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Properties

Label 5550.2.a.y.1.1
Level $5550$
Weight $2$
Character 5550.1
Self dual yes
Analytic conductor $44.317$
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] = [5550,2,Mod(1,5550)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5550, 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("5550.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.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: \(44.3169731218\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 5550.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} +6.00000 q^{19} +1.00000 q^{21} -5.00000 q^{22} -1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +9.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} +5.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} +1.00000 q^{37} +6.00000 q^{38} +2.00000 q^{39} +9.00000 q^{41} +1.00000 q^{42} -1.00000 q^{43} -5.00000 q^{44} -1.00000 q^{48} -6.00000 q^{49} +1.00000 q^{51} -2.00000 q^{52} -9.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} -6.00000 q^{57} +9.00000 q^{58} -10.0000 q^{59} -5.00000 q^{61} +3.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} +5.00000 q^{66} -16.0000 q^{67} -1.00000 q^{68} +1.00000 q^{72} -12.0000 q^{73} +1.00000 q^{74} +6.00000 q^{76} +5.00000 q^{77} +2.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} +9.00000 q^{82} -2.00000 q^{83} +1.00000 q^{84} -1.00000 q^{86} -9.00000 q^{87} -5.00000 q^{88} -2.00000 q^{89} +2.00000 q^{91} -3.00000 q^{93} -1.00000 q^{96} -17.0000 q^{97} -6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

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
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 − 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 1.00000 0.235702
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −5.00000 −1.06600
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) 9.00000 1.67126 0.835629 − 0.549294i \(-0.185103\pi\)
0.835629 + 0.549294i \(0.185103\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 1.00000 0.176777
\(33\) 5.00000 0.870388
\(34\) −1.00000 −0.171499
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 1.00000 0.164399
\(38\) 6.00000 0.973329
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 − 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) 1.00000 0.154303
\(43\) −1.00000 −0.152499 −0.0762493 − 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −1.00000 −0.144338
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) −2.00000 −0.277350
\(53\) −9.00000 −1.23625 −0.618123 − 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) −6.00000 −0.794719
\(58\) 9.00000 1.18176
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −5.00000 −0.640184 −0.320092 − 0.947386i \(-0.603714\pi\)
−0.320092 + 0.947386i \(0.603714\pi\)
\(62\) 3.00000 0.381000
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) −16.0000 −1.95471 −0.977356 − 0.211604i \(-0.932131\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.0000 −1.40449 −0.702247 − 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 5.00000 0.569803
\(78\) 2.00000 0.226455
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 9.00000 0.993884
\(83\) −2.00000 −0.219529 −0.109764 − 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −1.00000 −0.107833
\(87\) −9.00000 −0.964901
\(88\) −5.00000 −0.533002
\(89\) −2.00000 −0.212000 −0.106000 − 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) −3.00000 −0.311086
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −17.0000 −1.72609 −0.863044 − 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) −6.00000 −0.606092
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −12.0000 −1.19404 −0.597022 − 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 1.00000 0.0990148
\(103\) 4.00000 0.394132 0.197066 − 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −9.00000 −0.874157
\(107\) −18.0000 −1.74013 −0.870063 − 0.492941i \(-0.835922\pi\)
