Properties

Label 1450.2.a.i.1.1
Level $1450$
Weight $2$
Character 1450.1
Self dual yes
Analytic conductor $11.578$
Analytic rank $0$
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] = [1450,2,Mod(1,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.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: \(11.5783082931\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 58)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1450.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} +3.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} -1.00000 q^{11} +3.00000 q^{12} -3.00000 q^{13} +2.00000 q^{14} +1.00000 q^{16} +4.00000 q^{17} +6.00000 q^{18} -8.00000 q^{19} +6.00000 q^{21} -1.00000 q^{22} +3.00000 q^{24} -3.00000 q^{26} +9.00000 q^{27} +2.00000 q^{28} -1.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -3.00000 q^{33} +4.00000 q^{34} +6.00000 q^{36} +8.00000 q^{37} -8.00000 q^{38} -9.00000 q^{39} -2.00000 q^{41} +6.00000 q^{42} -7.00000 q^{43} -1.00000 q^{44} -11.0000 q^{47} +3.00000 q^{48} -3.00000 q^{49} +12.0000 q^{51} -3.00000 q^{52} -1.00000 q^{53} +9.00000 q^{54} +2.00000 q^{56} -24.0000 q^{57} -1.00000 q^{58} -4.00000 q^{59} +4.00000 q^{61} +3.00000 q^{62} +12.0000 q^{63} +1.00000 q^{64} -3.00000 q^{66} +4.00000 q^{67} +4.00000 q^{68} -2.00000 q^{71} +6.00000 q^{72} +12.0000 q^{73} +8.00000 q^{74} -8.00000 q^{76} -2.00000 q^{77} -9.00000 q^{78} -7.00000 q^{79} +9.00000 q^{81} -2.00000 q^{82} +6.00000 q^{84} -7.00000 q^{86} -3.00000 q^{87} -1.00000 q^{88} -6.00000 q^{89} -6.00000 q^{91} +9.00000 q^{93} -11.0000 q^{94} +3.00000 q^{96} +6.00000 q^{97} -3.00000 q^{98} -6.00000 q^{99} +O(q^{100})\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 3.00000 1.22474
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.00000 0.353553
\(9\) 6.00000 2.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 3.00000 0.866025
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 6.00000 1.41421
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 6.00000 1.30931
\(22\) −1.00000 −0.213201
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −3.00000 −0.588348
\(27\) 9.00000 1.73205
\(28\) 2.00000 0.377964
\(29\) −1.00000 −0.185695
\(30\) 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
\(33\) −3.00000 −0.522233
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −8.00000 −1.29777
\(39\) −9.00000 −1.44115
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 6.00000 0.925820
\(43\) −7.00000 −1.06749 −0.533745 0.845645i \(-0.679216\pi\)
−0.533745 + 0.845645i \(0.679216\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 0 0
\(47\) −11.0000 −1.60451 −0.802257 0.596978i \(-0.796368\pi\)
−0.802257 + 0.596978i \(0.796368\pi\)
\(48\) 3.00000 0.433013
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 12.0000 1.68034
\(52\) −3.00000 −0.416025
\(53\) −1.00000 −0.137361 −0.0686803 0.997639i \(-0.521879\pi\)
−0.0686803 + 0.997639i \(0.521879\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) 2.00000 0.267261
\(57\) −24.0000 −3.17888
\(58\) −1.00000 −0.131306
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 3.00000 0.381000
\(63\) 12.0000 1.51186
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 6.00000 0.707107
\(73\) 12.0000 1.40449 0.702247 0.711934i \(-0.252180\pi\)
0.702247 + 0.711934i \(0.252180\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −8.00000 −0.917663
\(77\) −2.00000 −0.227921
\(78\) −9.00000 −1.01905
\(79\) −7.00000 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −2.00000 −0.220863
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 6.00000 0.654654
\(85\) 0 0
\(86\) −7.00000 −0.754829
\(87\) −3.00000 −0.321634
\(88\) −1.00000 −0.106600
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 9.00000 0.933257
\(94\) −11.0000 −1.13456
\(95\) 0 0
\(96\) 3.00000 0.306186
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −3.00000 −0.303046
\(99\) −6.00000 −0.603023
\(100\) 0 0
