Properties

Label 2023.2.a.b.1.1
Level $2023$
Weight $2$
Character 2023.1
Self dual yes
Analytic conductor $16.154$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2023 = 7 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2023.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.1537363289\)
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 \(\chi\) \(=\) 2023.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +6.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.00000 q^{3} -1.00000 q^{4} +4.00000 q^{5} -3.00000 q^{6} -1.00000 q^{7} +3.00000 q^{8} +6.00000 q^{9} -4.00000 q^{10} -3.00000 q^{12} -2.00000 q^{13} +1.00000 q^{14} +12.0000 q^{15} -1.00000 q^{16} -6.00000 q^{18} -7.00000 q^{19} -4.00000 q^{20} -3.00000 q^{21} +4.00000 q^{23} +9.00000 q^{24} +11.0000 q^{25} +2.00000 q^{26} +9.00000 q^{27} +1.00000 q^{28} +3.00000 q^{29} -12.0000 q^{30} +7.00000 q^{31} -5.00000 q^{32} -4.00000 q^{35} -6.00000 q^{36} +10.0000 q^{37} +7.00000 q^{38} -6.00000 q^{39} +12.0000 q^{40} +3.00000 q^{42} -2.00000 q^{43} +24.0000 q^{45} -4.00000 q^{46} +3.00000 q^{47} -3.00000 q^{48} +1.00000 q^{49} -11.0000 q^{50} +2.00000 q^{52} -3.00000 q^{53} -9.00000 q^{54} -3.00000 q^{56} -21.0000 q^{57} -3.00000 q^{58} -9.00000 q^{59} -12.0000 q^{60} +8.00000 q^{61} -7.00000 q^{62} -6.00000 q^{63} +7.00000 q^{64} -8.00000 q^{65} -8.00000 q^{67} +12.0000 q^{69} +4.00000 q^{70} +2.00000 q^{71} +18.0000 q^{72} +6.00000 q^{73} -10.0000 q^{74} +33.0000 q^{75} +7.00000 q^{76} +6.00000 q^{78} +6.00000 q^{79} -4.00000 q^{80} +9.00000 q^{81} -1.00000 q^{83} +3.00000 q^{84} +2.00000 q^{86} +9.00000 q^{87} -8.00000 q^{89} -24.0000 q^{90} +2.00000 q^{91} -4.00000 q^{92} +21.0000 q^{93} -3.00000 q^{94} -28.0000 q^{95} -15.0000 q^{96} +2.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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 \(0.615027\pi\)
\(3\) 3.00000 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(4\) −1.00000 −0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) −3.00000 −1.22474
\(7\) −1.00000 −0.377964
\(8\) 3.00000 1.06066
\(9\) 6.00000 2.00000
\(10\) −4.00000 −1.26491
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −3.00000 −0.866025
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 1.00000 0.267261
\(15\) 12.0000 3.09839
\(16\) −1.00000 −0.250000
\(17\) 0 0
\(18\) −6.00000 −1.41421
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −4.00000 −0.894427
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 9.00000 1.83712
\(25\) 11.0000 2.20000
\(26\) 2.00000 0.392232
\(27\) 9.00000 1.73205
\(28\) 1.00000 0.188982
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −12.0000 −2.19089
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) −6.00000 −1.00000
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 7.00000 1.13555
\(39\) −6.00000 −0.960769
\(40\) 12.0000 1.89737
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 3.00000 0.462910
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 24.0000 3.57771
\(46\) −4.00000 −0.589768
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) −3.00000 −0.433013
\(49\) 1.00000 0.142857
\(50\) −11.0000 −1.55563
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −9.00000 −1.22474
\(55\) 0 0
\(56\) −3.00000 −0.400892
\(57\) −21.0000 −2.78152
\(58\) −3.00000 −0.393919
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) −12.0000 −1.54919
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −7.00000 −0.889001
\(63\) −6.00000 −0.755929
\(64\) 7.00000 0.875000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 12.0000 1.44463
\(70\) 4.00000 0.478091
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 18.0000 2.12132
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) −10.0000 −1.16248
\(75\) 33.0000 3.81051
\(76\) 7.00000 0.802955
\(77\) 0 0
\(78\) 6.00000 0.679366