−0.870063 + 0.492941i \(0.835922\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −5.00000 −0.478913 −0.239457 − 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) −1.00000 −0.0944911
\(113\) 13.0000 1.22294 0.611469 − 0.791269i \(-0.290579\pi\)
0.611469 + 0.791269i \(0.290579\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) 9.00000 0.835629
\(117\) −2.00000 −0.184900
\(118\) −10.0000 −0.920575
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) −5.00000 −0.452679
\(123\) −9.00000 −0.811503
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) −1.00000 −0.0890871
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.00000 0.0880451
\(130\) 0 0
\(131\) 10.0000 0.873704 0.436852 − 0.899533i \(-0.356093\pi\)
0.436852 + 0.899533i \(0.356093\pi\)
\(132\) 5.00000 0.435194
\(133\) −6.00000 −0.520266
\(134\) −16.0000 −1.38219
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 14.0000 1.19610 0.598050 − 0.801459i \(-0.295942\pi\)
0.598050 + 0.801459i \(0.295942\pi\)
\(138\) 0 0
\(139\) −19.0000 −1.61156 −0.805779 − 0.592216i \(-0.798253\pi\)
−0.805779 + 0.592216i \(0.798253\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.0000 0.836242
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 6.00000 0.494872
\(148\) 1.00000 0.0821995
\(149\) 14.0000 1.14692 0.573462 − 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) 18.0000 1.46482 0.732410 − 0.680864i \(-0.238396\pi\)
0.732410 + 0.680864i \(0.238396\pi\)
\(152\) 6.00000 0.486664
\(153\) −1.00000 −0.0808452
\(154\) 5.00000 0.402911
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) −7.00000 −0.558661 −0.279330 − 0.960195i \(-0.590112\pi\)
−0.279330 + 0.960195i \(0.590112\pi\)
\(158\) −12.0000 −0.954669
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −1.00000 −0.0783260 −0.0391630 − 0.999233i \(-0.512469\pi\)
−0.0391630 + 0.999233i \(0.512469\pi\)
\(164\) 9.00000 0.702782
\(165\) 0 0
\(166\) −2.00000 −0.155230
\(167\) −20.0000 −1.54765 −0.773823 − 0.633402i \(-0.781658\pi\)
−0.773823 + 0.633402i \(0.781658\pi\)
\(168\) 1.00000 0.0771517
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) −1.00000 −0.0762493
\(173\) 7.00000 0.532200 0.266100 − 0.963945i \(-0.414265\pi\)
0.266100 + 0.963945i \(0.414265\pi\)
\(174\) −9.00000 −0.682288
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 10.0000 0.751646
\(178\) −2.00000 −0.149906
\(179\) −16.0000 −1.19590 −0.597948 − 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 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
\(183\) 5.00000 0.369611
\(184\) 0 0
\(185\) 0 0
\(186\) −3.00000 −0.219971
\(187\) 5.00000 0.365636
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 11.0000 0.795932 0.397966 − 0.917400i \(-0.369716\pi\)
0.397966 + 0.917400i \(0.369716\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.00000 0.143963 0.0719816 − 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −17.0000 −1.22053
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −14.0000 −0.997459 −0.498729 − 0.866758i \(-0.666200\pi\)
−0.498729 + 0.866758i \(0.666200\pi\)
\(198\) −5.00000 −0.355335
\(199\) −12.0000 −0.850657 −0.425329 − 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 16.0000 1.12855
\(202\) −12.0000 −0.844317
\(203\) −9.00000 −0.631676
\(204\) 1.00000 0.0700140
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) −30.0000 −2.07514
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 − 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −9.00000 −0.618123
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) −3.00000 −0.203653
\(218\) −5.00000 −0.338643
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) −1.00000 −0.0671156
\(223\) 5.00000 0.334825 0.167412 − 0.985887i \(-0.446459\pi\)
0.167412 + 0.985887i \(0.446459\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 13.0000 0.864747
\(227\) 7.00000 0.464606 0.232303 − 0.972643i \(-0.425374\pi\)
0.232303 + 0.972643i \(0.425374\pi\)
\(228\) −6.00000 −0.397360
\(229\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 9.00000 0.590879
