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Properties

Label 6042.2.a.d.1.1
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +2.00000 q^{11} -1.00000 q^{12} +4.00000 q^{13} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{19} +2.00000 q^{20} -2.00000 q^{22} -2.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{27} +2.00000 q^{29} +2.00000 q^{30} +2.00000 q^{31} -1.00000 q^{32} -2.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +4.00000 q^{37} -1.00000 q^{38} -4.00000 q^{39} -2.00000 q^{40} +2.00000 q^{41} +8.00000 q^{43} +2.00000 q^{44} +2.00000 q^{45} +2.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +4.00000 q^{52} -1.00000 q^{53} +1.00000 q^{54} +4.00000 q^{55} -1.00000 q^{57} -2.00000 q^{58} -2.00000 q^{60} -2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +8.00000 q^{65} +2.00000 q^{66} -8.00000 q^{67} +2.00000 q^{68} +2.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -14.0000 q^{73} -4.00000 q^{74} +1.00000 q^{75} +1.00000 q^{76} +4.00000 q^{78} +10.0000 q^{79} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +4.00000 q^{85} -8.00000 q^{86} -2.00000 q^{87} -2.00000 q^{88} -10.0000 q^{89} -2.00000 q^{90} -2.00000 q^{92} -2.00000 q^{93} -12.0000 q^{94} +2.00000 q^{95} +1.00000 q^{96} +2.00000 q^{97} +7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
\(5\) 2.00000 0.894427 0.447214 − 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 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
\(10\) −2.00000 −0.632456
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −2.00000 −0.417029 −0.208514 − 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −4.00000 −0.784465
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 − 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 2.00000 0.365148
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.00000 −0.348155
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −1.00000 −0.162221
\(39\) −4.00000 −0.640513
\(40\) −2.00000 −0.316228
\(41\) 2.00000 0.312348 0.156174 − 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 − 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 2.00000 0.301511
\(45\) 2.00000 0.298142
\(46\) 2.00000 0.294884
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 4.00000 0.554700
\(53\) −1.00000 −0.137361
\(54\) 1.00000 0.136083
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) −2.00000 −0.262613
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −2.00000 −0.258199
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −2.00000 −0.254000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.00000 0.992278
\(66\) 2.00000 0.246183
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 − 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) −4.00000 −0.464991
\(75\) 1.00000 0.115470
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −8.00000 −0.862662
\(87\) −2.00000 −0.214423
\(88\) −2.00000 −0.213201
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) −2.00000 −0.207390
\(94\) −12.0000 −1.23771
\(95\) 2.00000 0.205196
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 − 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 7.00000 0.707107
\(99\) 2.00000 0.201008
\(100\) −1.00000 −0.100000
\(101\) 10.0000 0.995037 0.497519 − 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 2.00000 0.198030
\(103\) 18.0000 1.77359 0.886796 − 0.462160i \(-0.152926\pi\)
0.886796 + 0.462160i \(0.152926\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 1.00000 0.0971286
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −2.00000 −0.191565 −0.0957826 − 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −4.00000 −0.381385
\(111\) −4.00000 −0.379663
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 1.00000 0.0936586
\(115\) −4.00000 −0.373002
\(116\) 2.00000 0.185695
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) −7.00000 −0.636364
