Properties

Label 609.2.a.a.1.1
Level $609$
Weight $2$
Character 609.1
Self dual yes
Analytic conductor $4.863$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 609 = 3 \cdot 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 609.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.86288948310\)
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 \(\chi\) \(=\) 609.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} -1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} -1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{20} -1.00000 q^{21} -4.00000 q^{22} -3.00000 q^{24} -1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} +1.00000 q^{29} -2.00000 q^{30} -8.00000 q^{31} -5.00000 q^{32} -4.00000 q^{33} -2.00000 q^{34} -2.00000 q^{35} -1.00000 q^{36} -10.0000 q^{37} +4.00000 q^{38} +2.00000 q^{39} -6.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} +12.0000 q^{43} -4.00000 q^{44} -2.00000 q^{45} -8.00000 q^{47} +1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} -2.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -8.00000 q^{55} +3.00000 q^{56} +4.00000 q^{57} -1.00000 q^{58} +12.0000 q^{59} -2.00000 q^{60} -10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{63} +7.00000 q^{64} +4.00000 q^{65} +4.00000 q^{66} -12.0000 q^{67} -2.00000 q^{68} +2.00000 q^{70} -16.0000 q^{71} +3.00000 q^{72} +2.00000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +4.00000 q^{77} -2.00000 q^{78} +2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} -4.00000 q^{85} -12.0000 q^{86} -1.00000 q^{87} +12.0000 q^{88} -6.00000 q^{89} +2.00000 q^{90} -2.00000 q^{91} +8.00000 q^{93} +8.00000 q^{94} +8.00000 q^{95} +5.00000 q^{96} -6.00000 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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 −0.353553 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) −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
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 2.00000 0.632456
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.00000 −0.267261
\(15\) 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
\(19\) −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
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −3.00000 −0.612372
\(25\) −1.00000 −0.200000
\(26\) 2.00000 0.392232
\(27\) −1.00000 −0.192450
\(28\) −1.00000 −0.188982
\(29\) 1.00000 0.185695
\(30\) −2.00000 −0.365148
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −5.00000 −0.883883
\(33\) −4.00000 −0.696311
\(34\) −2.00000 −0.342997
\(35\) −2.00000 −0.338062
\(36\) −1.00000 −0.166667
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000 0.320256
\(40\) −6.00000 −0.948683
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 1.00000 0.154303
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) −4.00000 −0.603023
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 1.00000 0.141421
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) 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
\(56\) 3.00000 0.400892
\(57\) 4.00000 0.529813
\(58\) −1.00000 −0.131306
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) −2.00000 −0.258199
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000 1.01600
\(63\) 1.00000 0.125988
\(64\) 7.00000 0.875000
\(65\) 4.00000 0.496139
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0 0
\(70\) 2.00000 0.239046
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 3.00000 0.353553
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 10.0000 1.16248
\(75\) 1.00000 0.115470
\(76\) 4.00000 0.458831
\(77\) 4.00000 0.455842
\(78\) −2.00000 −0.226455
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 1.00000 0.109109
\(85\) −4.00000 −0.433861
\(86\) −12.0000 −1.29399
\(87\) −1.00000 −0.107211
\(88\) 12.0000 1.27920
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000 0.210819