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 12.0000 0.779484
\(238\) 1.00000 0.0648204
\(239\) −21.0000 −1.35838 −0.679189 − 0.733964i \(-0.737668\pi\)
−0.679189 + 0.733964i \(0.737668\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 − 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 14.0000 0.899954
\(243\) −1.00000 −0.0641500
\(244\) −5.00000 −0.320092
\(245\) 0 0
\(246\) −9.00000 −0.573819
\(247\) −12.0000 −0.763542
\(248\) 3.00000 0.190500
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 − 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 1.00000 0.0622573
\(259\) −1.00000 −0.0621370
\(260\) 0 0
\(261\) 9.00000 0.557086
\(262\) 10.0000 0.617802
\(263\) 15.0000 0.924940 0.462470 − 0.886635i \(-0.346963\pi\)
0.462470 + 0.886635i \(0.346963\pi\)
\(264\) 5.00000 0.307729
\(265\) 0 0
\(266\) −6.00000 −0.367884
\(267\) 2.00000 0.122398
\(268\) −16.0000 −0.977356
\(269\) 24.0000 1.46331 0.731653 − 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −2.00000 −0.121046
\(274\) 14.0000 0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000 0.480673 0.240337 − 0.970690i \(-0.422742\pi\)
0.240337 + 0.970690i \(0.422742\pi\)
\(278\) −19.0000 −1.13954
\(279\) 3.00000 0.179605
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 − 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 − 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 10.0000 0.591312
\(287\) −9.00000 −0.531253
\(288\) 1.00000 0.0589256
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) −12.0000 −0.702247
\(293\) −19.0000 −1.10999 −0.554996 − 0.831853i \(-0.687280\pi\)
−0.554996 + 0.831853i \(0.687280\pi\)
\(294\) 6.00000 0.349927
\(295\) 0 0
\(296\) 1.00000 0.0581238
\(297\) 5.00000 0.290129
\(298\) 14.0000 0.810998
\(299\) 0 0
\(300\) 0 0
\(301\) 1.00000 0.0576390
\(302\) 18.0000 1.03578
\(303\) 12.0000 0.689382
\(304\) 6.00000 0.344124
\(305\) 0 0
\(306\) −1.00000 −0.0571662
\(307\) 30.0000 1.71219 0.856095 − 0.516818i \(-0.172884\pi\)
0.856095 + 0.516818i \(0.172884\pi\)
\(308\) 5.00000 0.284901
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −25.0000 −1.41762 −0.708810 − 0.705399i \(-0.750768\pi\)
−0.708810 + 0.705399i \(0.750768\pi\)
\(312\) 2.00000 0.113228
\(313\) 18.0000 1.01742 0.508710 − 0.860938i \(-0.330123\pi\)
0.508710 + 0.860938i \(0.330123\pi\)
\(314\) −7.00000 −0.395033
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 7.00000 0.393159 0.196580 − 0.980488i \(-0.437017\pi\)
0.196580 + 0.980488i \(0.437017\pi\)
\(318\) 9.00000 0.504695
\(319\) −45.0000 −2.51952
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) −6.00000 −0.333849
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −1.00000 −0.0553849
\(327\) 5.00000 0.276501
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −2.00000 −0.109764
\(333\) 1.00000 0.0547997
\(334\) −20.0000 −1.09435
\(335\) 0 0
\(336\) 1.00000 0.0545545
\(337\) −6.00000 −0.326841 −0.163420 − 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) −9.00000 −0.489535
\(339\) −13.0000 −0.706063
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 6.00000 0.324443
\(343\) 13.0000 0.701934
\(344\) −1.00000 −0.0539164
\(345\) 0 0
\(346\) 7.00000 0.376322
\(347\) −12.0000 −0.644194 −0.322097 − 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) −9.00000 −0.482451
\(349\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) −5.00000 −0.266501
\(353\) −1.00000 −0.0532246 −0.0266123 − 0.999646i \(-0.508472\pi\)
−0.0266123 + 0.999646i \(0.508472\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) −1.00000 −0.0529256
\(358\) −16.0000 −0.845626
\(359\) 6.00000 0.316668 0.158334 − 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 14.0000 0.735824
\(363\) −14.0000 −0.734809
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 5.00000 0.261354
\(367\) 23.0000 1.20059 0.600295 − 0.799779i \(-0.295050\pi\)
0.600295 + 0.799779i \(0.295050\pi\)
\(368\) 0 0
\(369\) 9.00000 0.468521
\(370\) 0 0
\(371\) 9.00000 0.467257
\(372\) −3.00000 −0.155543
