.0000 −0.486828 −0.243414 − 0.969923i \(-0.578267\pi\)
−0.243414 + 0.969923i \(0.578267\pi\)
\(828\) −5.00000 −0.173762
\(829\) −25.0000 −0.868286 −0.434143 − 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 4.00000 0.138675
\(833\) 0 0
\(834\) 20.0000 0.692543
\(835\) 0 0
\(836\) 3.00000 0.103757
\(837\) −1.00000 −0.0345651
\(838\) 18.0000 0.621800
\(839\) 39.0000 1.34643 0.673215 − 0.739447i \(-0.264913\pi\)
0.673215 + 0.739447i \(0.264913\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 14.0000 0.482472
\(843\) −32.0000 −1.10214
\(844\) 7.00000 0.240950
\(845\) 0 0
\(846\) −8.00000 −0.275046
\(847\) 2.00000 0.0687208
\(848\) −5.00000 −0.171701
\(849\) −14.0000 −0.480479
\(850\) 0 0
\(851\) 20.0000 0.685591
\(852\) −5.00000 −0.171297
\(853\) 55.0000 1.88316 0.941582 − 0.336784i \(-0.109339\pi\)
0.941582 + 0.336784i \(0.109339\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 19.0000 0.649407
\(857\) 22.0000 0.751506 0.375753 − 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 12.0000 0.409673
\(859\) −32.0000 −1.09183 −0.545913 − 0.837842i \(-0.683817\pi\)
−0.545913 + 0.837842i \(0.683817\pi\)
\(860\) 0 0
\(861\) 10.0000 0.340799
\(862\) 0 0
\(863\) 29.0000 0.987171 0.493586 − 0.869697i \(-0.335686\pi\)
0.493586 + 0.869697i \(0.335686\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 19.0000 0.645646
\(867\) −17.0000 −0.577350
\(868\) 1.00000 0.0339422
\(869\) −9.00000 −0.305304
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) −2.00000 −0.0677285
\(873\) 10.0000 0.338449
\(874\) −5.00000 −0.169128
\(875\) 0 0
\(876\) 7.00000 0.236508
\(877\) −42.0000 −1.41824 −0.709120 − 0.705088i \(-0.750907\pi\)
−0.709120 + 0.705088i \(0.750907\pi\)
\(878\) −20.0000 −0.674967
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 − 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 6.00000 0.202031
\(883\) 35.0000 1.17784 0.588922 − 0.808190i \(-0.299553\pi\)
0.588922 + 0.808190i \(0.299553\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 29.0000 0.974274
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 4.00000 0.134231
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 20.0000 0.669650
\(893\) −8.00000 −0.267710
\(894\) −15.0000 −0.501675
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −20.0000 −0.667781
\(898\) −2.00000 −0.0667409
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) 0 0
\(902\) −30.0000 −0.998891
\(903\) −5.00000 −0.166390
\(904\) −15.0000 −0.498893
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 36.0000 1.19536 0.597680 − 0.801735i \(-0.296089\pi\)
0.597680 + 0.801735i \(0.296089\pi\)
\(908\) 9.00000 0.298675
\(909\) −15.0000 −0.497519
\(910\) 0 0
\(911\) −46.0000 −1.52405 −0.762024 − 0.647549i \(-0.775794\pi\)
−0.762024 + 0.647549i \(0.775794\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −6.00000 −0.198571
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) 15.0000 0.495614
\(917\) 2.00000 0.0660458
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) −16.0000 −0.526932
\(923\) −20.0000 −0.658308
\(924\) 3.00000 0.0986928
\(925\) 0 0
\(926\) −14.0000 −0.460069
\(927\) 0 0
\(928\) −2.00000 −0.0656532
\(929\) −1.00000 −0.0328089 −0.0164045 − 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 21.0000 0.687878
\(933\) 16.0000 0.523816
\(934\) 28.0000 0.916188
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 2.00000 0.0653372 0.0326686 − 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −2.00000 −0.0653023
\(939\) 6.00000 0.195803
\(940\) 0 0
\(941\) 16.0000 0.521585 0.260793 − 0.965395i \(-0.416016\pi\)
0.260793 + 0.965395i \(0.416016\pi\)
\(942\) 9.00000 0.293236
\(943\) 50.0000 1.62822
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 15.0000 0.487692
\(947\) −46.0000 −1.49480 −0.747400 − 0.664375i \(-0.768698\pi\)
−0.747400 + 0.664375i \(0.768698\pi\)
\(948\) 3.00000 0.0974355
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 28.0000 0.907009 0.453504 − 0.891254i \(-0.350174\pi\)
0.453504 + 0.891254i \(0.350174\pi\)
\(954\) 5.00000 0.161881
\(955\) 0 0
\(956\) −14.0000 −0.452792
\(957\) −6.00000 −0.193952
\(958\) 25.0000 0.807713
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 16.0000 0.515861
\(963\) −19.0000 −0.612266
\(964\) −12.0000 −0.386494
\(965\) 0 0
\(966\) −5.00000 −0.160872
\(967\) 8.00000 0.257263 0.128631 − 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 0 0
\(971\) −18.0000 −0.577647 −0.288824 − 0.957382i \(-0.593264\pi\)
−0.288824 + 0.957382i \(0.593264\pi\)
\(972\) 1.00000 0.0320750
\(973\) 20.0000 0.641171
\(974\) −4.00000 −0.128168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) −6.00000 −0.191957 −0.0959785 − 0.995383i \(-0.530598\pi\)
−0.0959785 + 0.995383i \(0.530598\pi\)
\(978\) −4.00000 −0.127906
\(979\) −3.00000 −0.0958804
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) −33.0000 −1.05307
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 10.0000 0.318788
\(985\) 0 0
\(986\) 0 0
\(987\) −8.00000 −0.254643
\(988\) −4.00000 −0.127257
\(989\) −25.0000 −0.794954
\(990\) 0 0
\(991\) −53.0000 −1.68360 −0.841800 − 0.539789i \(-0.818504\pi\)
−0.841800 + 0.539789i \(0.818504\pi\)
\(992\) 1.00000 0.0317500
\(993\) 8.00000 0.253872
\(994\) −5.00000 −0.158590
\(995\) 0 0
\(996\) 2.00000 0.0633724
\(997\) −6.00000 −0.190022 −0.0950110 − 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −22.0000 −0.696398
\(999\) −4.00000 −0.126554
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 4650.2.a.p.1.1 1
5.2 odd 4 930.2.d.a.559.1 ✓ 2
5.3 odd 4 930.2.d.a.559.2 yes 2
5.4 even 2 4650.2.a.bc.1.1 1
15.2 even 4 2790.2.d.f.559.2 2
15.8 even 4 2790.2.d.f.559.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.d.a.559.1 ✓ 2 5.2 odd 4
930.2.d.a.559.2 yes 2 5.3 odd 4
2790.2.d.f.559.1 2 15.8 even 4
2790.2.d.f.559.2 2 15.2 even 4
4650.2.a.p.1.1 1 1.1 even 1 trivial
4650.2.a.bc.1.1 1 5.4 even 2