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 8.00000 0.829561
\(94\) 8.00000 0.825137
\(95\) 8.00000 0.820783
\(96\) 5.00000 0.510310
\(97\) −6.00000 −0.609208 −0.304604 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) −1.00000 −0.101015
\(99\) 4.00000 0.402015
\(100\) 1.00000 0.100000
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 2.00000 0.198030
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −6.00000 −0.588348
\(105\) 2.00000 0.195180
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 8.00000 0.762770
\(111\) 10.0000 0.949158
\(112\) −1.00000 −0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) −2.00000 −0.184900
\(118\) −12.0000 −1.10469
\(119\) 2.00000 0.183340
\(120\) 6.00000 0.547723
\(121\) 5.00000 0.454545
\(122\) 10.0000 0.905357
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 12.0000 1.07331
\(126\) −1.00000 −0.0890871
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 3.00000 0.265165
\(129\) −12.0000 −1.05654
\(130\) −4.00000 −0.350823
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 4.00000 0.348155
\(133\) −4.00000 −0.346844
\(134\) 12.0000 1.03664
\(135\) 2.00000 0.172133
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.00000 0.169031
\(141\) 8.00000 0.673722
\(142\) 16.0000 1.34269
\(143\) −8.00000 −0.668994
\(144\) −1.00000 −0.0833333
\(145\) −2.00000 −0.166091
\(146\) −2.00000 −0.165521
\(147\) −1.00000 −0.0824786
\(148\) 10.0000 0.821995
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) −1.00000 −0.0816497
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −12.0000 −0.973329
\(153\) 2.00000 0.161690
\(154\) −4.00000 −0.322329
\(155\) 16.0000 1.28515
\(156\) −2.00000 −0.160128
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 10.0000 0.790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 6.00000 0.468521
\(165\) 8.00000 0.622799
\(166\) −4.00000 −0.310460
\(167\) −24.0000 −1.85718 −0.928588 0.371113i \(-0.878976\pi\)
−0.928588 + 0.371113i \(0.878976\pi\)
\(168\) −3.00000 −0.231455
\(169\) −9.00000 −0.692308
\(170\) 4.00000 0.306786
\(171\) −4.00000 −0.305888
\(172\) −12.0000 −0.914991
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 1.00000 0.0758098
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) −12.0000 −0.901975
\(178\) 6.00000 0.449719
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 2.00000 0.149071
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 2.00000 0.148250
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 20.0000 1.47043
\(186\) −8.00000 −0.586588
\(187\) 8.00000 0.585018
\(188\) 8.00000 0.583460
\(189\) −1.00000 −0.0727393
\(190\) −8.00000 −0.580381
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) −7.00000 −0.505181
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 6.00000 0.430775
\(195\) −4.00000 −0.286446
\(196\) −1.00000 −0.0714286
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) −4.00000 −0.284268
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) −3.00000 −0.212132
\(201\) 12.0000 0.846415
\(202\) −6.00000 −0.422159
\(203\) 1.00000 0.0701862
\(204\) 2.00000 0.140028
\(205\) 12.0000 0.838116
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −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
\(213\) 16.0000 1.09630
\(214\) 12.0000 0.820303
\(215\) −24.0000 −1.63679
\(216\) −3.00000 −0.204124
\(217\) −8.00000 −0.543075
\(218\) 18.0000 1.21911
\(219\) −2.00000 −0.135147
\(220\) 8.00000 0.539360
\(221\) −4.00000 −0.269069
\(222\) −10.0000 −0.671156
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) −5.00000 −0.334077
\(225\) −1.00000 −0.0666667
\(226\) −2.00000 −0.133038
\(227\) −12.0000 −0.796468 −0.398234 0.917284i \(-0.630377\pi\)
−0.398234 + 0.917284i \(0.630377\pi\)
\(228\) −4.00000 −0.264906
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 3.00000 0.196960
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 2.00000 0.130744
\(235\) 16.0000 1.04372
\(236\) −12.0000 −0.781133