\(373\) −26.0000 −1.34623 −0.673114 − 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) 5.00000 0.258544
\(375\) 0 0
\(376\) 0 0
\(377\) −18.0000 −0.927047
\(378\) 1.00000 0.0514344
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 11.0000 0.562809
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) −1.00000 −0.0508329
\(388\) −17.0000 −0.863044
\(389\) 19.0000 0.963338 0.481669 − 0.876353i \(-0.340031\pi\)
0.481669 + 0.876353i \(0.340031\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.00000 −0.303046
\(393\) −10.0000 −0.504433
\(394\) −14.0000 −0.705310
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) −18.0000 −0.903394 −0.451697 − 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) −12.0000 −0.601506
\(399\) 6.00000 0.300376
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 − 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 16.0000 0.798007
\(403\) −6.00000 −0.298881
\(404\) −12.0000 −0.597022
\(405\) 0 0
\(406\) −9.00000 −0.446663
\(407\) −5.00000 −0.247841
\(408\) 1.00000 0.0495074
\(409\) 6.00000 0.296681 0.148340 − 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) 4.00000 0.197066
\(413\) 10.0000 0.492068
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) 19.0000 0.930434
\(418\) −30.0000 −1.46735
\(419\) 20.0000 0.977064 0.488532 − 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 − 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) −5.00000 −0.243396
\(423\) 0 0
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) 0 0
\(427\) 5.00000 0.241967
\(428\) −18.0000 −0.870063
\(429\) −10.0000 −0.482805
\(430\) 0 0
\(431\) 15.0000 0.722525 0.361262 − 0.932464i \(-0.382346\pi\)
0.361262 + 0.932464i \(0.382346\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −36.0000 −1.73005 −0.865025 − 0.501729i \(-0.832697\pi\)
−0.865025 + 0.501729i \(0.832697\pi\)
\(434\) −3.00000 −0.144005
\(435\) 0 0
\(436\) −5.00000 −0.239457
\(437\) 0 0
\(438\) 12.0000 0.573382
\(439\) −27.0000 −1.28864 −0.644320 − 0.764756i \(-0.722859\pi\)
−0.644320 + 0.764756i \(0.722859\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 2.00000 0.0951303
\(443\) −2.00000 −0.0950229 −0.0475114 − 0.998871i \(-0.515129\pi\)
−0.0475114 + 0.998871i \(0.515129\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) 5.00000 0.236757
\(447\) −14.0000 −0.662177
\(448\) −1.00000 −0.0472456
\(449\) 24.0000 1.13263 0.566315 − 0.824189i \(-0.308369\pi\)
0.566315 + 0.824189i \(0.308369\pi\)
\(450\) 0 0
\(451\) −45.0000 −2.11897
\(452\) 13.0000 0.611469
\(453\) −18.0000 −0.845714
\(454\) 7.00000 0.328526
\(455\) 0 0
\(456\) −6.00000 −0.280976
\(457\) −1.00000 −0.0467780 −0.0233890 − 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 0 0
\(459\) 1.00000 0.0466760
\(460\) 0 0
\(461\) 35.0000 1.63011 0.815056 − 0.579382i \(-0.196706\pi\)
0.815056 + 0.579382i \(0.196706\pi\)
\(462\) −5.00000 −0.232621
\(463\) −26.0000 −1.20832 −0.604161 − 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) 9.00000 0.417815
\(465\) 0 0
\(466\) 8.00000 0.370593
\(467\) 21.0000 0.971764 0.485882 − 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 16.0000 0.738811
\(470\) 0 0
\(471\) 7.00000 0.322543
\(472\) −10.0000 −0.460287
\(473\) 5.00000 0.229900
\(474\) 12.0000 0.551178
\(475\) 0 0
\(476\) 1.00000 0.0458349
\(477\) −9.00000 −0.412082
\(478\) −21.0000 −0.960518
\(479\) −24.0000 −1.09659 −0.548294 − 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) 14.0000 0.636364
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −2.00000 −0.0906287 −0.0453143 − 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −5.00000 −0.226339
\(489\) 1.00000 0.0452216
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 − 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) −9.00000 −0.405751
\(493\) −9.00000 −0.405340
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) 0 0
\(498\) 2.00000 0.0896221
\(499\) −44.0000 −1.96971 −0.984855 − 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) 0 0
\(501\) 20.0000 0.893534
\(502\) −24.0000 −1.07117