\(101\) 8.00000 0.796030 0.398015 0.917379i \(-0.369699\pi\)
0.398015 + 0.917379i \(0.369699\pi\)
\(102\) 12.0000 1.18818
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −1.00000 −0.0971286
\(107\) 2.00000 0.193347 0.0966736 0.995316i \(-0.469180\pi\)
0.0966736 + 0.995316i \(0.469180\pi\)
\(108\) 9.00000 0.866025
\(109\) 1.00000 0.0957826 0.0478913 0.998853i \(-0.484750\pi\)
0.0478913 + 0.998853i \(0.484750\pi\)
\(110\) 0 0
\(111\) 24.0000 2.27798
\(112\) 2.00000 0.188982
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) −24.0000 −2.24781
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) −18.0000 −1.66410
\(118\) −4.00000 −0.368230
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 4.00000 0.362143
\(123\) −6.00000 −0.541002
\(124\) 3.00000 0.269408
\(125\) 0 0
\(126\) 12.0000 1.06904
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) −21.0000 −1.84895
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −3.00000 −0.261116
\(133\) −16.0000 −1.38738
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 4.00000 0.342997
\(137\) 20.0000 1.70872 0.854358 0.519685i \(-0.173951\pi\)
0.854358 + 0.519685i \(0.173951\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −33.0000 −2.77910
\(142\) −2.00000 −0.167836
\(143\) 3.00000 0.250873
\(144\) 6.00000 0.500000
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) −9.00000 −0.742307
\(148\) 8.00000 0.657596
\(149\) 3.00000 0.245770 0.122885 0.992421i \(-0.460785\pi\)
0.122885 + 0.992421i \(0.460785\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −8.00000 −0.648886
\(153\) 24.0000 1.94029
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) −9.00000 −0.720577
\(157\) −22.0000 −1.75579 −0.877896 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −7.00000 −0.556890
\(159\) −3.00000 −0.237915
\(160\) 0 0
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −19.0000 −1.48819 −0.744097 0.668071i \(-0.767120\pi\)
−0.744097 + 0.668071i \(0.767120\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 0 0
\(167\) 22.0000 1.70241 0.851206 0.524832i \(-0.175872\pi\)
0.851206 + 0.524832i \(0.175872\pi\)
\(168\) 6.00000 0.462910
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) −48.0000 −3.67065
\(172\) −7.00000 −0.533745
\(173\) 14.0000 1.06440 0.532200 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) −3.00000 −0.227429
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −12.0000 −0.901975
\(178\) −6.00000 −0.449719
\(179\) −14.0000 −1.04641 −0.523205 0.852207i \(-0.675264\pi\)
−0.523205 + 0.852207i \(0.675264\pi\)
\(180\) 0 0
\(181\) −13.0000 −0.966282 −0.483141 0.875542i \(-0.660504\pi\)
−0.483141 + 0.875542i \(0.660504\pi\)
\(182\) −6.00000 −0.444750
\(183\) 12.0000 0.887066
\(184\) 0 0
\(185\) 0 0
\(186\) 9.00000 0.659912
\(187\) −4.00000 −0.292509
\(188\) −11.0000 −0.802257
\(189\) 18.0000 1.30931
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 3.00000 0.216506
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −6.00000 −0.426401
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) 12.0000 0.846415
\(202\) 8.00000 0.562878
\(203\) −2.00000 −0.140372
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) 0 0
\(208\) −3.00000 −0.208013
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) −25.0000 −1.72107 −0.860535 0.509390i \(-0.829871\pi\)
−0.860535 + 0.509390i \(0.829871\pi\)
\(212\) −1.00000 −0.0686803
\(213\) −6.00000 −0.411113
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 9.00000 0.612372
\(217\) 6.00000 0.407307
\(218\) 1.00000 0.0677285
\(219\) 36.0000 2.43265
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 24.0000 1.61077
\(223\) −26.0000 −1.74109 −0.870544 0.492090i \(-0.836233\pi\)
−0.870544 + 0.492090i \(0.836233\pi\)
\(224\) 2.00000 0.133631
\(225\) 0 0
\(226\) −18.0000 −1.19734
\(227\) 18.0000 1.19470 0.597351 0.801980i \(-0.296220\pi\)
0.597351 + 0.801980i \(0.296220\pi\)
\(228\) −24.0000 −1.58944
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −6.00000 −0.394771