\(122\) 2.00000 0.181071
\(123\) −2.00000 −0.180334
\(124\) 2.00000 0.179605
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 − 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.00000 −0.704361
\(130\) −8.00000 −0.701646
\(131\) −22.0000 −1.92215 −0.961074 − 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) −2.00000 −0.174078
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) −2.00000 −0.172133
\(136\) −2.00000 −0.171499
\(137\) −16.0000 −1.36697 −0.683486 − 0.729964i \(-0.739537\pi\)
−0.683486 + 0.729964i \(0.739537\pi\)
\(138\) −2.00000 −0.170251
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) −12.0000 −1.00702
\(143\) 8.00000 0.668994
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 14.0000 1.15865
\(147\) 7.00000 0.577350
\(148\) 4.00000 0.328798
\(149\) −12.0000 −0.983078 −0.491539 − 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) −4.00000 −0.320256
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −10.0000 −0.795557
\(159\) 1.00000 0.0793052
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.00000 0.156174
\(165\) −4.00000 −0.311400
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 − 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −4.00000 −0.306786
\(171\) 1.00000 0.0764719
\(172\) 8.00000 0.609994
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 0 0
\(178\) 10.0000 0.749532
\(179\) 12.0000 0.896922 0.448461 − 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) 22.0000 1.63525 0.817624 − 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 2.00000 0.147442
\(185\) 8.00000 0.588172
\(186\) 2.00000 0.146647
\(187\) 4.00000 0.292509
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 10.0000 0.723575 0.361787 − 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −24.0000 −1.72756 −0.863779 − 0.503871i \(-0.831909\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −2.00000 −0.143592
\(195\) −8.00000 −0.572892
\(196\) −7.00000 −0.500000
\(197\) −8.00000 −0.569976 −0.284988 − 0.958531i \(-0.591990\pi\)
−0.284988 + 0.958531i \(0.591990\pi\)
\(198\) −2.00000 −0.142134
\(199\) −24.0000 −1.70131 −0.850657 − 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 1.00000 0.0707107
\(201\) 8.00000 0.564276
\(202\) −10.0000 −0.703598
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 4.00000 0.279372
\(206\) −18.0000 −1.25412
\(207\) −2.00000 −0.139010
\(208\) 4.00000 0.277350
\(209\) 2.00000 0.138343
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 − 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) −1.00000 −0.0686803
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 2.00000 0.135457
\(219\) 14.0000 0.946032
\(220\) 4.00000 0.269680
\(221\) 8.00000 0.538138
\(222\) 4.00000 0.268462
\(223\) 4.00000 0.267860 0.133930 − 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −6.00000 −0.399114
\(227\) −20.0000 −1.32745 −0.663723 − 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −18.0000 −1.18947 −0.594737 − 0.803921i \(-0.702744\pi\)
−0.594737 + 0.803921i \(0.702744\pi\)
\(230\) 4.00000 0.263752
\(231\) 0 0
\(232\) −2.00000 −0.131306
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) −4.00000 −0.261488
\(235\) 24.0000 1.56559
\(236\) 0 0
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 − 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) −2.00000 −0.129099
\(241\) −30.0000 −1.93247 −0.966235 − 0.257663i \(-0.917048\pi\)
−0.966235 + 0.257663i \(0.917048\pi\)
\(242\) 7.00000 0.449977
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) −14.0000 −0.894427
\(246\) 2.00000 0.127515
\(247\) 4.00000 0.254514
\(248\) −2.00000 −0.127000
\(249\) 0 0
\(250\) 12.0000 0.758947
\(251\) 8.00000 0.504956 0.252478 − 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 2.00000 0.125491
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 − 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 8.00000 0.498058