\(503\) −4.00000 −0.178351 −0.0891756 − 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) 24.0000 1.06378 0.531891 − 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) 1.00000 0.0441942
\(513\) −6.00000 −0.264906
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 1.00000 0.0440225
\(517\) 0 0
\(518\) −1.00000 −0.0439375
\(519\) −7.00000 −0.307266
\(520\) 0 0
\(521\) 3.00000 0.131432 0.0657162 − 0.997838i \(-0.479067\pi\)
0.0657162 + 0.997838i \(0.479067\pi\)
\(522\) 9.00000 0.393919
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 10.0000 0.436852
\(525\) 0 0
\(526\) 15.0000 0.654031
\(527\) −3.00000 −0.130682
\(528\) 5.00000 0.217597
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) −6.00000 −0.260133
\(533\) −18.0000 −0.779667
\(534\) 2.00000 0.0865485
\(535\) 0 0
\(536\) −16.0000 −0.691095
\(537\) 16.0000 0.690451
\(538\) 24.0000 1.03471
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 − 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −16.0000 −0.687259
\(543\) −14.0000 −0.600798
\(544\) −1.00000 −0.0428746
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) −17.0000 −0.726868 −0.363434 − 0.931620i \(-0.618396\pi\)
−0.363434 + 0.931620i \(0.618396\pi\)
\(548\) 14.0000 0.598050
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) 54.0000 2.30048
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 8.00000 0.339887
\(555\) 0 0
\(556\) −19.0000 −0.805779
\(557\) −18.0000 −0.762684 −0.381342 − 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 3.00000 0.127000
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) −5.00000 −0.211100
\(562\) −2.00000 −0.0843649
\(563\) −13.0000 −0.547885 −0.273942 − 0.961746i \(-0.588328\pi\)
−0.273942 + 0.961746i \(0.588328\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 38.0000 1.59304 0.796521 − 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) 35.0000 1.46470 0.732352 − 0.680926i \(-0.238422\pi\)
0.732352 + 0.680926i \(0.238422\pi\)
\(572\) 10.0000 0.418121
\(573\) −11.0000 −0.459532
\(574\) −9.00000 −0.375653
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −2.00000 −0.0832611 −0.0416305 − 0.999133i \(-0.513255\pi\)
−0.0416305 + 0.999133i \(0.513255\pi\)
\(578\) −16.0000 −0.665512
\(579\) −2.00000 −0.0831172
\(580\) 0 0
\(581\) 2.00000 0.0829740
\(582\) 17.0000 0.704673
\(583\) 45.0000 1.86371
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −19.0000 −0.784883
\(587\) −3.00000 −0.123823 −0.0619116 − 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 6.00000 0.247436
\(589\) 18.0000 0.741677
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 1.00000 0.0410997
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 5.00000 0.205152
\(595\) 0 0
\(596\) 14.0000 0.573462
\(597\) 12.0000 0.491127
\(598\) 0 0
\(599\) −4.00000 −0.163436 −0.0817178 − 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 − 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) 1.00000 0.0407570
\(603\) −16.0000 −0.651570
\(604\) 18.0000 0.732410
\(605\) 0 0
\(606\) 12.0000 0.487467
\(607\) −6.00000 −0.243532 −0.121766 − 0.992559i \(-0.538856\pi\)
−0.121766 + 0.992559i \(0.538856\pi\)
\(608\) 6.00000 0.243332
\(609\) 9.00000 0.364698
\(610\) 0 0
\(611\) 0 0
\(612\) −1.00000 −0.0404226
\(613\) 23.0000 0.928961 0.464481 − 0.885583i \(-0.346241\pi\)
0.464481 + 0.885583i \(0.346241\pi\)
\(614\) 30.0000 1.21070
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) 10.0000 0.402585 0.201292 − 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) −4.00000 −0.160904
\(619\) −11.0000 −0.442127 −0.221064 − 0.975259i \(-0.570953\pi\)
−0.221064 + 0.975259i \(0.570953\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −25.0000 −1.00241
\(623\) 2.00000 0.0801283
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 18.0000 0.719425
\(627\) 30.0000 1.19808
\(628\) −7.00000 −0.279330
\(629\) −1.00000 −0.0398726
\(630\) 0 0
\(631\) −13.0000 −0.517522 −0.258761 − 0.965941i \(-0.583314\pi\)
−0.258761 + 0.965941i \(0.583314\pi\)
\(632\) −12.0000 −0.477334
\(633\) 5.00000 0.198732
\(634\) 7.00000 0.278006