\(232\) −1.00000 −0.0656532
\(233\) 25.0000 1.63780 0.818902 0.573933i \(-0.194583\pi\)
0.818902 + 0.573933i \(0.194583\pi\)
\(234\) −18.0000 −1.17670
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) −21.0000 −1.36410
\(238\) 8.00000 0.518563
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 24.0000 1.52708
\(248\) 3.00000 0.190500
\(249\) 0 0
\(250\) 0 0
\(251\) −7.00000 −0.441836 −0.220918 0.975292i \(-0.570905\pi\)
−0.220918 + 0.975292i \(0.570905\pi\)
\(252\) 12.0000 0.755929
\(253\) 0 0
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −21.0000 −1.30994 −0.654972 0.755653i \(-0.727320\pi\)
−0.654972 + 0.755653i \(0.727320\pi\)
\(258\) −21.0000 −1.30740
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 12.0000 0.741362
\(263\) 17.0000 1.04826 0.524132 0.851637i \(-0.324390\pi\)
0.524132 + 0.851637i \(0.324390\pi\)
\(264\) −3.00000 −0.184637
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) −18.0000 −1.10158
\(268\) 4.00000 0.244339
\(269\) 20.0000 1.21942 0.609711 0.792624i \(-0.291286\pi\)
0.609711 + 0.792624i \(0.291286\pi\)
\(270\) 0 0
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 4.00000 0.242536
\(273\) −18.0000 −1.08941
\(274\) 20.0000 1.20824
\(275\) 0 0
\(276\) 0 0
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 0 0
\(279\) 18.0000 1.07763
\(280\) 0 0
\(281\) −13.0000 −0.775515 −0.387757 0.921761i \(-0.626750\pi\)
−0.387757 + 0.921761i \(0.626750\pi\)
\(282\) −33.0000 −1.96512
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 3.00000 0.177394
\(287\) −4.00000 −0.236113
\(288\) 6.00000 0.353553
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) 12.0000 0.702247
\(293\) 34.0000 1.98630 0.993151 0.116841i \(-0.0372769\pi\)
0.993151 + 0.116841i \(0.0372769\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) −9.00000 −0.522233
\(298\) 3.00000 0.173785
\(299\) 0 0
\(300\) 0 0
\(301\) −14.0000 −0.806947
\(302\) 10.0000 0.575435
\(303\) 24.0000 1.37876
\(304\) −8.00000 −0.458831
\(305\) 0 0
\(306\) 24.0000 1.37199
\(307\) 29.0000 1.65512 0.827559 0.561379i \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(308\) −2.00000 −0.113961
\(309\) 18.0000 1.02398
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) −9.00000 −0.509525
\(313\) −25.0000 −1.41308 −0.706542 0.707671i \(-0.749746\pi\)
−0.706542 + 0.707671i \(0.749746\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) −7.00000 −0.393781
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) −3.00000 −0.168232
\(319\) 1.00000 0.0559893
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −19.0000 −1.05231
\(327\) 3.00000 0.165900
\(328\) −2.00000 −0.110432
\(329\) −22.0000 −1.21290
\(330\) 0 0
\(331\) 3.00000 0.164895 0.0824475 0.996595i \(-0.473726\pi\)
0.0824475 + 0.996595i \(0.473726\pi\)
\(332\) 0 0
\(333\) 48.0000 2.63038
\(334\) 22.0000 1.20379
\(335\) 0 0
\(336\) 6.00000 0.327327
\(337\) −4.00000 −0.217894 −0.108947 0.994048i \(-0.534748\pi\)
−0.108947 + 0.994048i \(0.534748\pi\)
\(338\) −4.00000 −0.217571
\(339\) −54.0000 −2.93288
\(340\) 0 0
\(341\) −3.00000 −0.162459
\(342\) −48.0000 −2.59554
\(343\) −20.0000 −1.07990
\(344\) −7.00000 −0.377415
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) −3.00000 −0.160817
\(349\) −19.0000 −1.01705 −0.508523 0.861048i \(-0.669808\pi\)
−0.508523 + 0.861048i \(0.669808\pi\)
\(350\) 0 0
\(351\) −27.0000 −1.44115
\(352\) −1.00000 −0.0533002
\(353\) 2.00000 0.106449 0.0532246 0.998583i \(-0.483050\pi\)
0.0532246 + 0.998583i \(0.483050\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 24.0000 1.27021
\(358\) −14.0000 −0.739923
\(359\) 9.00000 0.475002 0.237501 0.971387i \(-0.423672\pi\)
0.237501 + 0.971387i \(0.423672\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) −13.0000 −0.683265
\(363\) −30.0000 −1.57459
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 12.0000 0.627250
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) −12.0000 −0.624695
\(370\) 0 0
