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Properties

Label 5550.2.a.bm.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: yes
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{16} -5.00000 q^{17} +1.00000 q^{18} -2.00000 q^{19} -2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{24} -1.00000 q^{26} +1.00000 q^{27} -8.00000 q^{29} -7.00000 q^{31} +1.00000 q^{32} -2.00000 q^{33} -5.00000 q^{34} +1.00000 q^{36} -1.00000 q^{37} -2.00000 q^{38} -1.00000 q^{39} -2.00000 q^{41} -2.00000 q^{44} -4.00000 q^{46} -3.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -5.00000 q^{51} -1.00000 q^{52} -3.00000 q^{53} +1.00000 q^{54} -2.00000 q^{57} -8.00000 q^{58} +7.00000 q^{59} +2.00000 q^{61} -7.00000 q^{62} +1.00000 q^{64} -2.00000 q^{66} -5.00000 q^{67} -5.00000 q^{68} -4.00000 q^{69} -1.00000 q^{71} +1.00000 q^{72} +15.0000 q^{73} -1.00000 q^{74} -2.00000 q^{76} -1.00000 q^{78} +13.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} -8.00000 q^{83} -8.00000 q^{87} -2.00000 q^{88} +10.0000 q^{89} -4.00000 q^{92} -7.00000 q^{93} -3.00000 q^{94} +1.00000 q^{96} +8.00000 q^{97} -7.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

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
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 − 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.00000 −1.21268 −0.606339 − 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 − 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.00000 −0.348155
\(34\) −5.00000 −0.857493
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −1.00000 −0.164399
\(38\) −2.00000 −0.324443
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −5.00000 −0.700140
\(52\) −1.00000 −0.138675
\(53\) −3.00000 −0.412082 −0.206041 − 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) −8.00000 −1.05045
\(59\) 7.00000 0.911322 0.455661 − 0.890153i \(-0.349403\pi\)
0.455661 + 0.890153i \(0.349403\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) −7.00000 −0.889001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.00000 −0.246183
\(67\) −5.00000 −0.610847 −0.305424 − 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) −5.00000 −0.606339
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 − 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 1.00000 0.117851
\(73\) 15.0000 1.75562 0.877809 − 0.479012i \(-0.159005\pi\)
0.877809 + 0.479012i \(0.159005\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) 13.0000 1.46261 0.731307 − 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −8.00000 −0.878114 −0.439057 − 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.00000 −0.857690
\(88\) −2.00000 −0.213201
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −4.00000 −0.417029
\(93\) −7.00000 −0.725866
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −7.00000 −0.707107
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −15.0000 −1.49256 −0.746278 − 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) −5.00000 −0.495074
\(103\) 7.00000 0.689730 0.344865 − 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −3.00000 −0.291386
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) −13.0000 −1.24517 −0.622587 − 0.782551i \(-0.713918\pi\)
−0.622587 + 0.782551i \(0.713918\pi\)
\(110\) 0 0
\(111\) −1.00000 −0.0949158
\(112\) 0 0
\(113\) 14.0000 1.31701 0.658505 − 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) −2.00000 −0.187317
\(115\) 0 0
\(116\) −8.00000 −0.742781
\(117\) −1.00000 −0.0924500
\(118\) 7.00000 0.644402
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) −2.00000 −0.180334
\(124\) −7.00000 −0.628619