\(635\) 0 0
\(636\) 9.00000 0.356873
\(637\) 12.0000 0.475457
\(638\) −45.0000 −1.78157
\(639\) 0 0
\(640\) 0 0
\(641\) −15.0000 −0.592464 −0.296232 − 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(642\) 18.0000 0.710403
\(643\) 19.0000 0.749287 0.374643 − 0.927169i \(-0.377765\pi\)
0.374643 + 0.927169i \(0.377765\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) −24.0000 −0.943537 −0.471769 − 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000 0.0392837
\(649\) 50.0000 1.96267
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) −1.00000 −0.0391630
\(653\) 12.0000 0.469596 0.234798 − 0.972044i \(-0.424557\pi\)
0.234798 + 0.972044i \(0.424557\pi\)
\(654\) 5.00000 0.195515
\(655\) 0 0
\(656\) 9.00000 0.351391
\(657\) −12.0000 −0.468165
\(658\) 0 0
\(659\) −32.0000 −1.24654 −0.623272 − 0.782006i \(-0.714197\pi\)
−0.623272 + 0.782006i \(0.714197\pi\)
\(660\) 0 0
\(661\) −11.0000 −0.427850 −0.213925 − 0.976850i \(-0.568625\pi\)
−0.213925 + 0.976850i \(0.568625\pi\)
\(662\) −8.00000 −0.310929
\(663\) −2.00000 −0.0776736
\(664\) −2.00000 −0.0776151
\(665\) 0 0
\(666\) 1.00000 0.0387492
\(667\) 0 0
\(668\) −20.0000 −0.773823
\(669\) −5.00000 −0.193311
\(670\) 0 0
\(671\) 25.0000 0.965114
\(672\) 1.00000 0.0385758
\(673\) 44.0000 1.69608 0.848038 − 0.529936i \(-0.177784\pi\)
0.848038 + 0.529936i \(0.177784\pi\)
\(674\) −6.00000 −0.231111
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 6.00000 0.230599 0.115299 − 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −13.0000 −0.499262
\(679\) 17.0000 0.652400
\(680\) 0 0
\(681\) −7.00000 −0.268241
\(682\) −15.0000 −0.574380
\(683\) −33.0000 −1.26271 −0.631355 − 0.775494i \(-0.717501\pi\)
−0.631355 + 0.775494i \(0.717501\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) 13.0000 0.496342
\(687\) 0 0
\(688\) −1.00000 −0.0381246
\(689\) 18.0000 0.685745
\(690\) 0 0
\(691\) −1.00000 −0.0380418 −0.0190209 − 0.999819i \(-0.506055\pi\)
−0.0190209 + 0.999819i \(0.506055\pi\)
\(692\) 7.00000 0.266100
\(693\) 5.00000 0.189934
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) −9.00000 −0.341144
\(697\) −9.00000 −0.340899
\(698\) 0 0
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 − 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 2.00000 0.0754851
\(703\) 6.00000 0.226294
\(704\) −5.00000 −0.188445
\(705\) 0 0
\(706\) −1.00000 −0.0376355
\(707\) 12.0000 0.451306
\(708\) 10.0000 0.375823
\(709\) −17.0000 −0.638448 −0.319224 − 0.947679i \(-0.603422\pi\)
−0.319224 + 0.947679i \(0.603422\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) −2.00000 −0.0749532
\(713\) 0 0
\(714\) −1.00000 −0.0374241
\(715\) 0 0
\(716\) −16.0000 −0.597948
\(717\) 21.0000 0.784259
\(718\) 6.00000 0.223918
\(719\) 10.0000 0.372937 0.186469 − 0.982461i \(-0.440296\pi\)
0.186469 + 0.982461i \(0.440296\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 17.0000 0.632674
\(723\) 18.0000 0.669427
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) 18.0000 0.667583 0.333792 − 0.942647i \(-0.391672\pi\)
0.333792 + 0.942647i \(0.391672\pi\)
\(728\) 2.00000 0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 1.00000 0.0369863
\(732\) 5.00000 0.184805
\(733\) 47.0000 1.73598 0.867992 − 0.496578i \(-0.165410\pi\)
0.867992 + 0.496578i \(0.165410\pi\)
\(734\) 23.0000 0.848945
\(735\) 0 0
\(736\) 0 0
\(737\) 80.0000 2.94684
\(738\) 9.00000 0.331295
\(739\) −7.00000 −0.257499 −0.128750 − 0.991677i \(-0.541096\pi\)
−0.128750 + 0.991677i \(0.541096\pi\)
\(740\) 0 0
\(741\) 12.0000 0.440831
\(742\) 9.00000 0.330400
\(743\) −1.00000 −0.0366864 −0.0183432 − 0.999832i \(-0.505839\pi\)
−0.0183432 + 0.999832i \(0.505839\pi\)
\(744\) −3.00000 −0.109985
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) −2.00000 −0.0731762
\(748\) 5.00000 0.182818
\(749\) 18.0000 0.657706
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) −18.0000 −0.655521
\(755\) 0 0
\(756\) 1.00000 0.0363696