\(237\) 0 0
\(238\) −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
\(241\) 18.0000 1.15948 0.579741 0.814801i \(-0.303154\pi\)
0.579741 + 0.814801i \(0.303154\pi\)
\(242\) −5.00000 −0.321412
\(243\) −1.00000 −0.0641500
\(244\) 10.0000 0.640184
\(245\) −2.00000 −0.127775
\(246\) −6.00000 −0.382546
\(247\) 8.00000 0.509028
\(248\) −24.0000 −1.52400
\(249\) −4.00000 −0.253490
\(250\) −12.0000 −0.758947
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) −1.00000 −0.0629941
\(253\) 0 0
\(254\) −16.0000 −1.00393
\(255\) 4.00000 0.250490
\(256\) −17.0000 −1.06250
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 12.0000 0.747087
\(259\) −10.0000 −0.621370
\(260\) −4.00000 −0.248069
\(261\) 1.00000 0.0618984
\(262\) 4.00000 0.247121
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) −12.0000 −0.738549
\(265\) −12.0000 −0.737154
\(266\) 4.00000 0.245256
\(267\) 6.00000 0.367194
\(268\) 12.0000 0.733017
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) −2.00000 −0.121716
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) −2.00000 −0.121268
\(273\) 2.00000 0.121046
\(274\) 6.00000 0.362473
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 4.00000 0.239904
\(279\) −8.00000 −0.478947
\(280\) −6.00000 −0.358569
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) −8.00000 −0.476393
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 16.0000 0.949425
\(285\) −8.00000 −0.473879
\(286\) 8.00000 0.473050
\(287\) −6.00000 −0.354169
\(288\) −5.00000 −0.294628
\(289\) −13.0000 −0.764706
\(290\) 2.00000 0.117444
\(291\) 6.00000 0.351726
\(292\) −2.00000 −0.117041
\(293\) −10.0000 −0.584206 −0.292103 0.956387i \(-0.594355\pi\)
−0.292103 + 0.956387i \(0.594355\pi\)
\(294\) 1.00000 0.0583212
\(295\) −24.0000 −1.39733
\(296\) −30.0000 −1.74371
\(297\) −4.00000 −0.232104
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 12.0000 0.691669
\(302\) 8.00000 0.460348
\(303\) −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 \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) −4.00000 −0.227921
\(309\) 8.00000 0.455104
\(310\) −16.0000 −0.908739
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) 6.00000 0.339683
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −22.0000 −1.24153
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 6.00000 0.336463
\(319\) 4.00000 0.223957
\(320\) −14.0000 −0.782624
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −8.00000 −0.445132
\(324\) −1.00000 −0.0555556
\(325\) 2.00000 0.110940
\(326\) −4.00000 −0.221540
\(327\) 18.0000 0.995402
\(328\) −18.0000 −0.993884
\(329\) −8.00000 −0.441054
\(330\) −8.00000 −0.440386
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −4.00000 −0.219529
\(333\) −10.0000 −0.547997
\(334\) 24.0000 1.31322
\(335\) 24.0000 1.31126
\(336\) 1.00000 0.0545545
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 9.00000 0.489535
\(339\) −2.00000 −0.108625
\(340\) 4.00000 0.216930
\(341\) −32.0000 −1.73290
\(342\) 4.00000 0.216295
\(343\) 1.00000 0.0539949
\(344\) 36.0000 1.94099
\(345\) 0 0
\(346\) −6.00000 −0.322562
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 1.00000 0.0536056
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 1.00000 0.0534522
\(351\) 2.00000 0.106752
\(352\) −20.0000 −1.06600
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 12.0000 0.637793
\(355\) 32.0000 1.69838
\(356\) 6.00000 0.317999
\(357\) −2.00000 −0.105851
\(358\) 4.00000 0.211407
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) −6.00000 −0.316228
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) −5.00000 −0.262432
\(364\) 2.00000 0.104828
\(365\) −4.00000 −0.209370
\(366\) −10.0000 −0.522708
\(367\) 24.0000 1.25279 0.626395 0.779506i \(-0.284530\pi\)
0.626395 + 0.779506i \(0.284530\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) −20.0000 −1.03975
\(371\) 6.00000 0.311504
\(372\) −8.00000 −0.414781