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 − 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −3.00000 −0.262111 −0.131056 − 0.991375i \(-0.541837\pi\)
−0.131056 + 0.991375i \(0.541837\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) −5.00000 −0.431934
\(135\) 0 0
\(136\) −5.00000 −0.428746
\(137\) −2.00000 −0.170872 −0.0854358 − 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −4.00000 −0.340503
\(139\) 9.00000 0.763370 0.381685 − 0.924292i \(-0.375344\pi\)
0.381685 + 0.924292i \(0.375344\pi\)
\(140\) 0 0
\(141\) −3.00000 −0.252646
\(142\) −1.00000 −0.0839181
\(143\) 2.00000 0.167248
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 15.0000 1.24141
\(147\) −7.00000 −0.577350
\(148\) −1.00000 −0.0821995
\(149\) 15.0000 1.22885 0.614424 − 0.788976i \(-0.289388\pi\)
0.614424 + 0.788976i \(0.289388\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −2.00000 −0.162221
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) −4.00000 −0.319235 −0.159617 − 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 13.0000 1.03422
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) 6.00000 0.469956 0.234978 − 0.972001i \(-0.424498\pi\)
0.234978 + 0.972001i \(0.424498\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) −2.00000 −0.154765 −0.0773823 − 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −2.00000 −0.152944
\(172\) 0 0
\(173\) −1.00000 −0.0760286 −0.0380143 − 0.999277i \(-0.512103\pi\)
−0.0380143 + 0.999277i \(0.512103\pi\)
\(174\) −8.00000 −0.606478
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 7.00000 0.526152
\(178\) 10.0000 0.749532
\(179\) 3.00000 0.224231 0.112115 − 0.993695i \(-0.464237\pi\)
0.112115 + 0.993695i \(0.464237\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −7.00000 −0.513265
\(187\) 10.0000 0.731272
\(188\) −3.00000 −0.218797
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 − 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) 2.00000 0.142494 0.0712470 − 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −2.00000 −0.142134
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −5.00000 −0.352673
\(202\) −15.0000 −1.05540
\(203\) 0 0
\(204\) −5.00000 −0.350070
\(205\) 0 0
\(206\) 7.00000 0.487713
\(207\) −4.00000 −0.278019
\(208\) −1.00000 −0.0693375
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 5.00000 0.344214 0.172107 − 0.985078i \(-0.444942\pi\)
0.172107 + 0.985078i \(0.444942\pi\)
\(212\) −3.00000 −0.206041
\(213\) −1.00000 −0.0685189
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −13.0000 −0.880471
\(219\) 15.0000 1.01361
\(220\) 0 0
\(221\) 5.00000 0.336336
\(222\) −1.00000 −0.0671156
\(223\) −18.0000 −1.20537 −0.602685 − 0.797980i \(-0.705902\pi\)
−0.602685 + 0.797980i \(0.705902\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) −23.0000 −1.52656 −0.763282 − 0.646066i \(-0.776413\pi\)
−0.763282 + 0.646066i \(0.776413\pi\)
\(228\) −2.00000 −0.132453
\(229\) 20.0000 1.32164 0.660819 − 0.750546i \(-0.270209\pi\)
0.660819 + 0.750546i \(0.270209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −8.00000 −0.525226
\(233\) 14.0000 0.917170 0.458585 − 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) 7.00000 0.455661
\(237\) 13.0000 0.844441
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 − 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 − 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 2.00000 0.127257
\(248\) −7.00000 −0.444500
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 − 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 2.00000 0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −17.0000 −1.06043 −0.530215 − 0.847863i \(-0.677889\pi\)
−0.530215 + 0.847863i \(0.677889\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −8.00000 −0.495188