\(757\) −22.0000 −0.799604 −0.399802 − 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 16.0000 0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) 33.0000 1.19625 0.598125 − 0.801403i \(-0.295913\pi\)
0.598125 + 0.801403i \(0.295913\pi\)
\(762\) 0 0
\(763\) 5.00000 0.181012
\(764\) 11.0000 0.397966
\(765\) 0 0
\(766\) 0 0
\(767\) 20.0000 0.722158
\(768\) −1.00000 −0.0360844
\(769\) 22.0000 0.793340 0.396670 − 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 2.00000 0.0719816
\(773\) −27.0000 −0.971123 −0.485561 − 0.874203i \(-0.661385\pi\)
−0.485561 + 0.874203i \(0.661385\pi\)
\(774\) −1.00000 −0.0359443
\(775\) 0 0
\(776\) −17.0000 −0.610264
\(777\) 1.00000 0.0358748
\(778\) 19.0000 0.681183
\(779\) 54.0000 1.93475
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −9.00000 −0.321634
\(784\) −6.00000 −0.214286
\(785\) 0 0
\(786\) −10.0000 −0.356688
\(787\) −36.0000 −1.28326 −0.641631 − 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) −14.0000 −0.498729
\(789\) −15.0000 −0.534014
\(790\) 0 0
\(791\) −13.0000 −0.462227
\(792\) −5.00000 −0.177667
\(793\) 10.0000 0.355110
\(794\) −18.0000 −0.638796
\(795\) 0 0
\(796\) −12.0000 −0.425329
\(797\) −40.0000 −1.41687 −0.708436 − 0.705775i \(-0.750599\pi\)
−0.708436 + 0.705775i \(0.750599\pi\)
\(798\) 6.00000 0.212398
\(799\) 0 0
\(800\) 0 0
\(801\) −2.00000 −0.0706665
\(802\) −6.00000 −0.211867
\(803\) 60.0000 2.11735
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) −6.00000 −0.211341
\(807\) −24.0000 −0.844840
\(808\) −12.0000 −0.422159
\(809\) −24.0000 −0.843795 −0.421898 − 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) −44.0000 −1.54505 −0.772524 − 0.634985i \(-0.781006\pi\)
−0.772524 + 0.634985i \(0.781006\pi\)
\(812\) −9.00000 −0.315838
\(813\) 16.0000 0.561144
\(814\) −5.00000 −0.175250
\(815\) 0 0
\(816\) 1.00000 0.0350070
\(817\) −6.00000 −0.209913
\(818\) 6.00000 0.209785
\(819\) 2.00000 0.0698857
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 − 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) −14.0000 −0.488306
\(823\) −40.0000 −1.39431 −0.697156 − 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 4.00000 0.139347
\(825\) 0 0
\(826\) 10.0000 0.347945
\(827\) 3.00000 0.104320 0.0521601 − 0.998639i \(-0.483389\pi\)
0.0521601 + 0.998639i \(0.483389\pi\)
\(828\) 0 0
\(829\) −17.0000 −0.590434 −0.295217 − 0.955430i \(-0.595392\pi\)
−0.295217 + 0.955430i \(0.595392\pi\)
\(830\) 0 0
\(831\) −8.00000 −0.277517
\(832\) −2.00000 −0.0693375
\(833\) 6.00000 0.207888
\(834\) 19.0000 0.657916
\(835\) 0 0
\(836\) −30.0000 −1.03757
\(837\) −3.00000 −0.103695
\(838\) 20.0000 0.690889
\(839\) 54.0000 1.86429 0.932144 − 0.362089i \(-0.117936\pi\)
0.932144 + 0.362089i \(0.117936\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 30.0000 1.03387
\(843\) 2.00000 0.0688837
\(844\) −5.00000 −0.172107
\(845\) 0 0
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) −9.00000 −0.309061
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 5.00000 0.171096
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 33.0000 1.12726 0.563629 − 0.826028i \(-0.309405\pi\)
0.563629 + 0.826028i \(0.309405\pi\)
\(858\) −10.0000 −0.341394
\(859\) −14.0000 −0.477674 −0.238837 − 0.971060i \(-0.576766\pi\)
−0.238837 + 0.971060i \(0.576766\pi\)
\(860\) 0 0
\(861\) 9.00000 0.306719
\(862\) 15.0000 0.510902
\(863\) −39.0000 −1.32758 −0.663788 − 0.747921i \(-0.731052\pi\)
−0.663788 + 0.747921i \(0.731052\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −36.0000 −1.22333
\(867\) 16.0000 0.543388
\(868\) −3.00000 −0.101827
\(869\) 60.0000 2.03536
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) −5.00000 −0.169321
\(873\) −17.0000 −0.575363
\(874\) 0 0
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) −47.0000 −1.58708 −0.793539 − 0.608520i \(-0.791764\pi\)
−0.793539 + 0.608520i \(0.791764\pi\)
\(878\) −27.0000 −0.911206
\(879\) 19.0000 0.640854
\(880\) 0 0
\(881\) 41.0000 1.38133 0.690663 − 0.723177i \(-0.257319\pi\)
0.690663 + 0.723177i \(0.257319\pi\)
\(882\) −6.00000 −0.202031
\(883\) 9.00000 0.302874 0.151437 − 0.988467i \(-0.451610\pi\)