\(259\) 0 0
\(260\) 8.00000 0.496139
\(261\) 2.00000 0.123797
\(262\) 22.0000 1.35916
\(263\) 22.0000 1.35658 0.678289 − 0.734795i \(-0.262722\pi\)
0.678289 + 0.734795i \(0.262722\pi\)
\(264\) 2.00000 0.123091
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −8.00000 −0.488678
\(269\) 14.0000 0.853595 0.426798 − 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(270\) 2.00000 0.121716
\(271\) −24.0000 −1.45790 −0.728948 − 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) 16.0000 0.966595
\(275\) −2.00000 −0.120605
\(276\) 2.00000 0.120386
\(277\) −10.0000 −0.600842 −0.300421 − 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −12.0000 −0.719712
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 − 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 12.0000 0.714590
\(283\) 4.00000 0.237775 0.118888 − 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 12.0000 0.712069
\(285\) −2.00000 −0.118470
\(286\) −8.00000 −0.473050
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) −2.00000 −0.117242
\(292\) −14.0000 −0.819288
\(293\) −2.00000 −0.116841 −0.0584206 − 0.998292i \(-0.518606\pi\)
−0.0584206 + 0.998292i \(0.518606\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) −4.00000 −0.232495
\(297\) −2.00000 −0.116052
\(298\) 12.0000 0.695141
\(299\) −8.00000 −0.462652
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) −10.0000 −0.575435
\(303\) −10.0000 −0.574485
\(304\) 1.00000 0.0573539
\(305\) −4.00000 −0.229039
\(306\) −2.00000 −0.114332
\(307\) −22.0000 −1.25561 −0.627803 − 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 0 0
\(309\) −18.0000 −1.02398
\(310\) −4.00000 −0.227185
\(311\) 12.0000 0.680458 0.340229 − 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 4.00000 0.226455
\(313\) 10.0000 0.565233 0.282617 − 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 4.00000 0.223957
\(320\) 2.00000 0.111803
\(321\) 0 0
\(322\) 0 0
\(323\) 2.00000 0.111283
\(324\) 1.00000 0.0555556
\(325\) −4.00000 −0.221880
\(326\) 4.00000 0.221540
\(327\) 2.00000 0.110600
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 4.00000 0.220193
\(331\) −30.0000 −1.64895 −0.824475 − 0.565899i \(-0.808529\pi\)
−0.824475 + 0.565899i \(0.808529\pi\)
\(332\) 0 0
\(333\) 4.00000 0.219199
\(334\) −16.0000 −0.875481
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 − 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) −3.00000 −0.163178
\(339\) −6.00000 −0.325875
\(340\) 4.00000 0.216930
\(341\) 4.00000 0.216612
\(342\) −1.00000 −0.0540738
\(343\) 0 0
\(344\) −8.00000 −0.431331
\(345\) 4.00000 0.215353
\(346\) −14.0000 −0.752645
\(347\) 18.0000 0.966291 0.483145 − 0.875540i \(-0.339494\pi\)
0.483145 + 0.875540i \(0.339494\pi\)
\(348\) −2.00000 −0.107211
\(349\) 2.00000 0.107058 0.0535288 − 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) −2.00000 −0.106600
\(353\) 28.0000 1.49029 0.745145 − 0.666903i \(-0.232380\pi\)
0.745145 + 0.666903i \(0.232380\pi\)
\(354\) 0 0
\(355\) 24.0000 1.27379
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 22.0000 1.16112 0.580558 − 0.814219i \(-0.302835\pi\)
0.580558 + 0.814219i \(0.302835\pi\)
\(360\) −2.00000 −0.105409
\(361\) 1.00000 0.0526316
\(362\) −22.0000 −1.15629
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) −28.0000 −1.46559
\(366\) −2.00000 −0.104542
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −2.00000 −0.104257
\(369\) 2.00000 0.104116
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) −2.00000 −0.103695
\(373\) −10.0000 −0.517780 −0.258890 − 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −4.00000 −0.206835
\(375\) 12.0000 0.619677
\(376\) −12.0000 −0.618853
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 − 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 2.00000 0.102598
\(381\) 2.00000 0.102463
\(382\) −10.0000 −0.511645
