Properties

Label 546.2.a.b.1.1
Level $546$
Weight $2$
Character 546.1
Self dual yes
Analytic conductor $4.360$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.35983195036\)
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 \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} +1.00000 q^{13} +1.00000 q^{14} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +4.00000 q^{22} -4.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -2.00000 q^{37} +4.00000 q^{38} +1.00000 q^{39} +2.00000 q^{40} +2.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +4.00000 q^{46} -12.0000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +1.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} +1.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} -2.00000 q^{60} -10.0000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -2.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} -4.00000 q^{69} -2.00000 q^{70} -8.00000 q^{71} -1.00000 q^{72} -6.00000 q^{73} +2.00000 q^{74} -1.00000 q^{75} -4.00000 q^{76} +4.00000 q^{77} -1.00000 q^{78} +8.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} +8.00000 q^{83} -1.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -1.00000 q^{91} -4.00000 q^{92} +12.0000 q^{94} +8.00000 q^{95} -1.00000 q^{96} +2.00000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 1.00000 0.267261
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −2.00000 −0.447214
\(21\) −1.00000 −0.218218
\(22\) 4.00000 0.852803
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 2.00000 0.365148
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 2.00000 0.342997
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 4.00000 0.648886
\(39\) 1.00000 0.160128
\(40\) 2.00000 0.316228
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 1.00000 0.154303
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.00000 −0.298142
\(46\) 4.00000 0.589768
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 1.00000 0.138675
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) −1.00000 −0.136083
\(55\) 8.00000 1.07872
\(56\) 1.00000 0.133631
\(57\) −4.00000 −0.529813
\(58\) 2.00000 0.262613
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −2.00000 −0.258199
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) −2.00000 −0.248069
\(66\) 4.00000 0.492366
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −2.00000 −0.242536
\(69\) −4.00000 −0.481543
\(70\) −2.00000 −0.239046
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) −1.00000 −0.117851
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) −4.00000 −0.458831
\(77\) 4.00000 0.455842
\(78\) −1.00000 −0.113228
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) −1.00000 −0.109109
\(85\) 4.00000 0.433861
\(86\) −4.00000 −0.431331
\(87\) −2.00000 −0.214423
\(88\) 4.00000 0.426401
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) −1.00000 −0.104828
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) 12.0000 1.23771
\(95\) 8.00000 0.820783
\(96\) −1.00000 −0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.00000 −0.402015
\(100\) −1.00000 −0.100000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 2.00000 0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 2.00000 0.195180
\(106\) −6.00000 −0.582772
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.00000 0.0962250
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) −8.00000 −0.762770
\(111\) −2.00000 −0.189832
\(112\) −1.00000 −0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 4.00000 0.374634
\(115\) 8.00000 0.746004
\(116\) −2.00000 −0.185695
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) 2.00000 0.183340
\(120\) 2.00000 0.182574
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 12.0000 1.07331
\(126\) 1.00000 0.0890871