\(371\) −2.00000 −0.103835
\(372\) 9.00000 0.466628
\(373\) 1.00000 0.0517780 0.0258890 0.999665i \(-0.491758\pi\)
0.0258890 + 0.999665i \(0.491758\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) −11.0000 −0.567282
\(377\) 3.00000 0.154508
\(378\) 18.0000 0.925820
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) 0 0
\(381\) 24.0000 1.22956
\(382\) 0 0
\(383\) 34.0000 1.73732 0.868659 0.495410i \(-0.164982\pi\)
0.868659 + 0.495410i \(0.164982\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) −42.0000 −2.13498
\(388\) 6.00000 0.304604
\(389\) 16.0000 0.811232 0.405616 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −3.00000 −0.151523
\(393\) 36.0000 1.81596
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −6.00000 −0.301511
\(397\) −19.0000 −0.953583 −0.476791 0.879017i \(-0.658200\pi\)
−0.476791 + 0.879017i \(0.658200\pi\)
\(398\) −2.00000 −0.100251
\(399\) −48.0000 −2.40301
\(400\) 0 0
\(401\) −5.00000 −0.249688 −0.124844 0.992176i \(-0.539843\pi\)
−0.124844 + 0.992176i \(0.539843\pi\)
\(402\) 12.0000 0.598506
\(403\) −9.00000 −0.448322
\(404\) 8.00000 0.398015
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) −8.00000 −0.396545
\(408\) 12.0000 0.594089
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 60.0000 2.95958
\(412\) 6.00000 0.295599
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) −3.00000 −0.147087
\(417\) 0 0
\(418\) 8.00000 0.391293
\(419\) −14.0000 −0.683945 −0.341972 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) −28.0000 −1.36464 −0.682318 0.731055i \(-0.739028\pi\)
−0.682318 + 0.731055i \(0.739028\pi\)
\(422\) −25.0000 −1.21698
\(423\) −66.0000 −3.20903
\(424\) −1.00000 −0.0485643
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) 8.00000 0.387147
\(428\) 2.00000 0.0966736
\(429\) 9.00000 0.434524
\(430\) 0 0
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 9.00000 0.433013
\(433\) 16.0000 0.768911 0.384455 0.923144i \(-0.374389\pi\)
0.384455 + 0.923144i \(0.374389\pi\)
\(434\) 6.00000 0.288009
\(435\) 0 0
\(436\) 1.00000 0.0478913
\(437\) 0 0
\(438\) 36.0000 1.72015
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) −18.0000 −0.857143
\(442\) −12.0000 −0.570782
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 24.0000 1.13899
\(445\) 0 0
\(446\) −26.0000 −1.23114
\(447\) 9.00000 0.425685
\(448\) 2.00000 0.0944911
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −18.0000 −0.846649
\(453\) 30.0000 1.40952
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) −24.0000 −1.12390
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 14.0000 0.654177
\(459\) 36.0000 1.68034
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) −6.00000 −0.279145
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 25.0000 1.15810
\(467\) −23.0000 −1.06431 −0.532157 0.846646i \(-0.678618\pi\)
−0.532157 + 0.846646i \(0.678618\pi\)
\(468\) −18.0000 −0.832050
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −66.0000 −3.04112
\(472\) −4.00000 −0.184115
\(473\) 7.00000 0.321860
\(474\) −21.0000 −0.964562
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) −6.00000 −0.274721
\(478\) 20.0000 0.914779
\(479\) −11.0000 −0.502603 −0.251301 0.967909i \(-0.580859\pi\)
−0.251301 + 0.967909i \(0.580859\pi\)
\(480\) 0 0
\(481\) −24.0000 −1.09431
\(482\) 17.0000 0.774329
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 4.00000 0.181071
\(489\) −57.0000 −2.57763
\(490\) 0 0
\(491\) 5.00000 0.225647 0.112823 0.993615i \(-0.464011\pi\)
0.112823 + 0.993615i \(0.464011\pi\)
\(492\) −6.00000 −0.270501
\(493\) −4.00000 −0.180151
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 3.00000 0.134704
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) −8.00000 −0.358129 −0.179065 0.983837i \(-0.557307\pi\)
−0.179065 + 0.983837i \(0.557307\pi\)
\(500\) 0 0
\(501\) 66.0000 2.94866
\(502\) −7.00000 −0.312425
\(503\) 19.0000 0.847168 0.423584 0.905857i \(-0.360772\pi\)
0.423584 + 0.905857i \(0.360772\pi\)