\(383\) −20.0000 −1.02195 −0.510976 − 0.859595i \(-0.670716\pi\)
−0.510976 + 0.859595i \(0.670716\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 24.0000 1.22157
\(387\) 8.00000 0.406663
\(388\) 2.00000 0.101535
\(389\) −10.0000 −0.507020 −0.253510 − 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 8.00000 0.405096
\(391\) −4.00000 −0.202289
\(392\) 7.00000 0.353553
\(393\) 22.0000 1.10975
\(394\) 8.00000 0.403034
\(395\) 20.0000 1.00631
\(396\) 2.00000 0.100504
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 24.0000 1.20301
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 18.0000 0.898877 0.449439 − 0.893311i \(-0.351624\pi\)
0.449439 + 0.893311i \(0.351624\pi\)
\(402\) −8.00000 −0.399004
\(403\) 8.00000 0.398508
\(404\) 10.0000 0.497519
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) 30.0000 1.48340 0.741702 − 0.670729i \(-0.234019\pi\)
0.741702 + 0.670729i \(0.234019\pi\)
\(410\) −4.00000 −0.197546
\(411\) 16.0000 0.789222
\(412\) 18.0000 0.886796
\(413\) 0 0
\(414\) 2.00000 0.0982946
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) −12.0000 −0.587643
\(418\) −2.00000 −0.0978232
\(419\) 12.0000 0.586238 0.293119 − 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 − 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) −10.0000 −0.486792
\(423\) 12.0000 0.583460
\(424\) 1.00000 0.0485643
\(425\) −2.00000 −0.0970143
\(426\) 12.0000 0.581402
\(427\) 0 0
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) −16.0000 −0.771589
\(431\) 24.0000 1.15604 0.578020 − 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −22.0000 −1.05725 −0.528626 − 0.848855i \(-0.677293\pi\)
−0.528626 + 0.848855i \(0.677293\pi\)
\(434\) 0 0
\(435\) −4.00000 −0.191785
\(436\) −2.00000 −0.0957826
\(437\) −2.00000 −0.0956730
\(438\) −14.0000 −0.668946
\(439\) 20.0000 0.954548 0.477274 − 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) −4.00000 −0.190693
\(441\) −7.00000 −0.333333
\(442\) −8.00000 −0.380521
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −4.00000 −0.189832
\(445\) −20.0000 −0.948091
\(446\) −4.00000 −0.189405
\(447\) 12.0000 0.567581
\(448\) 0 0
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 1.00000 0.0471405
\(451\) 4.00000 0.188353
\(452\) 6.00000 0.282216
\(453\) −10.0000 −0.469841
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) 1.00000 0.0468293
\(457\) 22.0000 1.02912 0.514558 − 0.857455i \(-0.327956\pi\)
0.514558 + 0.857455i \(0.327956\pi\)
\(458\) 18.0000 0.841085
\(459\) −2.00000 −0.0933520
\(460\) −4.00000 −0.186501
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 36.0000 1.67306 0.836531 − 0.547920i \(-0.184580\pi\)
0.836531 + 0.547920i \(0.184580\pi\)
\(464\) 2.00000 0.0928477
\(465\) −4.00000 −0.185496
\(466\) −8.00000 −0.370593
\(467\) 2.00000 0.0925490 0.0462745 − 0.998929i \(-0.485265\pi\)
0.0462745 + 0.998929i \(0.485265\pi\)
\(468\) 4.00000 0.184900
\(469\) 0 0
\(470\) −24.0000 −1.10704
\(471\) −2.00000 −0.0921551
\(472\) 0 0
\(473\) 16.0000 0.735681
\(474\) 10.0000 0.459315
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −1.00000 −0.0457869
\(478\) −6.00000 −0.274434
\(479\) −30.0000 −1.37073 −0.685367 − 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 2.00000 0.0912871
\(481\) 16.0000 0.729537
\(482\) 30.0000 1.36646
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) 4.00000 0.181631
\(486\) 1.00000 0.0453609
\(487\) 12.0000 0.543772 0.271886 − 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 2.00000 0.0905357
\(489\) 4.00000 0.180886
\(490\) 14.0000 0.632456
\(491\) −12.0000 −0.541552 −0.270776 − 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −2.00000 −0.0901670
\(493\) 4.00000 0.180151
\(494\) −4.00000 −0.179969
\(495\) 4.00000 0.179787
\(496\) 2.00000 0.0898027
\(497\) 0 0
\(498\) 0 0
\(499\) −28.0000 −1.25345 −0.626726 − 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) −12.0000 −0.536656