\(262\) −3.00000 −0.185341
\(263\) 16.0000 0.986602 0.493301 − 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) −2.00000 −0.123091
\(265\) 0 0
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −5.00000 −0.305424
\(269\) 25.0000 1.52428 0.762138 − 0.647414i \(-0.224150\pi\)
0.762138 + 0.647414i \(0.224150\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 − 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −5.00000 −0.303170
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) −4.00000 −0.240772
\(277\) 17.0000 1.02143 0.510716 − 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) 9.00000 0.539784
\(279\) −7.00000 −0.419079
\(280\) 0 0
\(281\) −11.0000 −0.656205 −0.328102 − 0.944642i \(-0.606409\pi\)
−0.328102 + 0.944642i \(0.606409\pi\)
\(282\) −3.00000 −0.178647
\(283\) −6.00000 −0.356663 −0.178331 − 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) −1.00000 −0.0593391
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) 15.0000 0.877809
\(293\) −3.00000 −0.175262 −0.0876309 − 0.996153i \(-0.527930\pi\)
−0.0876309 + 0.996153i \(0.527930\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −1.00000 −0.0581238
\(297\) −2.00000 −0.116052
\(298\) 15.0000 0.868927
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) 10.0000 0.575435
\(303\) −15.0000 −0.861727
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −5.00000 −0.285831
\(307\) 27.0000 1.54097 0.770486 − 0.637457i \(-0.220014\pi\)
0.770486 + 0.637457i \(0.220014\pi\)
\(308\) 0 0
\(309\) 7.00000 0.398216
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) 13.0000 0.731307
\(317\) −3.00000 −0.168497 −0.0842484 − 0.996445i \(-0.526849\pi\)
−0.0842484 + 0.996445i \(0.526849\pi\)
\(318\) −3.00000 −0.168232
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) 0 0
\(323\) 10.0000 0.556415
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 6.00000 0.332309
\(327\) −13.0000 −0.718902
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −18.0000 −0.989369 −0.494685 − 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) −8.00000 −0.439057
\(333\) −1.00000 −0.0547997
\(334\) −2.00000 −0.109435
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 − 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) −12.0000 −0.652714
\(339\) 14.0000 0.760376
\(340\) 0 0
\(341\) 14.0000 0.758143
\(342\) −2.00000 −0.108148
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.00000 −0.0537603
\(347\) −21.0000 −1.12734 −0.563670 − 0.826000i \(-0.690611\pi\)
−0.563670 + 0.826000i \(0.690611\pi\)
\(348\) −8.00000 −0.428845
\(349\) −22.0000 −1.17763 −0.588817 − 0.808267i \(-0.700406\pi\)
−0.588817 + 0.808267i \(0.700406\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) −2.00000 −0.106600
\(353\) 31.0000 1.64996 0.824982 − 0.565159i \(-0.191185\pi\)
0.824982 + 0.565159i \(0.191185\pi\)
\(354\) 7.00000 0.372046
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 3.00000 0.158555
\(359\) −29.0000 −1.53056 −0.765281 − 0.643697i \(-0.777400\pi\)
−0.765281 + 0.643697i \(0.777400\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) −22.0000 −1.15629
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 28.0000 1.46159 0.730794 − 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) −4.00000 −0.208514
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) 0 0
\(372\) −7.00000 −0.362933
\(373\) 32.0000 1.65690 0.828449 − 0.560065i \(-0.189224\pi\)
0.828449 + 0.560065i \(0.189224\pi\)
\(374\) 10.0000 0.517088
\(375\) 0 0
\(376\) −3.00000 −0.154713
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 − 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) −10.0000 −0.511645
\(383\) 14.0000 0.715367 0.357683 − 0.933843i \(-0.383567\pi\)