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.00000 0.352180
\(130\) 2.00000 0.175412
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 4.00000 0.346844
\(134\) −4.00000 −0.345547
\(135\) −2.00000 −0.172133
\(136\) 2.00000 0.171499
\(137\) 22.0000 1.87959 0.939793 0.341743i \(-0.111017\pi\)
0.939793 + 0.341743i \(0.111017\pi\)
\(138\) 4.00000 0.340503
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 2.00000 0.169031
\(141\) −12.0000 −1.01058
\(142\) 8.00000 0.671345
\(143\) −4.00000 −0.334497
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 6.00000 0.496564
\(147\) 1.00000 0.0824786
\(148\) −2.00000 −0.164399
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 1.00000 0.0816497
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 4.00000 0.324443
\(153\) −2.00000 −0.161690
\(154\) −4.00000 −0.322329
\(155\) 0 0
\(156\) 1.00000 0.0800641
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) −8.00000 −0.636446
\(159\) 6.00000 0.475831
\(160\) 2.00000 0.158114
\(161\) 4.00000 0.315244
\(162\) −1.00000 −0.0785674
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 2.00000 0.156174
\(165\) 8.00000 0.622799
\(166\) −8.00000 −0.620920
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 1.00000 0.0771517
\(169\) 1.00000 0.0769231
\(170\) −4.00000 −0.306786
\(171\) −4.00000 −0.305888
\(172\) 4.00000 0.304997
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 2.00000 0.151620
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 6.00000 0.449719
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) −2.00000 −0.149071
\(181\) −26.0000 −1.93256 −0.966282 0.257485i \(-0.917106\pi\)
−0.966282 + 0.257485i \(0.917106\pi\)
\(182\) 1.00000 0.0741249
\(183\) −10.0000 −0.739221
\(184\) 4.00000 0.294884
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 8.00000 0.585018
\(188\) −12.0000 −0.875190
\(189\) −1.00000 −0.0727393
\(190\) −8.00000 −0.580381
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 1.00000 0.0721688
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) −2.00000 −0.143592
\(195\) −2.00000 −0.143223
\(196\) 1.00000 0.0714286
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 4.00000 0.284268
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 1.00000 0.0707107
\(201\) 4.00000 0.282138
\(202\) 6.00000 0.422159
\(203\) 2.00000 0.140372
\(204\) −2.00000 −0.140028
\(205\) −4.00000 −0.279372
\(206\) 8.00000 0.557386
\(207\) −4.00000 −0.278019
\(208\) 1.00000 0.0693375
\(209\) 16.0000 1.10674
\(210\) −2.00000 −0.138013
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 6.00000 0.412082
\(213\) −8.00000 −0.548151
\(214\) 0 0
\(215\) −8.00000 −0.545595
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) −6.00000 −0.405442
\(220\) 8.00000 0.539360
\(221\) −2.00000 −0.134535
\(222\) 2.00000 0.134231
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 1.00000 0.0668153
\(225\) −1.00000 −0.0666667
\(226\) −10.0000 −0.665190
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) −4.00000 −0.264906
\(229\) −26.0000 −1.71813 −0.859064 0.511868i \(-0.828954\pi\)
−0.859064 + 0.511868i \(0.828954\pi\)
\(230\) −8.00000 −0.527504
\(231\) 4.00000 0.263181
\(232\) 2.00000 0.131306
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 24.0000 1.56559
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) −2.00000 −0.129641
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) −2.00000 −0.129099
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) −2.00000 −0.127775
\(246\) −2.00000 −0.127515
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 8.00000 0.506979
\(250\) −12.0000 −0.758947
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 16.0000 1.00591
\(254\) 8.00000 0.501965
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −4.00000 −0.249029