\(501\) −16.0000 −0.714827
\(502\) −8.00000 −0.357057
\(503\) 6.00000 0.267527 0.133763 − 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 20.0000 0.889988
\(506\) 4.00000 0.177822
\(507\) −3.00000 −0.133235
\(508\) −2.00000 −0.0887357
\(509\) 30.0000 1.32973 0.664863 − 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −6.00000 −0.264649
\(515\) 36.0000 1.58635
\(516\) −8.00000 −0.352180
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) −8.00000 −0.350823
\(521\) 42.0000 1.84005 0.920027 − 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) −2.00000 −0.0875376
\(523\) −26.0000 −1.13690 −0.568450 − 0.822718i \(-0.692457\pi\)
−0.568450 + 0.822718i \(0.692457\pi\)
\(524\) −22.0000 −0.961074
\(525\) 0 0
\(526\) −22.0000 −0.959246
\(527\) 4.00000 0.174243
\(528\) −2.00000 −0.0870388
\(529\) −19.0000 −0.826087
\(530\) 2.00000 0.0868744
\(531\) 0 0
\(532\) 0 0
\(533\) 8.00000 0.346518
\(534\) −10.0000 −0.432742
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) −12.0000 −0.517838
\(538\) −14.0000 −0.603583
\(539\) −14.0000 −0.603023
\(540\) −2.00000 −0.0860663
\(541\) 42.0000 1.80572 0.902861 − 0.429934i \(-0.141463\pi\)
0.902861 + 0.429934i \(0.141463\pi\)
\(542\) 24.0000 1.03089
\(543\) −22.0000 −0.944110
\(544\) −2.00000 −0.0857493
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −2.00000 −0.0855138 −0.0427569 − 0.999086i \(-0.513614\pi\)
−0.0427569 + 0.999086i \(0.513614\pi\)
\(548\) −16.0000 −0.683486
\(549\) −2.00000 −0.0853579
\(550\) 2.00000 0.0852803
\(551\) 2.00000 0.0852029
\(552\) −2.00000 −0.0851257
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −8.00000 −0.339581
\(556\) 12.0000 0.508913
\(557\) −10.0000 −0.423714 −0.211857 − 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) −2.00000 −0.0846668
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) −2.00000 −0.0843649
\(563\) 12.0000 0.505740 0.252870 − 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −12.0000 −0.505291
\(565\) 12.0000 0.504844
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −12.0000 −0.503509
\(569\) 30.0000 1.25767 0.628833 − 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 2.00000 0.0837708
\(571\) −36.0000 −1.50655 −0.753277 − 0.657704i \(-0.771528\pi\)
−0.753277 + 0.657704i \(0.771528\pi\)
\(572\) 8.00000 0.334497
\(573\) −10.0000 −0.417756
\(574\) 0 0
\(575\) 2.00000 0.0834058
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 − 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 13.0000 0.540729
\(579\) 24.0000 0.997406
\(580\) 4.00000 0.166091
\(581\) 0 0
\(582\) 2.00000 0.0829027
\(583\) −2.00000 −0.0828315
\(584\) 14.0000 0.579324
\(585\) 8.00000 0.330759
\(586\) 2.00000 0.0826192
\(587\) 42.0000 1.73353 0.866763 − 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 7.00000 0.288675
\(589\) 2.00000 0.0824086
\(590\) 0 0
\(591\) 8.00000 0.329076
\(592\) 4.00000 0.164399
\(593\) 30.0000 1.23195 0.615976 − 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −12.0000 −0.491539
\(597\) 24.0000 0.982255
\(598\) 8.00000 0.327144
\(599\) 32.0000 1.30748 0.653742 − 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −16.0000 −0.652654 −0.326327 − 0.945257i \(-0.605811\pi\)
−0.326327 + 0.945257i \(0.605811\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 10.0000 0.406894
\(605\) −14.0000 −0.569181
\(606\) 10.0000 0.406222
\(607\) −4.00000 −0.162355 −0.0811775 − 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) 48.0000 1.94187
\(612\) 2.00000 0.0808452
\(613\) 26.0000 1.05013 0.525065 − 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 22.0000 0.887848
\(615\) −4.00000 −0.161296
\(616\) 0 0
\(617\) 48.0000 1.93241 0.966204 − 0.257780i \(-0.0829910\pi\)
0.966204 + 0.257780i \(0.0829910\pi\)
\(618\) 18.0000 0.724066
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 4.00000 0.160644
\(621\) 2.00000 0.0802572