0.357683 + 0.933843i \(0.383567\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) −4.00000 −0.202808 −0.101404 − 0.994845i \(-0.532333\pi\)
−0.101404 + 0.994845i \(0.532333\pi\)
\(390\) 0 0
\(391\) 20.0000 1.01144
\(392\) −7.00000 −0.353553
\(393\) −3.00000 −0.151330
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) −20.0000 −1.00377 −0.501886 − 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) −8.00000 −0.401004
\(399\) 0 0
\(400\) 0 0
\(401\) −35.0000 −1.74782 −0.873908 − 0.486091i \(-0.838422\pi\)
−0.873908 + 0.486091i \(0.838422\pi\)
\(402\) −5.00000 −0.249377
\(403\) 7.00000 0.348695
\(404\) −15.0000 −0.746278
\(405\) 0 0
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) −5.00000 −0.247537
\(409\) −26.0000 −1.28562 −0.642809 − 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 7.00000 0.344865
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −1.00000 −0.0490290
\(417\) 9.00000 0.440732
\(418\) 4.00000 0.195646
\(419\) 38.0000 1.85642 0.928211 − 0.372055i \(-0.121347\pi\)
0.928211 + 0.372055i \(0.121347\pi\)
\(420\) 0 0
\(421\) 19.0000 0.926003 0.463002 − 0.886357i \(-0.346772\pi\)
0.463002 + 0.886357i \(0.346772\pi\)
\(422\) 5.00000 0.243396
\(423\) −3.00000 −0.145865
\(424\) −3.00000 −0.145693
\(425\) 0 0
\(426\) −1.00000 −0.0484502
\(427\) 0 0
\(428\) −8.00000 −0.386695
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 − 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) 37.0000 1.77811 0.889053 − 0.457804i \(-0.151364\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −13.0000 −0.622587
\(437\) 8.00000 0.382692
\(438\) 15.0000 0.716728
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 5.00000 0.237826
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) −1.00000 −0.0474579
\(445\) 0 0
\(446\) −18.0000 −0.852325
\(447\) 15.0000 0.709476
\(448\) 0 0
\(449\) −9.00000 −0.424736 −0.212368 − 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) 14.0000 0.658505
\(453\) 10.0000 0.469841
\(454\) −23.0000 −1.07944
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) −12.0000 −0.561336 −0.280668 − 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(458\) 20.0000 0.934539
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) −8.00000 −0.371391
\(465\) 0 0
\(466\) 14.0000 0.648537
\(467\) 21.0000 0.971764 0.485882 − 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 0 0
\(470\) 0 0
\(471\) −4.00000 −0.184310
\(472\) 7.00000 0.322201
\(473\) 0 0
\(474\) 13.0000 0.597110
\(475\) 0 0
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) 2.00000 0.0914779
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 1.00000 0.0455961
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −27.0000 −1.22349 −0.611743 − 0.791056i \(-0.709531\pi\)
−0.611743 + 0.791056i \(0.709531\pi\)
\(488\) 2.00000 0.0905357
\(489\) 6.00000 0.271329
\(490\) 0 0
\(491\) 10.0000 0.451294 0.225647 − 0.974209i \(-0.427550\pi\)
0.225647 + 0.974209i \(0.427550\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 40.0000 1.80151
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) 0 0
\(498\) −8.00000 −0.358489
\(499\) −2.00000 −0.0895323 −0.0447661 − 0.998997i \(-0.514254\pi\)
−0.0447661 + 0.998997i \(0.514254\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) −12.0000 −0.535586
\(503\) −24.0000 −1.07011 −0.535054 − 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.00000 0.355643
\(507\) −12.0000 −0.532939
\(508\) 2.00000 0.0887357
\(509\) −14.0000 −0.620539 −0.310270 − 0.950649i \(-0.600419\pi\)
−0.310270 + 0.950649i \(0.600419\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −2.00000 −0.0883022
\(514\) −17.0000 −0.749838
\(515\) 0 0
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) −1.00000 −0.0438951