\(259\) 2.00000 0.124274
\(260\) −2.00000 −0.124035
\(261\) −2.00000 −0.123797
\(262\) −12.0000 −0.741362
\(263\) −20.0000 −1.23325 −0.616626 0.787256i \(-0.711501\pi\)
−0.616626 + 0.787256i \(0.711501\pi\)
\(264\) 4.00000 0.246183
\(265\) −12.0000 −0.737154
\(266\) −4.00000 −0.245256
\(267\) −6.00000 −0.367194
\(268\) 4.00000 0.244339
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 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
\(273\) −1.00000 −0.0605228
\(274\) −22.0000 −1.32907
\(275\) 4.00000 0.241209
\(276\) −4.00000 −0.240772
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) −4.00000 −0.239904
\(279\) 0 0
\(280\) −2.00000 −0.119523
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 12.0000 0.714590
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) −8.00000 −0.474713
\(285\) 8.00000 0.473879
\(286\) 4.00000 0.236525
\(287\) −2.00000 −0.118056
\(288\) −1.00000 −0.0589256
\(289\) −13.0000 −0.764706
\(290\) −4.00000 −0.234888
\(291\) 2.00000 0.117242
\(292\) −6.00000 −0.351123
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) −4.00000 −0.232104
\(298\) −18.0000 −1.04271
\(299\) −4.00000 −0.231326
\(300\) −1.00000 −0.0577350
\(301\) −4.00000 −0.230556
\(302\) 16.0000 0.920697
\(303\) −6.00000 −0.344691
\(304\) −4.00000 −0.229416
\(305\) 20.0000 1.14520
\(306\) 2.00000 0.114332
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 4.00000 0.227921
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) −1.00000 −0.0566139
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) −22.0000 −1.24153
\(315\) 2.00000 0.112687
\(316\) 8.00000 0.450035
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) −6.00000 −0.336463
\(319\) 8.00000 0.447914
\(320\) −2.00000 −0.111803
\(321\) 0 0
\(322\) −4.00000 −0.222911
\(323\) 8.00000 0.445132
\(324\) 1.00000 0.0555556
\(325\) −1.00000 −0.0554700
\(326\) −12.0000 −0.664619
\(327\) 14.0000 0.774202
\(328\) −2.00000 −0.110432
\(329\) 12.0000 0.661581
\(330\) −8.00000 −0.440386
\(331\) −12.0000 −0.659580 −0.329790 0.944054i \(-0.606978\pi\)
−0.329790 + 0.944054i \(0.606978\pi\)
\(332\) 8.00000 0.439057
\(333\) −2.00000 −0.109599
\(334\) −12.0000 −0.656611
\(335\) −8.00000 −0.437087
\(336\) −1.00000 −0.0545545
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 10.0000 0.543125
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 4.00000 0.216295
\(343\) −1.00000 −0.0539949
\(344\) −4.00000 −0.215666
\(345\) 8.00000 0.430706
\(346\) −2.00000 −0.107521
\(347\) 16.0000 0.858925 0.429463 0.903085i \(-0.358703\pi\)
0.429463 + 0.903085i \(0.358703\pi\)
\(348\) −2.00000 −0.107211
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 1.00000 0.0533761
\(352\) 4.00000 0.213201
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) −6.00000 −0.317999
\(357\) 2.00000 0.105851
\(358\) 24.0000 1.26844
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 2.00000 0.105409
\(361\) −3.00000 −0.157895
\(362\) 26.0000 1.36653
\(363\) 5.00000 0.262432
\(364\) −1.00000 −0.0524142
\(365\) 12.0000 0.628109
\(366\) 10.0000 0.522708
\(367\) −24.0000 −1.25279 −0.626395 0.779506i \(-0.715470\pi\)
−0.626395 + 0.779506i \(0.715470\pi\)
\(368\) −4.00000 −0.208514
\(369\) 2.00000 0.104116
\(370\) −4.00000 −0.207950
\(371\) −6.00000 −0.311504
\(372\) 0 0
\(373\) −26.0000 −1.34623 −0.673114 0.739538i \(-0.735044\pi\)
−0.673114 + 0.739538i \(0.735044\pi\)
\(374\) −8.00000 −0.413670
\(375\) 12.0000 0.619677
\(376\) 12.0000 0.618853
\(377\) −2.00000 −0.103005
\(378\) 1.00000 0.0514344
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 8.00000 0.410391
\(381\) −8.00000 −0.409852
\(382\) 12.0000 0.613973
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −8.00000 −0.407718
\(386\) −2.00000 −0.101797
\(387\) 4.00000 0.203331
\(388\) 2.00000 0.101535