\(622\) −12.0000 −0.481156
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) −2.00000 −0.0798723
\(628\) 2.00000 0.0798087
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) −10.0000 −0.397779
\(633\) −10.0000 −0.397464
\(634\) −22.0000 −0.873732
\(635\) −4.00000 −0.158735
\(636\) 1.00000 0.0396526
\(637\) −28.0000 −1.10940
\(638\) −4.00000 −0.158362
\(639\) 12.0000 0.474713
\(640\) −2.00000 −0.0790569
\(641\) −22.0000 −0.868948 −0.434474 − 0.900684i \(-0.643066\pi\)
−0.434474 + 0.900684i \(0.643066\pi\)
\(642\) 0 0
\(643\) 40.0000 1.57745 0.788723 − 0.614749i \(-0.210743\pi\)
0.788723 + 0.614749i \(0.210743\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) −2.00000 −0.0786889
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 4.00000 0.156893
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −36.0000 −1.40879 −0.704394 − 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) −2.00000 −0.0782062
\(655\) −44.0000 −1.71922
\(656\) 2.00000 0.0780869
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) −4.00000 −0.155700
\(661\) 24.0000 0.933492 0.466746 − 0.884391i \(-0.345426\pi\)
0.466746 + 0.884391i \(0.345426\pi\)
\(662\) 30.0000 1.16598
\(663\) −8.00000 −0.310694
\(664\) 0 0
\(665\) 0 0
\(666\) −4.00000 −0.154997
\(667\) −4.00000 −0.154881
\(668\) 16.0000 0.619059
\(669\) −4.00000 −0.154649
\(670\) 16.0000 0.618134
\(671\) −4.00000 −0.154418
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 − 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) −20.0000 −0.770371
\(675\) 1.00000 0.0384900
\(676\) 3.00000 0.115385
\(677\) 26.0000 0.999261 0.499631 − 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 6.00000 0.230429
\(679\) 0 0
\(680\) −4.00000 −0.153393
\(681\) 20.0000 0.766402
\(682\) −4.00000 −0.153168
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 1.00000 0.0382360
\(685\) −32.0000 −1.22266
\(686\) 0 0
\(687\) 18.0000 0.686743
\(688\) 8.00000 0.304997
\(689\) −4.00000 −0.152388
\(690\) −4.00000 −0.152277
\(691\) −20.0000 −0.760836 −0.380418 − 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) 14.0000 0.532200
\(693\) 0 0
\(694\) −18.0000 −0.683271
\(695\) 24.0000 0.910372
\(696\) 2.00000 0.0758098
\(697\) 4.00000 0.151511
\(698\) −2.00000 −0.0757011
\(699\) −8.00000 −0.302588
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 − 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 4.00000 0.150970
\(703\) 4.00000 0.150863
\(704\) 2.00000 0.0753778
\(705\) −24.0000 −0.903892
\(706\) −28.0000 −1.05379
\(707\) 0 0
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 − 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −24.0000 −0.900704
\(711\) 10.0000 0.375029
\(712\) 10.0000 0.374766
\(713\) −4.00000 −0.149801
\(714\) 0 0
\(715\) 16.0000 0.598366
\(716\) 12.0000 0.448461
\(717\) −6.00000 −0.224074
\(718\) −22.0000 −0.821033
\(719\) −46.0000 −1.71551 −0.857755 − 0.514058i \(-0.828142\pi\)
−0.857755 + 0.514058i \(0.828142\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) 30.0000 1.11571
\(724\) 22.0000 0.817624
\(725\) −2.00000 −0.0742781
\(726\) −7.00000 −0.259794
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 28.0000 1.03633
\(731\) 16.0000 0.591781
\(732\) 2.00000 0.0739221
\(733\) −46.0000 −1.69905 −0.849524 − 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) 0 0
\(735\) 14.0000 0.516398
\(736\) 2.00000 0.0737210
\(737\) −16.0000 −0.589368
\(738\) −2.00000 −0.0736210
\(739\) 36.0000 1.32428 0.662141 − 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 8.00000 0.294086
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 24.0000 0.880475 0.440237 − 0.897881i \(-0.354894\pi\)
0.440237 + 0.897881i \(0.354894\pi\)
\(744\) 2.00000 0.0733236
\(745\) −24.0000 −0.879292
\(746\) 10.0000 0.366126
\(747\) 0 0
\(748\) 4.00000 0.146254
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 12.0000 0.437595