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) −8.00000 −0.350150
\(523\) 20.0000 0.874539 0.437269 − 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −3.00000 −0.131056
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) 35.0000 1.52462
\(528\) −2.00000 −0.0870388
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 7.00000 0.303774
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −5.00000 −0.215967
\(537\) 3.00000 0.129460
\(538\) 25.0000 1.07783
\(539\) 14.0000 0.603023
\(540\) 0 0
\(541\) 18.0000 0.773880 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) −16.0000 −0.687259
\(543\) −22.0000 −0.944110
\(544\) −5.00000 −0.214373
\(545\) 0 0
\(546\) 0 0
\(547\) 4.00000 0.171028 0.0855138 − 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) −4.00000 −0.170251
\(553\) 0 0
\(554\) 17.0000 0.722261
\(555\) 0 0
\(556\) 9.00000 0.381685
\(557\) −30.0000 −1.27114 −0.635570 − 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) −7.00000 −0.296334
\(559\) 0 0
\(560\) 0 0
\(561\) 10.0000 0.422200
\(562\) −11.0000 −0.464007
\(563\) −12.0000 −0.505740 −0.252870 − 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) −3.00000 −0.126323
\(565\) 0 0
\(566\) −6.00000 −0.252199
\(567\) 0 0
\(568\) −1.00000 −0.0419591
\(569\) 11.0000 0.461144 0.230572 − 0.973055i \(-0.425940\pi\)
0.230572 + 0.973055i \(0.425940\pi\)
\(570\) 0 0
\(571\) 9.00000 0.376638 0.188319 − 0.982108i \(-0.439696\pi\)
0.188319 + 0.982108i \(0.439696\pi\)
\(572\) 2.00000 0.0836242
\(573\) −10.0000 −0.417756
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 32.0000 1.33218 0.666089 − 0.745873i \(-0.267967\pi\)
0.666089 + 0.745873i \(0.267967\pi\)
\(578\) 8.00000 0.332756
\(579\) 10.0000 0.415586
\(580\) 0 0
\(581\) 0 0
\(582\) 8.00000 0.331611
\(583\) 6.00000 0.248495
\(584\) 15.0000 0.620704
\(585\) 0 0
\(586\) −3.00000 −0.123929
\(587\) −12.0000 −0.495293 −0.247647 − 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −7.00000 −0.288675
\(589\) 14.0000 0.576860
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) −1.00000 −0.0410997
\(593\) 14.0000 0.574911 0.287456 − 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −2.00000 −0.0820610
\(595\) 0 0
\(596\) 15.0000 0.614424
\(597\) −8.00000 −0.327418
\(598\) 4.00000 0.163572
\(599\) −15.0000 −0.612883 −0.306442 − 0.951889i \(-0.599138\pi\)
−0.306442 + 0.951889i \(0.599138\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 − 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) −5.00000 −0.203616
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) −15.0000 −0.609333
\(607\) 23.0000 0.933541 0.466771 − 0.884378i \(-0.345417\pi\)
0.466771 + 0.884378i \(0.345417\pi\)
\(608\) −2.00000 −0.0811107
\(609\) 0 0
\(610\) 0 0
\(611\) 3.00000 0.121367
\(612\) −5.00000 −0.202113
\(613\) 2.00000 0.0807792 0.0403896 − 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 27.0000 1.08963
\(615\) 0 0
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 − 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 7.00000 0.281581
\(619\) −7.00000 −0.281354 −0.140677 − 0.990056i \(-0.544928\pi\)
−0.140677 + 0.990056i \(0.544928\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −24.0000 −0.962312
\(623\) 0 0
\(624\) −1.00000 −0.0400320
\(625\) 0 0
\(626\) −8.00000 −0.319744
\(627\) 4.00000 0.159745
\(628\) −4.00000 −0.159617
\(629\) 5.00000 0.199363
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 13.0000 0.517112
\(633\) 5.00000 0.198732
\(634\) −3.00000 −0.119145
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 7.00000 0.277350
\(638\) 16.0000 0.633446
\(639\) −1.00000 −0.0395594
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 − 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) −8.00000 −0.315735