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 2.00000 0.101274
\(391\) 8.00000 0.404577
\(392\) −1.00000 −0.0505076
\(393\) 12.0000 0.605320
\(394\) 6.00000 0.302276
\(395\) −16.0000 −0.805047
\(396\) −4.00000 −0.201008
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 8.00000 0.401004
\(399\) 4.00000 0.200250
\(400\) −1.00000 −0.0500000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) −4.00000 −0.199502
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) −2.00000 −0.0993808
\(406\) −2.00000 −0.0992583
\(407\) 8.00000 0.396545
\(408\) 2.00000 0.0990148
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 4.00000 0.197546
\(411\) 22.0000 1.08518
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 4.00000 0.196589
\(415\) −16.0000 −0.785409
\(416\) −1.00000 −0.0490290
\(417\) 4.00000 0.195881
\(418\) −16.0000 −0.782586
\(419\) 20.0000 0.977064 0.488532 0.872546i \(-0.337533\pi\)
0.488532 + 0.872546i \(0.337533\pi\)
\(420\) 2.00000 0.0975900
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −4.00000 −0.194717
\(423\) −12.0000 −0.583460
\(424\) −6.00000 −0.291386
\(425\) 2.00000 0.0970143
\(426\) 8.00000 0.387601
\(427\) 10.0000 0.483934
\(428\) 0 0
\(429\) −4.00000 −0.193122
\(430\) 8.00000 0.385794
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) 1.00000 0.0481125
\(433\) −30.0000 −1.44171 −0.720854 0.693087i \(-0.756250\pi\)
−0.720854 + 0.693087i \(0.756250\pi\)
\(434\) 0 0
\(435\) 4.00000 0.191785
\(436\) 14.0000 0.670478
\(437\) 16.0000 0.765384
\(438\) 6.00000 0.286691
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) −8.00000 −0.381385
\(441\) 1.00000 0.0476190
\(442\) 2.00000 0.0951303
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 12.0000 0.568855
\(446\) 24.0000 1.13643
\(447\) 18.0000 0.851371
\(448\) −1.00000 −0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 1.00000 0.0471405
\(451\) −8.00000 −0.376705
\(452\) 10.0000 0.470360
\(453\) −16.0000 −0.751746
\(454\) 8.00000 0.375459
\(455\) 2.00000 0.0937614
\(456\) 4.00000 0.187317
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 26.0000 1.21490
\(459\) −2.00000 −0.0933520
\(460\) 8.00000 0.373002
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) −4.00000 −0.186097
\(463\) 24.0000 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) 4.00000 0.185098 0.0925490 0.995708i \(-0.470499\pi\)
0.0925490 + 0.995708i \(0.470499\pi\)
\(468\) 1.00000 0.0462250
\(469\) −4.00000 −0.184703
\(470\) −24.0000 −1.10704
\(471\) 22.0000 1.01371
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) −8.00000 −0.367452
\(475\) 4.00000 0.183533
\(476\) 2.00000 0.0916698
\(477\) 6.00000 0.274721
\(478\) 24.0000 1.09773
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 2.00000 0.0912871
\(481\) −2.00000 −0.0911922
\(482\) −2.00000 −0.0910975
\(483\) 4.00000 0.182006
\(484\) 5.00000 0.227273
\(485\) −4.00000 −0.181631
\(486\) −1.00000 −0.0453609
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 10.0000 0.452679
\(489\) 12.0000 0.542659
\(490\) 2.00000 0.0903508
\(491\) −8.00000 −0.361035 −0.180517 0.983572i \(-0.557777\pi\)
−0.180517 + 0.983572i \(0.557777\pi\)
\(492\) 2.00000 0.0901670
\(493\) 4.00000 0.180151
\(494\) 4.00000 0.179969
\(495\) 8.00000 0.359573
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) −8.00000 −0.358489
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 12.0000 0.536656
\(501\) 12.0000 0.536120
\(502\) 4.00000 0.178529
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 1.00000 0.0445435
\(505\) 12.0000 0.533993
\(506\) −16.0000 −0.711287
\(507\) 1.00000 0.0444116
\(508\) −8.00000 −0.354943
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) −4.00000 −0.177123
\(511\) 6.00000 0.265424
\(512\) −1.00000 −0.0441942
\(513\) −4.00000 −0.176604