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) −8.00000 −0.413670
\(375\) −12.0000 −0.619677
\(376\) −24.0000 −1.23771
\(377\) −2.00000 −0.103005
\(378\) 1.00000 0.0514344
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −8.00000 −0.410391
\(381\) −16.0000 −0.819705
\(382\) −8.00000 −0.409316
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) −3.00000 −0.153093
\(385\) −8.00000 −0.407718
\(386\) −2.00000 −0.101797
\(387\) 12.0000 0.609994
\(388\) 6.00000 0.304604
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 4.00000 0.202548
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 4.00000 0.201773
\(394\) −22.0000 −1.10834
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 24.0000 1.20301
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) −12.0000 −0.598506
\(403\) 16.0000 0.797017
\(404\) −6.00000 −0.298511
\(405\) −2.00000 −0.0993808
\(406\) −1.00000 −0.0496292
\(407\) −40.0000 −1.98273
\(408\) −6.00000 −0.297044
\(409\) 18.0000 0.890043 0.445021 0.895520i \(-0.353196\pi\)
0.445021 + 0.895520i \(0.353196\pi\)
\(410\) −12.0000 −0.592638
\(411\) 6.00000 0.295958
\(412\) 8.00000 0.394132
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −8.00000 −0.392705
\(416\) 10.0000 0.490290
\(417\) 4.00000 0.195881
\(418\) 16.0000 0.782586
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) −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
\(423\) −8.00000 −0.388973
\(424\) 18.0000 0.874157
\(425\) −2.00000 −0.0970143
\(426\) −16.0000 −0.775203
\(427\) −10.0000 −0.483934
\(428\) 12.0000 0.580042
\(429\) 8.00000 0.386244
\(430\) 24.0000 1.15738
\(431\) 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 \(0.677293\pi\)
\(434\) 8.00000 0.384012
\(435\) 2.00000 0.0958927
\(436\) 18.0000 0.862044
\(437\) 0 0
\(438\) 2.00000 0.0955637
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) −24.0000 −1.14416
\(441\) 1.00000 0.0476190
\(442\) 4.00000 0.190261
\(443\) 20.0000 0.950229 0.475114 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) −10.0000 −0.474579
\(445\) 12.0000 0.568855
\(446\) 16.0000 0.757622
\(447\) −6.00000 −0.283790
\(448\) 7.00000 0.330719
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 1.00000 0.0471405
\(451\) −24.0000 −1.13012
\(452\) −2.00000 −0.0940721
\(453\) 8.00000 0.375873
\(454\) 12.0000 0.563188
\(455\) 4.00000 0.187523
\(456\) 12.0000 0.561951
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −14.0000 −0.654177
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 4.00000 0.186097
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −1.00000 −0.0464238
\(465\) −16.0000 −0.741982
\(466\) −26.0000 −1.20443
\(467\) −4.00000 −0.185098 −0.0925490 0.995708i \(-0.529501\pi\)
−0.0925490 + 0.995708i \(0.529501\pi\)
\(468\) 2.00000 0.0924500
\(469\) −12.0000 −0.554109
\(470\) −16.0000 −0.738025
\(471\) −22.0000 −1.01371
\(472\) 36.0000 1.65703
\(473\) 48.0000 2.20704
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) −2.00000 −0.0916698
\(477\) 6.00000 0.274721
\(478\) 24.0000 1.09773
\(479\) 8.00000 0.365529 0.182765 0.983157i \(-0.441495\pi\)
0.182765 + 0.983157i \(0.441495\pi\)
\(480\) −10.0000 −0.456435
\(481\) 20.0000 0.911922
\(482\) −18.0000 −0.819878
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 12.0000 0.544892
\(486\) 1.00000 0.0453609
\(487\) 24.0000 1.08754 0.543772 0.839233i \(-0.316996\pi\)
0.543772 + 0.839233i \(0.316996\pi\)
\(488\) −30.0000 −1.35804
\(489\) −4.00000 −0.180886
\(490\) 2.00000 0.0903508
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) −6.00000 −0.270501
\(493\) 2.00000 0.0900755
\(494\) −8.00000 −0.359937
\(495\) −8.00000 −0.359573
\(496\) 8.00000 0.359211
\(497\) −16.0000 −0.717698
\(498\) 4.00000 0.179244
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) −12.0000 −0.536656
\(501\) 24.0000 1.07224
\(502\) 12.0000 0.535586
\(503\) 16.0000 0.713405 0.356702 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 3.00000 0.133631