\(643\) −14.0000 −0.552106 −0.276053 − 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 10.0000 0.393445
\(647\) 32.0000 1.25805 0.629025 − 0.777385i \(-0.283454\pi\)
0.629025 + 0.777385i \(0.283454\pi\)
\(648\) 1.00000 0.0392837
\(649\) −14.0000 −0.549548
\(650\) 0 0
\(651\) 0 0
\(652\) 6.00000 0.234978
\(653\) −24.0000 −0.939193 −0.469596 − 0.882881i \(-0.655601\pi\)
−0.469596 + 0.882881i \(0.655601\pi\)
\(654\) −13.0000 −0.508340
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 15.0000 0.585206
\(658\) 0 0
\(659\) −48.0000 −1.86981 −0.934907 − 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) 13.0000 0.505641 0.252821 − 0.967513i \(-0.418642\pi\)
0.252821 + 0.967513i \(0.418642\pi\)
\(662\) −18.0000 −0.699590
\(663\) 5.00000 0.194184
\(664\) −8.00000 −0.310460
\(665\) 0 0
\(666\) −1.00000 −0.0387492
\(667\) 32.0000 1.23904
\(668\) −2.00000 −0.0773823
\(669\) −18.0000 −0.695920
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 − 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −2.00000 −0.0770371
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) 27.0000 1.03769 0.518847 − 0.854867i \(-0.326361\pi\)
0.518847 + 0.854867i \(0.326361\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) 0 0
\(681\) −23.0000 −0.881362
\(682\) 14.0000 0.536088
\(683\) −25.0000 −0.956598 −0.478299 − 0.878197i \(-0.658747\pi\)
−0.478299 + 0.878197i \(0.658747\pi\)
\(684\) −2.00000 −0.0764719
\(685\) 0 0
\(686\) 0 0
\(687\) 20.0000 0.763048
\(688\) 0 0
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) −1.00000 −0.0380143
\(693\) 0 0
\(694\) −21.0000 −0.797149
\(695\) 0 0
\(696\) −8.00000 −0.303239
\(697\) 10.0000 0.378777
\(698\) −22.0000 −0.832712
\(699\) 14.0000 0.529529
\(700\) 0 0
\(701\) −28.0000 −1.05755 −0.528773 − 0.848763i \(-0.677348\pi\)
−0.528773 + 0.848763i \(0.677348\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 2.00000 0.0754314
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 31.0000 1.16670
\(707\) 0 0
\(708\) 7.00000 0.263076
\(709\) −6.00000 −0.225335 −0.112667 − 0.993633i \(-0.535939\pi\)
−0.112667 + 0.993633i \(0.535939\pi\)
\(710\) 0 0
\(711\) 13.0000 0.487538
\(712\) 10.0000 0.374766
\(713\) 28.0000 1.04861
\(714\) 0 0
\(715\) 0 0
\(716\) 3.00000 0.112115
\(717\) 2.00000 0.0746914
\(718\) −29.0000 −1.08227
\(719\) 19.0000 0.708580 0.354290 − 0.935136i \(-0.384723\pi\)
0.354290 + 0.935136i \(0.384723\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −15.0000 −0.558242
\(723\) −2.00000 −0.0743808
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) 12.0000 0.445055 0.222528 − 0.974926i \(-0.428569\pi\)
0.222528 + 0.974926i \(0.428569\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 2.00000 0.0739221
\(733\) 34.0000 1.25582 0.627909 − 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 28.0000 1.03350
\(735\) 0 0
\(736\) −4.00000 −0.147442
\(737\) 10.0000 0.368355
\(738\) −2.00000 −0.0736210
\(739\) 21.0000 0.772497 0.386249 − 0.922395i \(-0.373771\pi\)
0.386249 + 0.922395i \(0.373771\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) −29.0000 −1.06391 −0.531953 − 0.846774i \(-0.678542\pi\)
−0.531953 + 0.846774i \(0.678542\pi\)
\(744\) −7.00000 −0.256632
\(745\) 0 0
\(746\) 32.0000 1.17160
\(747\) −8.00000 −0.292705
\(748\) 10.0000 0.365636
\(749\) 0 0
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 − 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) −3.00000 −0.109399
\(753\) −12.0000 −0.437304
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) 5.00000 0.181728 0.0908640 − 0.995863i \(-0.471037\pi\)
0.0908640 + 0.995863i \(0.471037\pi\)
\(758\) −20.0000 −0.726433