\(504\) 12.0000 0.534522
\(505\) 0 0
\(506\) 0 0
\(507\) −12.0000 −0.532939
\(508\) 8.00000 0.354943
\(509\) 21.0000 0.930809 0.465404 0.885098i \(-0.345909\pi\)
0.465404 + 0.885098i \(0.345909\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 1.00000 0.0441942
\(513\) −72.0000 −3.17888
\(514\) −21.0000 −0.926270
\(515\) 0 0
\(516\) −21.0000 −0.924473
\(517\) 11.0000 0.483779
\(518\) 16.0000 0.703000
\(519\) 42.0000 1.84360
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) −6.00000 −0.262613
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 17.0000 0.741235
\(527\) 12.0000 0.522728
\(528\) −3.00000 −0.130558
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −24.0000 −1.04151
\(532\) −16.0000 −0.693688
\(533\) 6.00000 0.259889
\(534\) −18.0000 −0.778936
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) −42.0000 −1.81243
\(538\) 20.0000 0.862261
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) 13.0000 0.558398
\(543\) −39.0000 −1.67365
\(544\) 4.00000 0.171499
\(545\) 0 0
\(546\) −18.0000 −0.770329
\(547\) 2.00000 0.0855138 0.0427569 0.999086i \(-0.486386\pi\)
0.0427569 + 0.999086i \(0.486386\pi\)
\(548\) 20.0000 0.854358
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) −14.0000 −0.595341
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 18.0000 0.762001
\(559\) 21.0000 0.888205
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) −13.0000 −0.548372
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) −33.0000 −1.38955
\(565\) 0 0
\(566\) 0 0
\(567\) 18.0000 0.755929
\(568\) −2.00000 −0.0839181
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) −36.0000 −1.50655 −0.753277 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 3.00000 0.125436
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) 44.0000 1.83174 0.915872 0.401470i \(-0.131501\pi\)
0.915872 + 0.401470i \(0.131501\pi\)
\(578\) −1.00000 −0.0415945
\(579\) −30.0000 −1.24676
\(580\) 0 0
\(581\) 0 0
\(582\) 18.0000 0.746124
\(583\) 1.00000 0.0414158
\(584\) 12.0000 0.496564
\(585\) 0 0
\(586\) 34.0000 1.40453
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −9.00000 −0.371154
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 8.00000 0.328798
\(593\) 17.0000 0.698106 0.349053 0.937103i \(-0.386503\pi\)
0.349053 + 0.937103i \(0.386503\pi\)
\(594\) −9.00000 −0.369274
\(595\) 0 0
\(596\) 3.00000 0.122885
\(597\) −6.00000 −0.245564
\(598\) 0 0
\(599\) 13.0000 0.531166 0.265583 0.964088i \(-0.414436\pi\)
0.265583 + 0.964088i \(0.414436\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −14.0000 −0.570597
\(603\) 24.0000 0.977356
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) 24.0000 0.974933
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) −8.00000 −0.324443
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) 33.0000 1.33504
\(612\) 24.0000 0.970143
\(613\) 27.0000 1.09052 0.545260 0.838267i \(-0.316431\pi\)
0.545260 + 0.838267i \(0.316431\pi\)
\(614\) 29.0000 1.17034
\(615\) 0 0
\(616\) −2.00000 −0.0805823
\(617\) −12.0000 −0.483102 −0.241551 0.970388i \(-0.577656\pi\)
−0.241551 + 0.970388i \(0.577656\pi\)
\(618\) 18.0000 0.724066
\(619\) 31.0000 1.24600 0.622998 0.782224i \(-0.285915\pi\)
0.622998 + 0.782224i \(0.285915\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0000 −0.480770
\(624\) −9.00000 −0.360288
\(625\) 0 0
\(626\) −25.0000 −0.999201
\(627\) 24.0000 0.958468
\(628\) −22.0000 −0.877896
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) 22.0000 0.875806 0.437903 0.899022i \(-0.355721\pi\)
0.437903 + 0.899022i \(0.355721\pi\)
\(632\) −7.00000 −0.278445
\(633\) −75.0000 −2.98098
\(634\) 0 0
\(635\) 0 0
\(636\) −3.00000 −0.118958
\(637\) 9.00000 0.356593
\(638\) 1.00000 0.0395904
\(639\) −12.0000 −0.474713
\(640\) 0 0
\(641\) 36.0000 1.42191 0.710957 0.703235i \(-0.248262\pi\)
0.710957 + 0.703235i \(0.248262\pi\)
\(642\) 6.00000 0.236801
\(643\) −2.00000 −0.0788723 −0.0394362 0.999222i \(-0.512556\pi\)