\(753\) −8.00000 −0.291536
\(754\) −8.00000 −0.291343
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 42.0000 1.52652 0.763258 − 0.646094i \(-0.223599\pi\)
0.763258 + 0.646094i \(0.223599\pi\)
\(758\) 16.0000 0.581146
\(759\) 4.00000 0.145191
\(760\) −2.00000 −0.0725476
\(761\) −8.00000 −0.290000 −0.145000 − 0.989432i \(-0.546318\pi\)
−0.145000 + 0.989432i \(0.546318\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 0 0
\(764\) 10.0000 0.361787
\(765\) 4.00000 0.144620
\(766\) 20.0000 0.722629
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 42.0000 1.51456 0.757279 − 0.653091i \(-0.226528\pi\)
0.757279 + 0.653091i \(0.226528\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) −24.0000 −0.863779
\(773\) 42.0000 1.51064 0.755318 − 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) −8.00000 −0.287554
\(775\) −2.00000 −0.0718421
\(776\) −2.00000 −0.0717958
\(777\) 0 0
\(778\) 10.0000 0.358517
\(779\) 2.00000 0.0716574
\(780\) −8.00000 −0.286446
\(781\) 24.0000 0.858788
\(782\) 4.00000 0.143040
\(783\) −2.00000 −0.0714742
\(784\) −7.00000 −0.250000
\(785\) 4.00000 0.142766
\(786\) −22.0000 −0.784714
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −8.00000 −0.284988
\(789\) −22.0000 −0.783221
\(790\) −20.0000 −0.711568
\(791\) 0 0
\(792\) −2.00000 −0.0710669
\(793\) −8.00000 −0.284088
\(794\) −18.0000 −0.638796
\(795\) 2.00000 0.0709327
\(796\) −24.0000 −0.850657
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) −10.0000 −0.353333
\(802\) −18.0000 −0.635602
\(803\) −28.0000 −0.988099
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) −14.0000 −0.492823
\(808\) −10.0000 −0.351799
\(809\) 12.0000 0.421898 0.210949 − 0.977497i \(-0.432345\pi\)
0.210949 + 0.977497i \(0.432345\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −2.00000 −0.0702295 −0.0351147 − 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) −8.00000 −0.280400
\(815\) −8.00000 −0.280228
\(816\) −2.00000 −0.0700140
\(817\) 8.00000 0.279885
\(818\) −30.0000 −1.04893
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) −30.0000 −1.04701 −0.523504 − 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) −16.0000 −0.558064
\(823\) −48.0000 −1.67317 −0.836587 − 0.547833i \(-0.815453\pi\)
−0.836587 + 0.547833i \(0.815453\pi\)
\(824\) −18.0000 −0.627060
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) −2.00000 −0.0695048
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 4.00000 0.138675
\(833\) −14.0000 −0.485071
\(834\) 12.0000 0.415526
\(835\) 32.0000 1.10741
\(836\) 2.00000 0.0691714
\(837\) −2.00000 −0.0691301
\(838\) −12.0000 −0.414533
\(839\) −32.0000 −1.10476 −0.552381 − 0.833592i \(-0.686281\pi\)
−0.552381 + 0.833592i \(0.686281\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 18.0000 0.620321
\(843\) −2.00000 −0.0688837
\(844\) 10.0000 0.344214
\(845\) 6.00000 0.206406
\(846\) −12.0000 −0.412568
\(847\) 0 0
\(848\) −1.00000 −0.0343401
\(849\) −4.00000 −0.137280
\(850\) 2.00000 0.0685994
\(851\) −8.00000 −0.274236
\(852\) −12.0000 −0.411113
\(853\) −46.0000 −1.57501 −0.787505 − 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) −38.0000 −1.29806 −0.649028 − 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 8.00000 0.273115
\(859\) −48.0000 −1.63774 −0.818869 − 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) 16.0000 0.545595
\(861\) 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
\(865\) 28.0000 0.952029
\(866\) 22.0000 0.747590
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 20.0000 0.678454
\(870\) 4.00000 0.135613
\(871\) −32.0000 −1.08428
\(872\) 2.00000 0.0677285
\(873\) 2.00000 0.0676897
\(874\) 2.00000 0.0676510
\(875\) 0 0
\(876\) 14.0000 0.473016
\(877\) 36.0000 1.21563 0.607817 − 0.794077i \(-0.292045\pi\)
0.607817 + 0.794077i \(0.292045\pi\)
\(878\) −20.0000 −0.674967
\(879\) 2.00000 0.0674583
\(880\) 4.00000 0.134840