\(759\) 8.00000 0.290382
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 − 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) 2.00000 0.0724524
\(763\) 0 0
\(764\) −10.0000 −0.361787
\(765\) 0 0
\(766\) 14.0000 0.505841
\(767\) −7.00000 −0.252755
\(768\) 1.00000 0.0360844
\(769\) −40.0000 −1.44244 −0.721218 − 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) −17.0000 −0.612240
\(772\) 10.0000 0.359908
\(773\) −30.0000 −1.07903 −0.539513 − 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) 0 0
\(778\) −4.00000 −0.143407
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) 20.0000 0.715199
\(783\) −8.00000 −0.285897
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) −3.00000 −0.107006
\(787\) −5.00000 −0.178231 −0.0891154 − 0.996021i \(-0.528404\pi\)
−0.0891154 + 0.996021i \(0.528404\pi\)
\(788\) 2.00000 0.0712470
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) −2.00000 −0.0710221
\(794\) −20.0000 −0.709773
\(795\) 0 0
\(796\) −8.00000 −0.283552
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 15.0000 0.530662
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) −35.0000 −1.23589
\(803\) −30.0000 −1.05868
\(804\) −5.00000 −0.176336
\(805\) 0 0
\(806\) 7.00000 0.246564
\(807\) 25.0000 0.880042
\(808\) −15.0000 −0.527698
\(809\) 50.0000 1.75791 0.878953 − 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 0 0
\(811\) 23.0000 0.807639 0.403820 − 0.914839i \(-0.367682\pi\)
0.403820 + 0.914839i \(0.367682\pi\)
\(812\) 0 0
\(813\) −16.0000 −0.561144
\(814\) 2.00000 0.0701000
\(815\) 0 0
\(816\) −5.00000 −0.175035
\(817\) 0 0
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) 0 0
\(821\) 27.0000 0.942306 0.471153 − 0.882051i \(-0.343838\pi\)
0.471153 + 0.882051i \(0.343838\pi\)
\(822\) −2.00000 −0.0697580
\(823\) 26.0000 0.906303 0.453152 − 0.891434i \(-0.350300\pi\)
0.453152 + 0.891434i \(0.350300\pi\)
\(824\) 7.00000 0.243857
\(825\) 0 0
\(826\) 0 0
\(827\) −9.00000 −0.312961 −0.156480 − 0.987681i \(-0.550015\pi\)
−0.156480 + 0.987681i \(0.550015\pi\)
\(828\) −4.00000 −0.139010
\(829\) 2.00000 0.0694629 0.0347314 − 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 17.0000 0.589723
\(832\) −1.00000 −0.0346688
\(833\) 35.0000 1.21268
\(834\) 9.00000 0.311645
\(835\) 0 0
\(836\) 4.00000 0.138343
\(837\) −7.00000 −0.241955
\(838\) 38.0000 1.31269
\(839\) 9.00000 0.310715 0.155357 − 0.987858i \(-0.450347\pi\)
0.155357 + 0.987858i \(0.450347\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 19.0000 0.654783
\(843\) −11.0000 −0.378860
\(844\) 5.00000 0.172107
\(845\) 0 0
\(846\) −3.00000 −0.103142
\(847\) 0 0
\(848\) −3.00000 −0.103020
\(849\) −6.00000 −0.205919
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) −1.00000 −0.0342594
\(853\) −30.0000 −1.02718 −0.513590 − 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) 21.0000 0.717346 0.358673 − 0.933463i \(-0.383229\pi\)
0.358673 + 0.933463i \(0.383229\pi\)
\(858\) 2.00000 0.0682789
\(859\) −48.0000 −1.63774 −0.818869 − 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 24.0000 0.816970 0.408485 − 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 37.0000 1.25731
\(867\) 8.00000 0.271694
\(868\) 0 0
\(869\) −26.0000 −0.881990
\(870\) 0 0
\(871\) 5.00000 0.169419
\(872\) −13.0000 −0.440236
\(873\) 8.00000 0.270759
\(874\) 8.00000 0.270604
\(875\) 0 0
\(876\) 15.0000 0.506803
\(877\) −6.00000 −0.202606 −0.101303 − 0.994856i \(-0.532301\pi\)
−0.101303 + 0.994856i \(0.532301\pi\)
\(878\) 0 0
\(879\) −3.00000 −0.101187
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 − 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) −7.00000 −0.235702
\(883\) 4.00000 0.134611 0.0673054 − 0.997732i \(-0.478560\pi\)