−0.0394362 + 0.999222i \(0.512556\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −32.0000 −1.25902
\(647\) 20.0000 0.786281 0.393141 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) 9.00000 0.353553
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) −19.0000 −0.744097
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 3.00000 0.117309
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 72.0000 2.80899
\(658\) −22.0000 −0.857649
\(659\) 21.0000 0.818044 0.409022 0.912525i \(-0.365870\pi\)
0.409022 + 0.912525i \(0.365870\pi\)
\(660\) 0 0
\(661\) 46.0000 1.78919 0.894596 0.446875i \(-0.147463\pi\)
0.894596 + 0.446875i \(0.147463\pi\)
\(662\) 3.00000 0.116598
\(663\) −36.0000 −1.39812
\(664\) 0 0
\(665\) 0 0
\(666\) 48.0000 1.85996
\(667\) 0 0
\(668\) 22.0000 0.851206
\(669\) −78.0000 −3.01565
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 6.00000 0.231455
\(673\) −1.00000 −0.0385472 −0.0192736 0.999814i \(-0.506135\pi\)
−0.0192736 + 0.999814i \(0.506135\pi\)
\(674\) −4.00000 −0.154074
\(675\) 0 0
\(676\) −4.00000 −0.153846
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) −54.0000 −2.07386
\(679\) 12.0000 0.460518
\(680\) 0 0
\(681\) 54.0000 2.06928
\(682\) −3.00000 −0.114876
\(683\) 16.0000 0.612223 0.306111 0.951996i \(-0.400972\pi\)
0.306111 + 0.951996i \(0.400972\pi\)
\(684\) −48.0000 −1.83533
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 42.0000 1.60240
\(688\) −7.00000 −0.266872
\(689\) 3.00000 0.114291
\(690\) 0 0
\(691\) 4.00000 0.152167 0.0760836 0.997101i \(-0.475758\pi\)
0.0760836 + 0.997101i \(0.475758\pi\)
\(692\) 14.0000 0.532200
\(693\) −12.0000 −0.455842
\(694\) −18.0000 −0.683271
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) −8.00000 −0.303022
\(698\) −19.0000 −0.719161
\(699\) 75.0000 2.83676
\(700\) 0 0
\(701\) 31.0000 1.17085 0.585427 0.810725i \(-0.300927\pi\)
0.585427 + 0.810725i \(0.300927\pi\)
\(702\) −27.0000 −1.01905
\(703\) −64.0000 −2.41381
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 2.00000 0.0752710
\(707\) 16.0000 0.601742
\(708\) −12.0000 −0.450988
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) −42.0000 −1.57512
\(712\) −6.00000 −0.224860
\(713\) 0 0
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) −14.0000 −0.523205
\(717\) 60.0000 2.24074
\(718\) 9.00000 0.335877
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 45.0000 1.67473
\(723\) 51.0000 1.89671
\(724\) −13.0000 −0.483141
\(725\) 0 0
\(726\) −30.0000 −1.11340
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) −6.00000 −0.222375
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −28.0000 −1.03562
\(732\) 12.0000 0.443533
\(733\) 48.0000 1.77292 0.886460 0.462805i \(-0.153157\pi\)
0.886460 + 0.462805i \(0.153157\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 0 0
\(737\) −4.00000 −0.147342
\(738\) −12.0000 −0.441726
\(739\) 1.00000 0.0367856 0.0183928 0.999831i \(-0.494145\pi\)
0.0183928 + 0.999831i \(0.494145\pi\)
\(740\) 0 0
\(741\) 72.0000 2.64499
\(742\) −2.00000 −0.0734223
\(743\) 28.0000 1.02722 0.513610 0.858024i \(-0.328308\pi\)
0.513610 + 0.858024i \(0.328308\pi\)
\(744\) 9.00000 0.329956
\(745\) 0 0
\(746\) 1.00000 0.0366126
\(747\) 0 0
\(748\) −4.00000 −0.146254
\(749\) 4.00000 0.146157
\(750\) 0 0
\(751\) 32.0000 1.16770 0.583848 0.811863i \(-0.301546\pi\)
0.583848 + 0.811863i \(0.301546\pi\)
\(752\) −11.0000 −0.401129
\(753\) −21.0000 −0.765283
\(754\) 3.00000 0.109254
\(755\) 0 0
\(756\) 18.0000 0.654654
\(757\) 28.0000 1.01768 0.508839 0.860862i \(-0.330075\pi\)
0.508839 + 0.860862i \(0.330075\pi\)
\(758\) −28.0000 −1.01701
\(759\) 0 0
\(760\) 0 0
\(761\) 34.0000 1.23250 0.616250 0.787551i \(-0.288651\pi\)
0.616250 + 0.787551i \(0.288651\pi\)
\(762\) 24.0000 0.869428
\(763\) 2.00000 0.0724049
\(764\) 0 0
\(765\) 0 0
\(766\) 34.0000 1.22847
\(767\) 12.0000 0.433295
\(768\) 3.00000 0.108253
\(769\) 48.0000 1.73092 0.865462 0.500974i \(-0.167025\pi\)
0.865462 + 0.500974i \(0.167025\pi\)