\(881\) 20.0000 0.673817 0.336909 − 0.941537i \(-0.390619\pi\)
0.336909 + 0.941537i \(0.390619\pi\)
\(882\) 7.00000 0.235702
\(883\) −28.0000 −0.942275 −0.471138 − 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 8.00000 0.269069
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 4.00000 0.134231
\(889\) 0 0
\(890\) 20.0000 0.670402
\(891\) 2.00000 0.0670025
\(892\) 4.00000 0.133930
\(893\) 12.0000 0.401565
\(894\) −12.0000 −0.401340
\(895\) 24.0000 0.802232
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 30.0000 1.00111
\(899\) 4.00000 0.133407
\(900\) −1.00000 −0.0333333
\(901\) −2.00000 −0.0666297
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 44.0000 1.46261
\(906\) 10.0000 0.332228
\(907\) 2.00000 0.0664089 0.0332045 − 0.999449i \(-0.489429\pi\)
0.0332045 + 0.999449i \(0.489429\pi\)
\(908\) −20.0000 −0.663723
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 − 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) −1.00000 −0.0331133
\(913\) 0 0
\(914\) −22.0000 −0.727695
\(915\) 4.00000 0.132236
\(916\) −18.0000 −0.594737
\(917\) 0 0
\(918\) 2.00000 0.0660098
\(919\) 48.0000 1.58337 0.791687 − 0.610927i \(-0.209203\pi\)
0.791687 + 0.610927i \(0.209203\pi\)
\(920\) 4.00000 0.131876
\(921\) 22.0000 0.724925
\(922\) 12.0000 0.395199
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −36.0000 −1.18303
\(927\) 18.0000 0.591198
\(928\) −2.00000 −0.0656532
\(929\) −50.0000 −1.64045 −0.820223 − 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 4.00000 0.131165
\(931\) −7.00000 −0.229416
\(932\) 8.00000 0.262049
\(933\) −12.0000 −0.392862
\(934\) −2.00000 −0.0654420
\(935\) 8.00000 0.261628
\(936\) −4.00000 −0.130744
\(937\) −42.0000 −1.37208 −0.686040 − 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 24.0000 0.782794
\(941\) 30.0000 0.977972 0.488986 − 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 2.00000 0.0651635
\(943\) −4.00000 −0.130258
\(944\) 0 0
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 2.00000 0.0649913 0.0324956 − 0.999472i \(-0.489654\pi\)
0.0324956 + 0.999472i \(0.489654\pi\)
\(948\) −10.0000 −0.324785
\(949\) −56.0000 −1.81784
\(950\) 1.00000 0.0324443
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) 34.0000 1.10137 0.550684 − 0.834714i \(-0.314367\pi\)
0.550684 + 0.834714i \(0.314367\pi\)
\(954\) 1.00000 0.0323762
\(955\) 20.0000 0.647185
\(956\) 6.00000 0.194054
\(957\) −4.00000 −0.129302
\(958\) 30.0000 0.969256
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −27.0000 −0.870968
\(962\) −16.0000 −0.515861
\(963\) 0 0
\(964\) −30.0000 −0.966235
\(965\) −48.0000 −1.54517
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 − 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 7.00000 0.224989
\(969\) −2.00000 −0.0642493
\(970\) −4.00000 −0.128432
\(971\) −24.0000 −0.770197 −0.385098 − 0.922876i \(-0.625832\pi\)
−0.385098 + 0.922876i \(0.625832\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 4.00000 0.128103
\(976\) −2.00000 −0.0640184
\(977\) −38.0000 −1.21573 −0.607864 − 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(978\) −4.00000 −0.127906
\(979\) −20.0000 −0.639203
\(980\) −14.0000 −0.447214
\(981\) −2.00000 −0.0638551
\(982\) 12.0000 0.382935
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 2.00000 0.0637577
\(985\) −16.0000 −0.509802
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) 4.00000 0.127257
\(989\) −16.0000 −0.508770
\(990\) −4.00000 −0.127128
\(991\) −24.0000 −0.762385 −0.381193 − 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) −2.00000 −0.0635001
\(993\) 30.0000 0.952021
\(994\) 0 0
\(995\) −48.0000 −1.52170
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 28.0000 0.886325
\(999\) −4.00000 −0.126554
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6042.2.a.d.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.d.1.1 ✓ 1 1.1 even 1 trivial