\(514\) 18.0000 0.793946
\(515\) 16.0000 0.705044
\(516\) 4.00000 0.176090
\(517\) 48.0000 2.11104
\(518\) −2.00000 −0.0878750
\(519\) 2.00000 0.0877903
\(520\) 2.00000 0.0877058
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 2.00000 0.0875376
\(523\) 36.0000 1.57417 0.787085 0.616844i \(-0.211589\pi\)
0.787085 + 0.616844i \(0.211589\pi\)
\(524\) 12.0000 0.524222
\(525\) 1.00000 0.0436436
\(526\) 20.0000 0.872041
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 2.00000 0.0866296
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −4.00000 −0.172774
\(537\) −24.0000 −1.03568
\(538\) 6.00000 0.258678
\(539\) −4.00000 −0.172292
\(540\) −2.00000 −0.0860663
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 24.0000 1.03089
\(543\) −26.0000 −1.11577
\(544\) 2.00000 0.0857493
\(545\) −28.0000 −1.19939
\(546\) 1.00000 0.0427960
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) 22.0000 0.939793
\(549\) −10.0000 −0.426790
\(550\) −4.00000 −0.170561
\(551\) 8.00000 0.340811
\(552\) 4.00000 0.170251
\(553\) −8.00000 −0.340195
\(554\) −22.0000 −0.934690
\(555\) 4.00000 0.169791
\(556\) 4.00000 0.169638
\(557\) −38.0000 −1.61011 −0.805056 0.593199i \(-0.797865\pi\)
−0.805056 + 0.593199i \(0.797865\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 2.00000 0.0845154
\(561\) 8.00000 0.337760
\(562\) −22.0000 −0.928014
\(563\) −4.00000 −0.168580 −0.0842900 0.996441i \(-0.526862\pi\)
−0.0842900 + 0.996441i \(0.526862\pi\)
\(564\) −12.0000 −0.505291
\(565\) −20.0000 −0.841406
\(566\) 28.0000 1.17693
\(567\) −1.00000 −0.0419961
\(568\) 8.00000 0.335673
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) −8.00000 −0.335083
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) −4.00000 −0.167248
\(573\) −12.0000 −0.501307
\(574\) 2.00000 0.0834784
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) 18.0000 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(578\) 13.0000 0.540729
\(579\) 2.00000 0.0831172
\(580\) 4.00000 0.166091
\(581\) −8.00000 −0.331896
\(582\) −2.00000 −0.0829027
\(583\) −24.0000 −0.993978
\(584\) 6.00000 0.248282
\(585\) −2.00000 −0.0826898
\(586\) −14.0000 −0.578335
\(587\) −24.0000 −0.990586 −0.495293 0.868726i \(-0.664939\pi\)
−0.495293 + 0.868726i \(0.664939\pi\)
\(588\) 1.00000 0.0412393
\(589\) 0 0
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) −2.00000 −0.0821995
\(593\) −22.0000 −0.903432 −0.451716 0.892162i \(-0.649188\pi\)
−0.451716 + 0.892162i \(0.649188\pi\)
\(594\) 4.00000 0.164122
\(595\) −4.00000 −0.163984
\(596\) 18.0000 0.737309
\(597\) −8.00000 −0.327418
\(598\) 4.00000 0.163572
\(599\) −12.0000 −0.490307 −0.245153 0.969484i \(-0.578838\pi\)
−0.245153 + 0.969484i \(0.578838\pi\)
\(600\) 1.00000 0.0408248
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 4.00000 0.163028
\(603\) 4.00000 0.162893
\(604\) −16.0000 −0.651031
\(605\) −10.0000 −0.406558
\(606\) 6.00000 0.243733
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 4.00000 0.162221
\(609\) 2.00000 0.0810441
\(610\) −20.0000 −0.809776
\(611\) −12.0000 −0.485468
\(612\) −2.00000 −0.0808452
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 12.0000 0.484281
\(615\) −4.00000 −0.161296
\(616\) −4.00000 −0.161165
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) 8.00000 0.321807
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) −24.0000 −0.962312
\(623\) 6.00000 0.240385
\(624\) 1.00000 0.0400320
\(625\) −19.0000 −0.760000
\(626\) 6.00000 0.239808
\(627\) 16.0000 0.638978
\(628\) 22.0000 0.877896
\(629\) 4.00000 0.159490
\(630\) −2.00000 −0.0796819
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) −8.00000 −0.318223
\(633\) 4.00000 0.158986
\(634\) −2.00000 −0.0794301
\(635\) 16.0000 0.634941
\(636\) 6.00000 0.237915
\(637\) 1.00000 0.0396214