\(770\) 0 0
\(771\) −63.0000 −2.26889
\(772\) −10.0000 −0.359908
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −42.0000 −1.50966
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 48.0000 1.72199
\(778\) 16.0000 0.573628
\(779\) 16.0000 0.573259
\(780\) 0 0
\(781\) 2.00000 0.0715656
\(782\) 0 0
\(783\) −9.00000 −0.321634
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) −46.0000 −1.63972 −0.819861 0.572562i \(-0.805950\pi\)
−0.819861 + 0.572562i \(0.805950\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 51.0000 1.81565
\(790\) 0 0
\(791\) −36.0000 −1.28001
\(792\) −6.00000 −0.213201
\(793\) −12.0000 −0.426132
\(794\) −19.0000 −0.674285
\(795\) 0 0
\(796\) −2.00000 −0.0708881
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) −48.0000 −1.69918
\(799\) −44.0000 −1.55661
\(800\) 0 0
\(801\) −36.0000 −1.27200
\(802\) −5.00000 −0.176556
\(803\) −12.0000 −0.423471
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −9.00000 −0.317011
\(807\) 60.0000 2.11210
\(808\) 8.00000 0.281439
\(809\) 24.0000 0.843795 0.421898 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) 6.00000 0.210688 0.105344 0.994436i \(-0.466406\pi\)
0.105344 + 0.994436i \(0.466406\pi\)
\(812\) −2.00000 −0.0701862
\(813\) 39.0000 1.36779
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) 12.0000 0.420084
\(817\) 56.0000 1.95919
\(818\) 22.0000 0.769212
\(819\) −36.0000 −1.25794
\(820\) 0 0
\(821\) −37.0000 −1.29131 −0.645654 0.763630i \(-0.723415\pi\)
−0.645654 + 0.763630i \(0.723415\pi\)
\(822\) 60.0000 2.09274
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) −23.0000 −0.799788 −0.399894 0.916561i \(-0.630953\pi\)
−0.399894 + 0.916561i \(0.630953\pi\)
\(828\) 0 0
\(829\) −28.0000 −0.972480 −0.486240 0.873825i \(-0.661632\pi\)
−0.486240 + 0.873825i \(0.661632\pi\)
\(830\) 0 0
\(831\) −42.0000 −1.45696
\(832\) −3.00000 −0.104006
\(833\) −12.0000 −0.415775
\(834\) 0 0
\(835\) 0 0
\(836\) 8.00000 0.276686
\(837\) 27.0000 0.933257
\(838\) −14.0000 −0.483622
\(839\) −45.0000 −1.55357 −0.776786 0.629764i \(-0.783151\pi\)
−0.776786 + 0.629764i \(0.783151\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −28.0000 −0.964944
\(843\) −39.0000 −1.34323
\(844\) −25.0000 −0.860535
\(845\) 0 0
\(846\) −66.0000 −2.26913
\(847\) −20.0000 −0.687208
\(848\) −1.00000 −0.0343401
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) −6.00000 −0.205557
\(853\) −30.0000 −1.02718 −0.513590 0.858036i \(-0.671685\pi\)
−0.513590 + 0.858036i \(0.671685\pi\)
\(854\) 8.00000 0.273754
\(855\) 0 0
\(856\) 2.00000 0.0683586
\(857\) 11.0000 0.375753 0.187876 0.982193i \(-0.439840\pi\)
0.187876 + 0.982193i \(0.439840\pi\)
\(858\) 9.00000 0.307255
\(859\) −51.0000 −1.74010 −0.870049 0.492966i \(-0.835913\pi\)
−0.870049 + 0.492966i \(0.835913\pi\)
\(860\) 0 0
\(861\) −12.0000 −0.408959
\(862\) −20.0000 −0.681203
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 9.00000 0.306186
\(865\) 0 0
\(866\) 16.0000 0.543702
\(867\) −3.00000 −0.101885
\(868\) 6.00000 0.203653
\(869\) 7.00000 0.237459
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 1.00000 0.0338643
\(873\) 36.0000 1.21842
\(874\) 0 0
\(875\) 0 0
\(876\) 36.0000 1.21633
\(877\) 23.0000 0.776655 0.388327 0.921521i \(-0.373053\pi\)
0.388327 + 0.921521i \(0.373053\pi\)
\(878\) −8.00000 −0.269987
\(879\) 102.000 3.44037
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) −18.0000 −0.606092
\(883\) 42.0000 1.41341 0.706706 0.707507i \(-0.250180\pi\)
0.706706 + 0.707507i \(0.250180\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 1.00000 0.0335767 0.0167884 0.999859i \(-0.494656\pi\)
0.0167884 + 0.999859i \(0.494656\pi\)
\(888\) 24.0000 0.805387
\(889\) 16.0000 0.536623
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) −26.0000 −0.870544
\(893\) 88.0000 2.94481
\(894\) 9.00000 0.301005
\(895\) 0 0
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) 22.0000 0.734150
\(899\) −3.00000 −0.100056