0.151437 + 0.988467i \(0.451610\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) −2.00000 −0.0671913
\(887\) 35.0000 1.17518 0.587592 − 0.809157i \(-0.300076\pi\)
0.587592 + 0.809157i \(0.300076\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 0 0
\(890\) 0 0
\(891\) −5.00000 −0.167506
\(892\) 5.00000 0.167412
\(893\) 0 0
\(894\) −14.0000 −0.468230
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 24.0000 0.800890
\(899\) 27.0000 0.900500
\(900\) 0 0
\(901\) 9.00000 0.299833
\(902\) −45.0000 −1.49834
\(903\) −1.00000 −0.0332779
\(904\) 13.0000 0.432374
\(905\) 0 0
\(906\) −18.0000 −0.598010
\(907\) −4.00000 −0.132818 −0.0664089 − 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(908\) 7.00000 0.232303
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) −6.00000 −0.198680
\(913\) 10.0000 0.330952
\(914\) −1.00000 −0.0330771
\(915\) 0 0
\(916\) 0 0
\(917\) −10.0000 −0.330229
\(918\) 1.00000 0.0330049
\(919\) −56.0000 −1.84727 −0.923635 − 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −30.0000 −0.988534
\(922\) 35.0000 1.15266
\(923\) 0 0
\(924\) −5.00000 −0.164488
\(925\) 0 0
\(926\) −26.0000 −0.854413
\(927\) 4.00000 0.131377
\(928\) 9.00000 0.295439
\(929\) −1.00000 −0.0328089 −0.0164045 − 0.999865i \(-0.505222\pi\)
−0.0164045 + 0.999865i \(0.505222\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 8.00000 0.262049
\(933\) 25.0000 0.818463
\(934\) 21.0000 0.687141
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) −56.0000 −1.82944 −0.914720 − 0.404088i \(-0.867589\pi\)
−0.914720 + 0.404088i \(0.867589\pi\)
\(938\) 16.0000 0.522419
\(939\) −18.0000 −0.587408
\(940\) 0 0
\(941\) −22.0000 −0.717180 −0.358590 − 0.933495i \(-0.616742\pi\)
−0.358590 + 0.933495i \(0.616742\pi\)
\(942\) 7.00000 0.228072
\(943\) 0 0
\(944\) −10.0000 −0.325472
\(945\) 0 0
\(946\) 5.00000 0.162564
\(947\) −1.00000 −0.0324956 −0.0162478 − 0.999868i \(-0.505172\pi\)
−0.0162478 + 0.999868i \(0.505172\pi\)
\(948\) 12.0000 0.389742
\(949\) 24.0000 0.779073
\(950\) 0 0
\(951\) −7.00000 −0.226991
\(952\) 1.00000 0.0324102
\(953\) 4.00000 0.129573 0.0647864 − 0.997899i \(-0.479363\pi\)
0.0647864 + 0.997899i \(0.479363\pi\)
\(954\) −9.00000 −0.291386
\(955\) 0 0
\(956\) −21.0000 −0.679189
\(957\) 45.0000 1.45464
\(958\) −24.0000 −0.775405
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −2.00000 −0.0644826
\(963\) −18.0000 −0.580042
\(964\) −18.0000 −0.579741
\(965\) 0 0
\(966\) 0 0
\(967\) −36.0000 −1.15768 −0.578841 − 0.815440i \(-0.696495\pi\)
−0.578841 + 0.815440i \(0.696495\pi\)
\(968\) 14.0000 0.449977
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) 29.0000 0.930654 0.465327 − 0.885139i \(-0.345937\pi\)
0.465327 + 0.885139i \(0.345937\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 19.0000 0.609112
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −5.00000 −0.160046
\(977\) 27.0000 0.863807 0.431903 − 0.901920i \(-0.357842\pi\)
0.431903 + 0.901920i \(0.357842\pi\)
\(978\) 1.00000 0.0319765
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) −5.00000 −0.159638
\(982\) −36.0000 −1.14881
\(983\) 49.0000 1.56286 0.781429 − 0.623995i \(-0.214491\pi\)
0.781429 + 0.623995i \(0.214491\pi\)
\(984\) −9.00000 −0.286910
\(985\) 0 0
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) −12.0000 −0.381771
\(989\) 0 0
\(990\) 0 0
\(991\) −31.0000 −0.984747 −0.492374 − 0.870384i \(-0.663871\pi\)
−0.492374 + 0.870384i \(0.663871\pi\)
\(992\) 3.00000 0.0952501
\(993\) 8.00000 0.253872
\(994\) 0 0
\(995\) 0 0
\(996\) 2.00000 0.0633724
\(997\) −6.00000 −0.190022 −0.0950110 − 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −44.0000 −1.39280
\(999\) −1.00000 −0.0316386
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 5550.2.a.y.1.1 1
5.4 even 2 1110.2.a.f.1.1 ✓ 1
15.14 odd 2 3330.2.a.y.1.1 1
20.19 odd 2 8880.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.f.1.1 ✓ 1 5.4 even 2
3330.2.a.y.1.1 1 15.14 odd 2
5550.2.a.y.1.1 1 1.1 even 1 trivial
8880.2.a.d.1.1 1 20.19 odd 2