0.0673054 + 0.997732i \(0.478560\pi\)
\(884\) 5.00000 0.168168
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −27.0000 −0.906571 −0.453286 − 0.891365i \(-0.649748\pi\)
−0.453286 + 0.891365i \(0.649748\pi\)
\(888\) −1.00000 −0.0335578
\(889\) 0 0
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) −18.0000 −0.602685
\(893\) 6.00000 0.200782
\(894\) 15.0000 0.501675
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) −9.00000 −0.300334
\(899\) 56.0000 1.86770
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) 0 0
\(906\) 10.0000 0.332228
\(907\) 24.0000 0.796907 0.398453 − 0.917189i \(-0.369547\pi\)
0.398453 + 0.917189i \(0.369547\pi\)
\(908\) −23.0000 −0.763282
\(909\) −15.0000 −0.497519
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) −2.00000 −0.0662266
\(913\) 16.0000 0.529523
\(914\) −12.0000 −0.396925
\(915\) 0 0
\(916\) 20.0000 0.660819
\(917\) 0 0
\(918\) −5.00000 −0.165025
\(919\) −24.0000 −0.791687 −0.395843 − 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) 27.0000 0.889680
\(922\) −12.0000 −0.395199
\(923\) 1.00000 0.0329154
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 7.00000 0.229910
\(928\) −8.00000 −0.262613
\(929\) −32.0000 −1.04989 −0.524943 − 0.851137i \(-0.675913\pi\)
−0.524943 + 0.851137i \(0.675913\pi\)
\(930\) 0 0
\(931\) 14.0000 0.458831
\(932\) 14.0000 0.458585
\(933\) −24.0000 −0.785725
\(934\) 21.0000 0.687141
\(935\) 0 0
\(936\) −1.00000 −0.0326860
\(937\) 43.0000 1.40475 0.702374 − 0.711808i \(-0.252123\pi\)
0.702374 + 0.711808i \(0.252123\pi\)
\(938\) 0 0
\(939\) −8.00000 −0.261070
\(940\) 0 0
\(941\) 45.0000 1.46696 0.733479 − 0.679712i \(-0.237895\pi\)
0.733479 + 0.679712i \(0.237895\pi\)
\(942\) −4.00000 −0.130327
\(943\) 8.00000 0.260516
\(944\) 7.00000 0.227831
\(945\) 0 0
\(946\) 0 0
\(947\) −31.0000 −1.00736 −0.503682 − 0.863889i \(-0.668022\pi\)
−0.503682 + 0.863889i \(0.668022\pi\)
\(948\) 13.0000 0.422220
\(949\) −15.0000 −0.486921
\(950\) 0 0
\(951\) −3.00000 −0.0972817
\(952\) 0 0
\(953\) 16.0000 0.518291 0.259145 − 0.965838i \(-0.416559\pi\)
0.259145 + 0.965838i \(0.416559\pi\)
\(954\) −3.00000 −0.0971286
\(955\) 0 0
\(956\) 2.00000 0.0646846
\(957\) 16.0000 0.517207
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 1.00000 0.0322413
\(963\) −8.00000 −0.257796
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) −5.00000 −0.160789 −0.0803946 − 0.996763i \(-0.525618\pi\)
−0.0803946 + 0.996763i \(0.525618\pi\)
\(968\) −7.00000 −0.224989
\(969\) 10.0000 0.321246
\(970\) 0 0
\(971\) −46.0000 −1.47621 −0.738105 − 0.674686i \(-0.764279\pi\)
−0.738105 + 0.674686i \(0.764279\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −27.0000 −0.865136
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −29.0000 −0.927792 −0.463896 − 0.885890i \(-0.653549\pi\)
−0.463896 + 0.885890i \(0.653549\pi\)
\(978\) 6.00000 0.191859
\(979\) −20.0000 −0.639203
\(980\) 0 0
\(981\) −13.0000 −0.415058
\(982\) 10.0000 0.319113
\(983\) −48.0000 −1.53096 −0.765481 − 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 40.0000 1.27386
\(987\) 0 0
\(988\) 2.00000 0.0636285
\(989\) 0 0
\(990\) 0 0
\(991\) −29.0000 −0.921215 −0.460608 − 0.887604i \(-0.652368\pi\)
−0.460608 + 0.887604i \(0.652368\pi\)
\(992\) −7.00000 −0.222250
\(993\) −18.0000 −0.571213
\(994\) 0 0
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) 9.00000 0.285033 0.142516 − 0.989792i \(-0.454481\pi\)
0.142516 + 0.989792i \(0.454481\pi\)
\(998\) −2.00000 −0.0633089
\(999\) −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.bm.1.1 yes 1
5.4 even 2 5550.2.a.e.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5550.2.a.e.1.1 ✓ 1 5.4 even 2
5550.2.a.bm.1.1 yes 1 1.1 even 1 trivial