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) −4.00000 −0.447214
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −1.00000 −0.109764 −0.0548821 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) 2.00000 0.215666
\(87\) 9.00000 0.964901
\(88\) 0 0
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) −24.0000 −2.52982
\(91\) 2.00000 0.209657
\(92\) −4.00000 −0.417029
\(93\) 21.0000 2.17760
\(94\) −3.00000 −0.309426
\(95\) −28.0000 −2.87274
\(96\) −15.0000 −1.53093
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) −11.0000 −1.10000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) −7.00000 −0.689730 −0.344865 0.938652i \(-0.612075\pi\)
−0.344865 + 0.938652i \(0.612075\pi\)
\(104\) −6.00000 −0.588348
\(105\) −12.0000 −1.17108
\(106\) 3.00000 0.291386
\(107\) −16.0000 −1.54678 −0.773389 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) −9.00000 −0.866025
\(109\) −9.00000 −0.862044 −0.431022 0.902342i \(-0.641847\pi\)
−0.431022 + 0.902342i \(0.641847\pi\)
\(110\) 0 0
\(111\) 30.0000 2.84747
\(112\) 1.00000 0.0944911
\(113\) −2.00000 −0.188144 −0.0940721 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 21.0000 1.96683
\(115\) 16.0000 1.49201
\(116\) −3.00000 −0.278543
\(117\) −12.0000 −1.10940
\(118\) 9.00000 0.828517
\(119\) 0 0
\(120\) 36.0000 3.28634
\(121\) −11.0000 −1.00000
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) −7.00000 −0.628619
\(125\) 24.0000 2.14663
\(126\) 6.00000 0.534522
\(127\) −4.00000 −0.354943 −0.177471 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 3.00000 0.265165
\(129\) −6.00000 −0.528271
\(130\) 8.00000 0.701646
\(131\) 19.0000 1.66004 0.830019 0.557735i \(-0.188330\pi\)
0.830019 + 0.557735i \(0.188330\pi\)
\(132\) 0 0
\(133\) 7.00000 0.606977
\(134\) 8.00000 0.691095
\(135\) 36.0000 3.09839
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) −12.0000 −1.02151
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 4.00000 0.338062
\(141\) 9.00000 0.757937
\(142\) −2.00000 −0.167836
\(143\) 0 0
\(144\) −6.00000 −0.500000
\(145\) 12.0000 0.996546
\(146\) −6.00000 −0.496564
\(147\) 3.00000 0.247436
\(148\) −10.0000 −0.821995
\(149\) −7.00000 −0.573462 −0.286731 0.958011i \(-0.592569\pi\)
−0.286731 + 0.958011i \(0.592569\pi\)
\(150\) −33.0000 −2.69444
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) −21.0000 −1.70332
\(153\) 0 0
\(154\) 0 0
\(155\) 28.0000 2.24901
\(156\) 6.00000 0.480384
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) −6.00000 −0.477334
\(159\) −9.00000 −0.713746
\(160\) −20.0000 −1.58114
\(161\) −4.00000 −0.315244
\(162\) −9.00000 −0.707107
\(163\) 18.0000 1.40987 0.704934 0.709273i \(-0.250976\pi\)
0.704934 + 0.709273i \(0.250976\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 1.00000 0.0776151
\(167\) −1.00000 −0.0773823 −0.0386912 0.999251i \(-0.512319\pi\)
−0.0386912 + 0.999251i \(0.512319\pi\)
\(168\) −9.00000 −0.694365
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −42.0000 −3.21182
\(172\) 2.00000 0.152499
\(173\) −20.0000 −1.52057 −0.760286 0.649589i \(-0.774941\pi\)
−0.760286 + 0.649589i \(0.774941\pi\)
\(174\) −9.00000 −0.682288
\(175\) −11.0000 −0.831522
\(176\) 0 0
\(177\) −27.0000 −2.02944
\(178\) 8.00000 0.599625
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) −24.0000 −1.78885
\(181\) 16.0000 1.18927 0.594635 0.803996i \(-0.297296\pi\)
0.594635 + 0.803996i \(0.297296\pi\)
\(182\) −2.00000 −0.148250
\(183\) 24.0000 1.77413
\(184\) 12.0000 0.884652
\(185\) 40.0000 2.94086
\(186\) −21.0000 −1.53979
\(187\) 0 0
\(188\) −3.00000 −0.218797
\(189\) −9.00000 −0.654654
\(190\) 28.0000 2.03133
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) 21.0000 1.51554
\(193\) 23.0000 1.65558 0.827788 0.561041i \(-0.189599\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) −2.00000 −0.143592
\(195\) −24.0000 −1.71868