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 2.00000 0.0665927
\(903\) −42.0000 −1.39767
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) 30.0000 0.996683
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 18.0000 0.597351
\(909\) 48.0000 1.59206
\(910\) 0 0
\(911\) −3.00000 −0.0993944 −0.0496972 0.998764i \(-0.515826\pi\)
−0.0496972 + 0.998764i \(0.515826\pi\)
\(912\) −24.0000 −0.794719
\(913\) 0 0
\(914\) 10.0000 0.330771
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 24.0000 0.792550
\(918\) 36.0000 1.18818
\(919\) −6.00000 −0.197922 −0.0989609 0.995091i \(-0.531552\pi\)
−0.0989609 + 0.995091i \(0.531552\pi\)
\(920\) 0 0
\(921\) 87.0000 2.86675
\(922\) −30.0000 −0.987997
\(923\) 6.00000 0.197492
\(924\) −6.00000 −0.197386
\(925\) 0 0
\(926\) 16.0000 0.525793
\(927\) 36.0000 1.18240
\(928\) −1.00000 −0.0328266
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 25.0000 0.818902
\(933\) 0 0
\(934\) −23.0000 −0.752583
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 8.00000 0.261209
\(939\) −75.0000 −2.44753
\(940\) 0 0
\(941\) −15.0000 −0.488986 −0.244493 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) −66.0000 −2.15040
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 7.00000 0.227590
\(947\) −51.0000 −1.65728 −0.828639 0.559784i \(-0.810884\pi\)
−0.828639 + 0.559784i \(0.810884\pi\)
\(948\) −21.0000 −0.682048
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) 0 0
\(952\) 8.00000 0.259281
\(953\) −15.0000 −0.485898 −0.242949 0.970039i \(-0.578115\pi\)
−0.242949 + 0.970039i \(0.578115\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 3.00000 0.0969762
\(958\) −11.0000 −0.355394
\(959\) 40.0000 1.29167
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) −24.0000 −0.773791
\(963\) 12.0000 0.386695
\(964\) 17.0000 0.547533
\(965\) 0 0
\(966\) 0 0
\(967\) −59.0000 −1.89731 −0.948656 0.316310i \(-0.897556\pi\)
−0.948656 + 0.316310i \(0.897556\pi\)
\(968\) −10.0000 −0.321412
\(969\) −96.0000 −3.08396
\(970\) 0 0
\(971\) −52.0000 −1.66876 −0.834380 0.551190i \(-0.814174\pi\)
−0.834380 + 0.551190i \(0.814174\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 4.00000 0.128037
\(977\) −37.0000 −1.18373 −0.591867 0.806035i \(-0.701609\pi\)
−0.591867 + 0.806035i \(0.701609\pi\)
\(978\) −57.0000 −1.82266
\(979\) 6.00000 0.191761
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 5.00000 0.159556
\(983\) −39.0000 −1.24391 −0.621953 0.783054i \(-0.713661\pi\)
−0.621953 + 0.783054i \(0.713661\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) −66.0000 −2.10080
\(988\) 24.0000 0.763542
\(989\) 0 0
\(990\) 0 0
\(991\) 26.0000 0.825917 0.412959 0.910750i \(-0.364495\pi\)
0.412959 + 0.910750i \(0.364495\pi\)
\(992\) 3.00000 0.0952501
\(993\) 9.00000 0.285606
\(994\) −4.00000 −0.126872
\(995\) 0 0
\(996\) 0 0
\(997\) −16.0000 −0.506725 −0.253363 0.967371i \(-0.581537\pi\)
−0.253363 + 0.967371i \(0.581537\pi\)
\(998\) −8.00000 −0.253236
\(999\) 72.0000 2.27798
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 1450.2.a.i.1.1 1
5.2 odd 4 1450.2.b.f.349.2 2
5.3 odd 4 1450.2.b.f.349.1 2
5.4 even 2 58.2.a.a.1.1 1
15.14 odd 2 522.2.a.k.1.1 1
20.19 odd 2 464.2.a.f.1.1 1
35.34 odd 2 2842.2.a.d.1.1 1
40.19 odd 2 1856.2.a.b.1.1 1
40.29 even 2 1856.2.a.p.1.1 1
55.54 odd 2 7018.2.a.c.1.1 1
60.59 even 2 4176.2.a.bh.1.1 1
65.64 even 2 9802.2.a.d.1.1 1
145.99 odd 4 1682.2.b.e.1681.1 2
145.104 odd 4 1682.2.b.e.1681.2 2
145.144 even 2 1682.2.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
58.2.a.a.1.1 1 5.4 even 2
464.2.a.f.1.1 1 20.19 odd 2
522.2.a.k.1.1 1 15.14 odd 2
1450.2.a.i.1.1 1 1.1 even 1 trivial
1450.2.b.f.349.1 2 5.3 odd 4
1450.2.b.f.349.2 2 5.2 odd 4
1682.2.a.j.1.1 1 145.144 even 2
1682.2.b.e.1681.1 2 145.99 odd 4
1682.2.b.e.1681.2 2 145.104 odd 4
1856.2.a.b.1.1 1 40.19 odd 2
1856.2.a.p.1.1 1 40.29 even 2
2842.2.a.d.1.1 1 35.34 odd 2
4176.2.a.bh.1.1 1 60.59 even 2
7018.2.a.c.1.1 1 55.54 odd 2
9802.2.a.d.1.1 1 65.64 even 2