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) −16.0000 −0.709885
\(509\) 22.0000 0.975133 0.487566 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) −4.00000 −0.177123
\(511\) 2.00000 0.0884748
\(512\) 11.0000 0.486136
\(513\) 4.00000 0.176604
\(514\) 6.00000 0.264649
\(515\) 16.0000 0.705044
\(516\) 12.0000 0.528271
\(517\) −32.0000 −1.40736
\(518\) 10.0000 0.439375
\(519\) −6.00000 −0.263371
\(520\) 12.0000 0.526235
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −1.00000 −0.0437688
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 4.00000 0.174741
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) −16.0000 −0.696971
\(528\) 4.00000 0.174078
\(529\) −23.0000 −1.00000
\(530\) 12.0000 0.521247
\(531\) 12.0000 0.520756
\(532\) 4.00000 0.173422
\(533\) 12.0000 0.519778
\(534\) −6.00000 −0.259645
\(535\) 24.0000 1.03761
\(536\) −36.0000 −1.55496
\(537\) 4.00000 0.172613
\(538\) 18.0000 0.776035
\(539\) 4.00000 0.172292
\(540\) −2.00000 −0.0860663
\(541\) 30.0000 1.28980 0.644900 0.764267i \(-0.276899\pi\)
0.644900 + 0.764267i \(0.276899\pi\)
\(542\) −8.00000 −0.343629
\(543\) −6.00000 −0.257485
\(544\) −10.0000 −0.428746
\(545\) 36.0000 1.54207
\(546\) −2.00000 −0.0855921
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 6.00000 0.256307
\(549\) −10.0000 −0.426790
\(550\) 4.00000 0.170561
\(551\) −4.00000 −0.170406
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −20.0000 −0.848953
\(556\) 4.00000 0.169638
\(557\) −2.00000 −0.0847427 −0.0423714 0.999102i \(-0.513491\pi\)
−0.0423714 + 0.999102i \(0.513491\pi\)
\(558\) 8.00000 0.338667
\(559\) −24.0000 −1.01509
\(560\) 2.00000 0.0845154
\(561\) −8.00000 −0.337760
\(562\) −10.0000 −0.421825
\(563\) −20.0000 −0.842900 −0.421450 0.906852i \(-0.638479\pi\)
−0.421450 + 0.906852i \(0.638479\pi\)
\(564\) −8.00000 −0.336861
\(565\) −4.00000 −0.168281
\(566\) 4.00000 0.168133
\(567\) 1.00000 0.0419961
\(568\) −48.0000 −2.01404
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 8.00000 0.335083
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 8.00000 0.334497
\(573\) −8.00000 −0.334205
\(574\) 6.00000 0.250435
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) 13.0000 0.540729
\(579\) −2.00000 −0.0831172
\(580\) 2.00000 0.0830455
\(581\) 4.00000 0.165948
\(582\) −6.00000 −0.248708
\(583\) 24.0000 0.993978
\(584\) 6.00000 0.248282
\(585\) 4.00000 0.165380
\(586\) 10.0000 0.413096
\(587\) 44.0000 1.81607 0.908037 0.418890i \(-0.137581\pi\)
0.908037 + 0.418890i \(0.137581\pi\)
\(588\) 1.00000 0.0412393
\(589\) 32.0000 1.31854
\(590\) 24.0000 0.988064
\(591\) −22.0000 −0.904959
\(592\) 10.0000 0.410997
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 4.00000 0.164122
\(595\) −4.00000 −0.163984
\(596\) −6.00000 −0.245770
\(597\) 24.0000 0.982255
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 3.00000 0.122474
\(601\) −30.0000 −1.22373 −0.611863 0.790964i \(-0.709580\pi\)
−0.611863 + 0.790964i \(0.709580\pi\)
\(602\) −12.0000 −0.489083
\(603\) −12.0000 −0.488678
\(604\) 8.00000 0.325515
\(605\) −10.0000 −0.406558
\(606\) 6.00000 0.243733
\(607\) 40.0000 1.62355 0.811775 0.583970i \(-0.198502\pi\)
0.811775 + 0.583970i \(0.198502\pi\)
\(608\) 20.0000 0.811107
\(609\) −1.00000 −0.0405220
\(610\) −20.0000 −0.809776
\(611\) 16.0000 0.647291
\(612\) −2.00000 −0.0808452
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) −12.0000 −0.484281
\(615\) −12.0000 −0.483887
\(616\) 12.0000 0.483494
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) −8.00000 −0.321807
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) 32.0000 1.28308
\(623\) −6.00000 −0.240385
\(624\) −2.00000 −0.0800641
\(625\) −19.0000 −0.760000
\(626\) −10.0000 −0.399680
\(627\) 16.0000 0.638978
\(628\) −22.0000 −0.877896
\(629\) −20.0000 −0.797452
\(630\) 2.00000 0.0796819