\(638\) −8.00000 −0.316723
\(639\) −8.00000 −0.316475
\(640\) 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
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 4.00000 0.157622
\(645\) −8.00000 −0.315000
\(646\) −8.00000 −0.314756
\(647\) −48.0000 −1.88707 −0.943537 0.331266i \(-0.892524\pi\)
−0.943537 + 0.331266i \(0.892524\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) 1.00000 0.0392232
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) 30.0000 1.17399 0.586995 0.809590i \(-0.300311\pi\)
0.586995 + 0.809590i \(0.300311\pi\)
\(654\) −14.0000 −0.547443
\(655\) −24.0000 −0.937758
\(656\) 2.00000 0.0780869
\(657\) −6.00000 −0.234082
\(658\) −12.0000 −0.467809
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 8.00000 0.311400
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 12.0000 0.466393
\(663\) −2.00000 −0.0776736
\(664\) −8.00000 −0.310460
\(665\) −8.00000 −0.310227
\(666\) 2.00000 0.0774984
\(667\) 8.00000 0.309761
\(668\) 12.0000 0.464294
\(669\) −24.0000 −0.927894
\(670\) 8.00000 0.309067
\(671\) 40.0000 1.54418
\(672\) 1.00000 0.0385758
\(673\) 18.0000 0.693849 0.346925 0.937893i \(-0.387226\pi\)
0.346925 + 0.937893i \(0.387226\pi\)
\(674\) −18.0000 −0.693334
\(675\) −1.00000 −0.0384900
\(676\) 1.00000 0.0384615
\(677\) 34.0000 1.30673 0.653363 0.757045i \(-0.273358\pi\)
0.653363 + 0.757045i \(0.273358\pi\)
\(678\) −10.0000 −0.384048
\(679\) −2.00000 −0.0767530
\(680\) −4.00000 −0.153393
\(681\) −8.00000 −0.306561
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) −4.00000 −0.152944
\(685\) −44.0000 −1.68115
\(686\) 1.00000 0.0381802
\(687\) −26.0000 −0.991962
\(688\) 4.00000 0.152499
\(689\) 6.00000 0.228582
\(690\) −8.00000 −0.304555
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 2.00000 0.0760286
\(693\) 4.00000 0.151947
\(694\) −16.0000 −0.607352
\(695\) −8.00000 −0.303457
\(696\) 2.00000 0.0758098
\(697\) −4.00000 −0.151511
\(698\) 34.0000 1.28692
\(699\) 26.0000 0.983410
\(700\) 1.00000 0.0377964
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 8.00000 0.301726
\(704\) −4.00000 −0.150756
\(705\) 24.0000 0.903892
\(706\) 14.0000 0.526897
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) −16.0000 −0.600469
\(711\) 8.00000 0.300023
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) −2.00000 −0.0748481
\(715\) 8.00000 0.299183
\(716\) −24.0000 −0.896922
\(717\) −24.0000 −0.896296
\(718\) −24.0000 −0.895672
\(719\) 8.00000 0.298350 0.149175 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 8.00000 0.297936
\(722\) 3.00000 0.111648
\(723\) 2.00000 0.0743808
\(724\) −26.0000 −0.966282
\(725\) 2.00000 0.0742781
\(726\) −5.00000 −0.185567
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 1.00000 0.0370625
\(729\) 1.00000 0.0370370
\(730\) −12.0000 −0.444140
\(731\) −8.00000 −0.295891
\(732\) −10.0000 −0.369611
\(733\) 30.0000 1.10808 0.554038 0.832492i \(-0.313086\pi\)
0.554038 + 0.832492i \(0.313086\pi\)
\(734\) 24.0000 0.885856
\(735\) −2.00000 −0.0737711
\(736\) 4.00000 0.147442
\(737\) −16.0000 −0.589368
\(738\) −2.00000 −0.0736210
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 4.00000 0.147043
\(741\) −4.00000 −0.146944
\(742\) 6.00000 0.220267
\(743\) 40.0000 1.46746 0.733729 0.679442i \(-0.237778\pi\)
0.733729 + 0.679442i \(0.237778\pi\)
\(744\) 0 0
\(745\) −36.0000 −1.31894
\(746\) 26.0000 0.951928
\(747\) 8.00000 0.292705
\(748\) 8.00000 0.292509
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −12.0000 −0.437595
\(753\) −4.00000 −0.145768
\(754\) 2.00000 0.0728357
\(755\) 32.0000 1.16460
\(756\) −1.00000 −0.0363696
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) −28.0000 −1.01701
\(759\) 16.0000 0.580763
\(760\) −8.00000 −0.290191
\(761\) −14.0000 −0.507500 −0.253750 0.967270i \(-0.581664\pi\)