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 2.00000 0.0794301
\(635\) −32.0000 −1.26988
\(636\) 6.00000 0.237915
\(637\) −2.00000 −0.0792429
\(638\) −4.00000 −0.158362
\(639\) −16.0000 −0.632950
\(640\) −6.00000 −0.237171
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −12.0000 −0.473602
\(643\) 36.0000 1.41970 0.709851 0.704352i \(-0.248762\pi\)
0.709851 + 0.704352i \(0.248762\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 8.00000 0.314756
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 3.00000 0.117851
\(649\) 48.0000 1.88416
\(650\) −2.00000 −0.0784465
\(651\) 8.00000 0.313545
\(652\) −4.00000 −0.156652
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) −18.0000 −0.703856
\(655\) 8.00000 0.312586
\(656\) 6.00000 0.234261
\(657\) 2.00000 0.0780274
\(658\) 8.00000 0.311872
\(659\) −20.0000 −0.779089 −0.389545 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) −8.00000 −0.311400
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 20.0000 0.777322
\(663\) 4.00000 0.155347
\(664\) 12.0000 0.465690
\(665\) 8.00000 0.310227
\(666\) 10.0000 0.387492
\(667\) 0 0
\(668\) 24.0000 0.928588
\(669\) 16.0000 0.618596
\(670\) −24.0000 −0.927201
\(671\) −40.0000 −1.54418
\(672\) 5.00000 0.192879
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) −34.0000 −1.30963
\(675\) 1.00000 0.0384900
\(676\) 9.00000 0.346154
\(677\) 38.0000 1.46046 0.730229 0.683202i \(-0.239413\pi\)
0.730229 + 0.683202i \(0.239413\pi\)
\(678\) 2.00000 0.0768095
\(679\) −6.00000 −0.230259
\(680\) −12.0000 −0.460179
\(681\) 12.0000 0.459841
\(682\) 32.0000 1.22534
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 4.00000 0.152944
\(685\) 12.0000 0.458496
\(686\) −1.00000 −0.0381802
\(687\) −14.0000 −0.534133
\(688\) −12.0000 −0.457496
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) −12.0000 −0.456502 −0.228251 0.973602i \(-0.573301\pi\)
−0.228251 + 0.973602i \(0.573301\pi\)
\(692\) −6.00000 −0.228086
\(693\) 4.00000 0.151947
\(694\) −20.0000 −0.759190
\(695\) 8.00000 0.303457
\(696\) −3.00000 −0.113715
\(697\) −12.0000 −0.454532
\(698\) −14.0000 −0.529908
\(699\) −26.0000 −0.983410
\(700\) 1.00000 0.0377964
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) −2.00000 −0.0754851
\(703\) 40.0000 1.50863
\(704\) 28.0000 1.05529
\(705\) −16.0000 −0.602595
\(706\) 6.00000 0.225813
\(707\) 6.00000 0.225653
\(708\) 12.0000 0.450988
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) −32.0000 −1.20094
\(711\) 0 0
\(712\) −18.0000 −0.674579
\(713\) 0 0
\(714\) 2.00000 0.0748481
\(715\) 16.0000 0.598366
\(716\) 4.00000 0.149487
\(717\) 24.0000 0.896296
\(718\) 16.0000 0.597115
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 2.00000 0.0745356
\(721\) −8.00000 −0.297936
\(722\) 3.00000 0.111648
\(723\) −18.0000 −0.669427
\(724\) −6.00000 −0.222988
\(725\) −1.00000 −0.0371391
\(726\) 5.00000 0.185567
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −6.00000 −0.222375
\(729\) 1.00000 0.0370370
\(730\) 4.00000 0.148047
\(731\) 24.0000 0.887672
\(732\) −10.0000 −0.369611
\(733\) −26.0000 −0.960332 −0.480166 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) −24.0000 −0.885856
\(735\) 2.00000 0.0737711
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) 6.00000 0.220863
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) −20.0000 −0.735215
\(741\) −8.00000 −0.293887
\(742\) −6.00000 −0.220267
\(743\) 32.0000 1.17397 0.586983 0.809599i \(-0.300316\pi\)
0.586983 + 0.809599i \(0.300316\pi\)
\(744\) 24.0000 0.879883
\(745\) −12.0000 −0.439646
\(746\) −22.0000 −0.805477
\(747\) 4.00000 0.146352
\(748\) −8.00000 −0.292509
\(749\) −12.0000 −0.438470
\(750\) 12.0000 0.438178
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 8.00000 0.291730
\(753\) 12.0000 0.437304
\(754\) 2.00000 0.0728357
\(755\) 16.0000 0.582300
\(756\) 1.00000 0.0363696
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 4.00000 0.145287
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 16.0000 0.579619
\(763\) −18.0000 −0.651644
\(764\) −8.00000 −0.289430
\(765\) −4.00000 −0.144620
\(766\) −16.0000 −0.578103
\(767\) −24.0000 −0.866590
\(768\) 17.0000 0.613435
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 8.00000 0.288300
\(771\) 6.00000 0.216085
\(772\) −2.00000 −0.0719816
\(773\) −10.0000 −0.359675 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(774\) −12.0000 −0.431331
\(775\) 8.00000 0.287368
\(776\) −18.0000 −0.646162
\(777\) 10.0000 0.358748
\(778\) −6.00000 −0.215110
\(779\) 24.0000 0.859889
\(780\) 4.00000 0.143223
\(781\) −64.0000 −2.29010
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) −1.00000 −0.0357143
\(785\) −44.0000 −1.57043
\(786\) −4.00000 −0.142675
\(787\) −28.0000 −0.998092 −0.499046 0.866575i \(-0.666316\pi\)
−0.499046 + 0.866575i \(0.666316\pi\)
\(788\) −22.0000 −0.783718
\(789\) 0 0
\(790\) 0 0
\(791\) 2.00000 0.0711118
\(792\) 12.0000 0.426401
\(793\) 20.0000 0.710221
\(794\) 18.0000 0.638796
\(795\) 12.0000 0.425596
\(796\) 24.0000 0.850657
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) −4.00000 −0.141598
\(799\) −16.0000 −0.566039
\(800\) 5.00000 0.176777
\(801\) −6.00000 −0.212000
\(802\) −2.00000 −0.0706225
\(803\) 8.00000 0.282314
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 18.0000 0.633630
\(808\) 18.0000 0.633238
\(809\) −38.0000 −1.33601 −0.668004 0.744157i \(-0.732851\pi\)
−0.668004 + 0.744157i \(0.732851\pi\)
\(810\) 2.00000 0.0702728
\(811\) −36.0000 −1.26413 −0.632065 0.774915i \(-0.717793\pi\)
−0.632065 + 0.774915i \(0.717793\pi\)
\(812\) −1.00000 −0.0350931
\(813\) −8.00000 −0.280572
\(814\) 40.0000 1.40200
\(815\) −8.00000 −0.280228
\(816\) 2.00000 0.0700140
\(817\) −48.0000 −1.67931
\(818\) −18.0000 −0.629355
\(819\) −2.00000 −0.0698857
\(820\) −12.0000 −0.419058
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) −6.00000 −0.209274
\(823\) −8.00000 −0.278862 −0.139431 0.990232i \(-0.544527\pi\)
−0.139431 + 0.990232i \(0.544527\pi\)
\(824\) −24.0000 −0.836080
\(825\) 4.00000 0.139262
\(826\) −12.0000 −0.417533
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 8.00000 0.277684
\(831\) 10.0000 0.346896
\(832\) −14.0000 −0.485363
\(833\) 2.00000 0.0692959
\(834\) −4.00000 −0.138509
\(835\) 48.0000 1.66111
\(836\) 16.0000 0.553372
\(837\) 8.00000 0.276520
\(838\) 12.0000 0.414533
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 6.00000 0.207020
\(841\) 1.00000 0.0344828
\(842\) 26.0000 0.896019
\(843\) −10.0000 −0.344418
\(844\) −4.00000 −0.137686
\(845\) 18.0000 0.619219
\(846\) 8.00000 0.275046
\(847\) 5.00000 0.171802
\(848\) −6.00000 −0.206041
\(849\) 4.00000 0.137280
\(850\) 2.00000 0.0685994
\(851\) 0 0
\(852\) −16.0000 −0.548151
\(853\) −2.00000 −0.0684787 −0.0342393 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) 10.0000 0.342193
\(855\) 8.00000 0.273594
\(856\) −36.0000 −1.23045
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) −8.00000 −0.273115
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 24.0000 0.818393
\(861\) 6.00000 0.204479
\(862\) −24.0000 −0.817443
\(863\) −40.0000 −1.36162 −0.680808 0.732462i \(-0.738371\pi\)
−0.680808 + 0.732462i \(0.738371\pi\)
\(864\) 5.00000 0.170103
\(865\) −12.0000 −0.408012
\(866\) 22.0000 0.747590
\(867\) 13.0000 0.441503
\(868\) 8.00000 0.271538
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) 24.0000 0.813209
\(872\) −54.0000 −1.82867
\(873\) −6.00000 −0.203069
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 2.00000 0.0675737
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) 8.00000 0.269987
\(879\) 10.0000 0.337292
\(880\) 8.00000 0.269680
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 4.00000 0.134535
\(885\) 24.0000 0.806751
\(886\) −20.0000 −0.671913
\(887\) −32.0000 −1.07445 −0.537227 0.843437i \(-0.680528\pi\)
−0.537227 + 0.843437i \(0.680528\pi\)