−0.253750 + 0.967270i \(0.581664\pi\)
\(762\) 8.00000 0.289809
\(763\) −14.0000 −0.506834
\(764\) −12.0000 −0.434145
\(765\) 4.00000 0.144620
\(766\) 12.0000 0.433578
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 34.0000 1.22607 0.613036 0.790055i \(-0.289948\pi\)
0.613036 + 0.790055i \(0.289948\pi\)
\(770\) 8.00000 0.288300
\(771\) −18.0000 −0.648254
\(772\) 2.00000 0.0719816
\(773\) −34.0000 −1.22290 −0.611448 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) 2.00000 0.0717496
\(778\) −14.0000 −0.501924
\(779\) −8.00000 −0.286630
\(780\) −2.00000 −0.0716115
\(781\) 32.0000 1.14505
\(782\) −8.00000 −0.286079
\(783\) −2.00000 −0.0714742
\(784\) 1.00000 0.0357143
\(785\) −44.0000 −1.57043
\(786\) −12.0000 −0.428026
\(787\) 44.0000 1.56843 0.784215 0.620489i \(-0.213066\pi\)
0.784215 + 0.620489i \(0.213066\pi\)
\(788\) −6.00000 −0.213741
\(789\) −20.0000 −0.712019
\(790\) 16.0000 0.569254
\(791\) −10.0000 −0.355559
\(792\) 4.00000 0.142134
\(793\) −10.0000 −0.355110
\(794\) −14.0000 −0.496841
\(795\) −12.0000 −0.425596
\(796\) −8.00000 −0.283552
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) −4.00000 −0.141598
\(799\) 24.0000 0.849059
\(800\) 1.00000 0.0353553
\(801\) −6.00000 −0.212000
\(802\) 10.0000 0.353112
\(803\) 24.0000 0.846942
\(804\) 4.00000 0.141069
\(805\) −8.00000 −0.281963
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) 6.00000 0.211079
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 2.00000 0.0702728
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 2.00000 0.0701862
\(813\) −24.0000 −0.841717
\(814\) −8.00000 −0.280400
\(815\) −24.0000 −0.840683
\(816\) −2.00000 −0.0700140
\(817\) −16.0000 −0.559769
\(818\) −26.0000 −0.909069
\(819\) −1.00000 −0.0349428
\(820\) −4.00000 −0.139686
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) −22.0000 −0.767338
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) 8.00000 0.278693
\(825\) 4.00000 0.139262
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) −4.00000 −0.139010
\(829\) 30.0000 1.04194 0.520972 0.853574i \(-0.325570\pi\)
0.520972 + 0.853574i \(0.325570\pi\)
\(830\) 16.0000 0.555368
\(831\) 22.0000 0.763172
\(832\) 1.00000 0.0346688
\(833\) −2.00000 −0.0692959
\(834\) −4.00000 −0.138509
\(835\) −24.0000 −0.830554
\(836\) 16.0000 0.553372
\(837\) 0 0
\(838\) −20.0000 −0.690889
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) −2.00000 −0.0690066
\(841\) −25.0000 −0.862069
\(842\) 26.0000 0.896019
\(843\) 22.0000 0.757720
\(844\) 4.00000 0.137686
\(845\) −2.00000 −0.0688021
\(846\) 12.0000 0.412568
\(847\) −5.00000 −0.171802
\(848\) 6.00000 0.206041
\(849\) −28.0000 −0.960958
\(850\) −2.00000 −0.0685994
\(851\) 8.00000 0.274236
\(852\) −8.00000 −0.274075
\(853\) −26.0000 −0.890223 −0.445112 0.895475i \(-0.646836\pi\)
−0.445112 + 0.895475i \(0.646836\pi\)
\(854\) −10.0000 −0.342193
\(855\) 8.00000 0.273594
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 4.00000 0.136558
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) −8.00000 −0.272798
\(861\) −2.00000 −0.0681598
\(862\) 40.0000 1.36241
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −4.00000 −0.136004
\(866\) 30.0000 1.01944
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) −4.00000 −0.135613
\(871\) 4.00000 0.135535
\(872\) −14.0000 −0.474100
\(873\) 2.00000 0.0676897
\(874\) −16.0000 −0.541208
\(875\) −12.0000 −0.405674
\(876\) −6.00000 −0.202721
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 32.0000 1.07995
\(879\) 14.0000 0.472208
\(880\) 8.00000 0.269680
\(881\) 22.0000 0.741199 0.370599 0.928793i \(-0.379152\pi\)
0.370599 + 0.928793i \(0.379152\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 2.00000 0.0671156
\(889\) 8.00000 0.268311
\(890\) −12.0000 −0.402241
\(891\) −4.00000 −0.134005
\(892\) −24.0000 −0.803579
\(893\) 48.0000 1.60626
\(894\) −18.0000 −0.602010
\(895\) 48.0000 1.60446
\(896\) 1.00000 0.0334077
\(897\) −4.00000 −0.133556
\(898\) 18.0000 0.600668
\(899\) 0 0
\(900\) −1.00000 −0.0333333
\(901\) −12.0000 −0.399778
\(902\) 8.00000 0.266371
\(903\) −4.00000 −0.133112
\(904\) −10.0000 −0.332595
\(905\) 52.0000 1.72854
\(906\) 16.0000 0.531564
\(907\) −36.0000 −1.19536 −0.597680 0.801735i \(-0.703911\pi\)
−0.597680 + 0.801735i \(0.703911\pi\)
\(908\) −8.00000 −0.265489
\(909\) −6.00000 −0.199007
\(910\) −2.00000 −0.0662994
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) −4.00000 −0.132453
\(913\) −32.0000 −1.05905
\(914\) −18.0000 −0.595387
\(915\) 20.0000 0.661180
\(916\) −26.0000 −0.859064
\(917\) −12.0000 −0.396275
\(918\) 2.00000 0.0660098
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −8.00000 −0.263752
\(921\) −12.0000 −0.395413
\(922\) 2.00000 0.0658665
\(923\) −8.00000 −0.263323
\(924\) 4.00000 0.131590
\(925\) 2.00000 0.0657596
\(926\) −24.0000 −0.788689
\(927\) −8.00000 −0.262754
\(928\) 2.00000 0.0656532
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 26.0000 0.851658
\(933\) 24.0000 0.785725
\(934\) −4.00000 −0.130884
\(935\) −16.0000 −0.523256
\(936\) −1.00000 −0.0326860
\(937\) 42.0000 1.37208 0.686040 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 4.00000 0.130605
\(939\) −6.00000 −0.195803
\(940\) 24.0000 0.782794
\(941\) −34.0000 −1.10837 −0.554184 0.832394i \(-0.686970\pi\)
−0.554184 + 0.832394i \(0.686970\pi\)
\(942\) −22.0000 −0.716799
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 16.0000 0.520205
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 8.00000 0.259828
\(949\) −6.00000 −0.194768
\(950\) −4.00000 −0.129777
\(951\) 2.00000 0.0648544
\(952\) −2.00000 −0.0648204
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) −6.00000 −0.194257
\(955\) 24.0000 0.776622
\(956\) −24.0000 −0.776215
\(957\) 8.00000 0.258603
\(958\) −12.0000 −0.387702
\(959\) −22.0000 −0.710417
\(960\) −2.00000 −0.0645497
\(961\) −31.0000 −1.00000
\(962\) 2.00000 0.0644826
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) −4.00000 −0.128765
\(966\) −4.00000 −0.128698
\(967\) −24.0000 −0.771788 −0.385894 0.922543i \(-0.626107\pi\)
−0.385894 + 0.922543i \(0.626107\pi\)
\(968\) −5.00000 −0.160706
\(969\) 8.00000 0.256997
\(970\) 4.00000 0.128432
\(971\) 12.0000 0.385098 0.192549 0.981287i \(-0.438325\pi\)
0.192549 + 0.981287i \(0.438325\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) −32.0000 −1.02535
\(975\) −1.00000 −0.0320256
\(976\) −10.0000 −0.320092
\(977\) −42.0000 −1.34370 −0.671850 0.740688i \(-0.734500\pi\)
−0.671850 + 0.740688i \(0.734500\pi\)
\(978\) −12.0000 −0.383718
\(979\) 24.0000 0.767043
\(980\) −2.00000 −0.0638877
\(981\) 14.0000 0.446986
\(982\) 8.00000 0.255290
\(983\) −4.00000 −0.127580 −0.0637901 0.997963i \(-0.520319\pi\)
−0.0637901 + 0.997963i \(0.520319\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 12.0000 0.382352
\(986\) −4.00000 −0.127386
\(987\) 12.0000 0.381964
\(988\) −4.00000 −0.127257
\(989\) −16.0000 −0.508770
\(990\) −8.00000 −0.254257
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) 0 0
\(993\) −12.0000 −0.380808
\(994\) −8.00000 −0.253745
\(995\) 16.0000 0.507234
\(996\) 8.00000 0.253490
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) 20.0000 0.633089
\(999\) −2.00000 −0.0632772
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 546.2.a.b.1.1 1
3.2 odd 2 1638.2.a.s.1.1 1
4.3 odd 2 4368.2.a.d.1.1 1
7.6 odd 2 3822.2.a.h.1.1 1
13.12 even 2 7098.2.a.be.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.b.1.1 1 1.1 even 1 trivial
1638.2.a.s.1.1 1 3.2 odd 2
3822.2.a.h.1.1 1 7.6 odd 2
4368.2.a.d.1.1 1 4.3 odd 2
7098.2.a.be.1.1 1 13.12 even 2