\(888\) 30.0000 1.00673
\(889\) 16.0000 0.536623
\(890\) −12.0000 −0.402241
\(891\) 4.00000 0.134005
\(892\) 16.0000 0.535720
\(893\) 32.0000 1.07084
\(894\) 6.00000 0.200670
\(895\) 8.00000 0.267411
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) −8.00000 −0.266815
\(900\) 1.00000 0.0333333
\(901\) 12.0000 0.399778
\(902\) 24.0000 0.799113
\(903\) −12.0000 −0.399335
\(904\) 6.00000 0.199557
\(905\) −12.0000 −0.398893
\(906\) −8.00000 −0.265782
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 12.0000 0.398234
\(909\) 6.00000 0.199007
\(910\) −4.00000 −0.132599
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) −4.00000 −0.132453
\(913\) 16.0000 0.529523
\(914\) −10.0000 −0.330771
\(915\) −20.0000 −0.661180
\(916\) −14.0000 −0.462573
\(917\) −4.00000 −0.132092
\(918\) 2.00000 0.0660098
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) −30.0000 −0.987997
\(923\) 32.0000 1.05329
\(924\) 4.00000 0.131590
\(925\) 10.0000 0.328798
\(926\) 16.0000 0.525793
\(927\) −8.00000 −0.262754
\(928\) −5.00000 −0.164133
\(929\) −6.00000 −0.196854 −0.0984268 0.995144i \(-0.531381\pi\)
−0.0984268 + 0.995144i \(0.531381\pi\)
\(930\) 16.0000 0.524661
\(931\) −4.00000 −0.131095
\(932\) −26.0000 −0.851658
\(933\) 32.0000 1.04763
\(934\) 4.00000 0.130884
\(935\) −16.0000 −0.523256
\(936\) −6.00000 −0.196116
\(937\) −54.0000 −1.76410 −0.882052 0.471153i \(-0.843838\pi\)
−0.882052 + 0.471153i \(0.843838\pi\)
\(938\) 12.0000 0.391814
\(939\) −10.0000 −0.326338
\(940\) −16.0000 −0.521862
\(941\) 6.00000 0.195594 0.0977972 0.995206i \(-0.468820\pi\)
0.0977972 + 0.995206i \(0.468820\pi\)
\(942\) 22.0000 0.716799
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 2.00000 0.0650600
\(946\) −48.0000 −1.56061
\(947\) 28.0000 0.909878 0.454939 0.890523i \(-0.349661\pi\)
0.454939 + 0.890523i \(0.349661\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) −4.00000 −0.129777
\(951\) 2.00000 0.0648544
\(952\) 6.00000 0.194461
\(953\) 26.0000 0.842223 0.421111 0.907009i \(-0.361640\pi\)
0.421111 + 0.907009i \(0.361640\pi\)
\(954\) −6.00000 −0.194257
\(955\) −16.0000 −0.517748
\(956\) 24.0000 0.776215
\(957\) −4.00000 −0.129302
\(958\) −8.00000 −0.258468
\(959\) −6.00000 −0.193750
\(960\) 14.0000 0.451848
\(961\) 33.0000 1.06452
\(962\) −20.0000 −0.644826
\(963\) −12.0000 −0.386695
\(964\) −18.0000 −0.579741
\(965\) −4.00000 −0.128765
\(966\) 0 0
\(967\) −56.0000 −1.80084 −0.900419 0.435023i \(-0.856740\pi\)
−0.900419 + 0.435023i \(0.856740\pi\)
\(968\) 15.0000 0.482118
\(969\) 8.00000 0.256997
\(970\) −12.0000 −0.385297
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) 1.00000 0.0320750
\(973\) −4.00000 −0.128234
\(974\) −24.0000 −0.769010
\(975\) −2.00000 −0.0640513
\(976\) 10.0000 0.320092
\(977\) 2.00000 0.0639857 0.0319928 0.999488i \(-0.489815\pi\)
0.0319928 + 0.999488i \(0.489815\pi\)
\(978\) 4.00000 0.127906
\(979\) −24.0000 −0.767043
\(980\) 2.00000 0.0638877
\(981\) −18.0000 −0.574696
\(982\) −20.0000 −0.638226
\(983\) −32.0000 −1.02064 −0.510321 0.859984i \(-0.670473\pi\)
−0.510321 + 0.859984i \(0.670473\pi\)
\(984\) 18.0000 0.573819
\(985\) −44.0000 −1.40196
\(986\) −2.00000 −0.0636930
\(987\) 8.00000 0.254643
\(988\) −8.00000 −0.254514
\(989\) 0 0
\(990\) 8.00000 0.254257
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 40.0000 1.27000
\(993\) 20.0000 0.634681
\(994\) 16.0000 0.507489
\(995\) 48.0000 1.52170
\(996\) 4.00000 0.126745
\(997\) −2.00000 −0.0633406 −0.0316703 0.999498i \(-0.510083\pi\)
−0.0316703 + 0.999498i \(0.510083\pi\)
\(998\) −4.00000 −0.126618
\(999\) 10.0000 0.316386
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 609.2.a.a.1.1 1
3.2 odd 2 1827.2.a.d.1.1 1
4.3 odd 2 9744.2.a.l.1.1 1
7.6 odd 2 4263.2.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
609.2.a.a.1.1 1 1.1 even 1 trivial
1827.2.a.d.1.1 1 3.2 odd 2
4263.2.a.c.1.1 1 7.6 odd 2
9744.2.a.l.1.1 1 4.3 odd 2