\(196\) −1.00000 −0.0714286
\(197\) −15.0000 −1.06871 −0.534353 0.845262i \(-0.679445\pi\)
−0.534353 + 0.845262i \(0.679445\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 33.0000 2.33345
\(201\) −24.0000 −1.69283
\(202\) −2.00000 −0.140720
\(203\) −3.00000 −0.210559
\(204\) 0 0
\(205\) 0 0
\(206\) 7.00000 0.487713
\(207\) 24.0000 1.66812
\(208\) 2.00000 0.138675
\(209\) 0 0
\(210\) 12.0000 0.828079
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 3.00000 0.206041
\(213\) 6.00000 0.411113
\(214\) 16.0000 1.09374
\(215\) −8.00000 −0.545595
\(216\) 27.0000 1.83712
\(217\) −7.00000 −0.475191
\(218\) 9.00000 0.609557
\(219\) 18.0000 1.21633
\(220\) 0 0
\(221\) 0 0
\(222\) −30.0000 −2.01347
\(223\) −15.0000 −1.00447 −0.502237 0.864730i \(-0.667490\pi\)
−0.502237 + 0.864730i \(0.667490\pi\)
\(224\) 5.00000 0.334077
\(225\) 66.0000 4.40000
\(226\) 2.00000 0.133038
\(227\) −9.00000 −0.597351 −0.298675 0.954355i \(-0.596545\pi\)
−0.298675 + 0.954355i \(0.596545\pi\)
\(228\) 21.0000 1.39076
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −16.0000 −1.05501
\(231\) 0 0
\(232\) 9.00000 0.590879
\(233\) 9.00000 0.589610 0.294805 0.955557i \(-0.404745\pi\)
0.294805 + 0.955557i \(0.404745\pi\)
\(234\) 12.0000 0.784465
\(235\) 12.0000 0.782794
\(236\) 9.00000 0.585850
\(237\) 18.0000 1.16923
\(238\) 0 0
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) −12.0000 −0.774597
\(241\) −12.0000 −0.772988 −0.386494 0.922292i \(-0.626314\pi\)
−0.386494 + 0.922292i \(0.626314\pi\)
\(242\) 11.0000 0.707107
\(243\) 0 0
\(244\) −8.00000 −0.512148
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) 14.0000 0.890799
\(248\) 21.0000 1.33350
\(249\) −3.00000 −0.190117
\(250\) −24.0000 −1.51789
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 6.00000 0.377964
\(253\) 0 0
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 6.00000 0.373544
\(259\) −10.0000 −0.621370
\(260\) 8.00000 0.496139
\(261\) 18.0000 1.11417
\(262\) −19.0000 −1.17382
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) −7.00000 −0.429198
\(267\) −24.0000 −1.46878
\(268\) 8.00000 0.488678
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −36.0000 −2.19089
\(271\) 17.0000 1.03268 0.516338 0.856385i \(-0.327295\pi\)
0.516338 + 0.856385i \(0.327295\pi\)
\(272\) 0 0
\(273\) 6.00000 0.363137
\(274\) −9.00000 −0.543710
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) 5.00000 0.300421 0.150210 0.988654i \(-0.452005\pi\)
0.150210 + 0.988654i \(0.452005\pi\)
\(278\) 20.0000 1.19952
\(279\) 42.0000 2.51447
\(280\) −12.0000 −0.717137
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) −9.00000 −0.535942
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) −2.00000 −0.118678
\(285\) −84.0000 −4.97573
\(286\) 0 0
\(287\) 0 0
\(288\) −30.0000 −1.76777
\(289\) 0 0
\(290\) −12.0000 −0.704664
\(291\) 6.00000 0.351726
\(292\) −6.00000 −0.351123
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) −3.00000 −0.174964
\(295\) −36.0000 −2.09600
\(296\) 30.0000 1.74371
\(297\) 0 0
\(298\) 7.00000 0.405499
\(299\) −8.00000 −0.462652
\(300\) −33.0000 −1.90526
\(301\) 2.00000 0.115278
\(302\) −2.00000 −0.115087
\(303\) 6.00000 0.344691
\(304\) 7.00000 0.401478
\(305\) 32.0000 1.83231
\(306\) 0 0
\(307\) 15.0000 0.856095 0.428048 0.903756i \(-0.359202\pi\)
0.428048 + 0.903756i \(0.359202\pi\)
\(308\) 0 0
\(309\) −21.0000 −1.19465
\(310\) −28.0000 −1.59029
\(311\) 31.0000 1.75785 0.878924 0.476961i \(-0.158262\pi\)
0.878924 + 0.476961i \(0.158262\pi\)
\(312\) −18.0000 −1.01905
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 14.0000 0.790066
\(315\) −24.0000 −1.35225
\(316\) −6.00000 −0.337526
\(317\) −31.0000 −1.74113 −0.870567 0.492050i \(-0.836248\pi\)
−0.870567 + 0.492050i \(0.836248\pi\)
\(318\) 9.00000 0.504695
\(319\) 0 0
\(320\) 28.0000 1.56525
\(321\) −48.0000 −2.67910
\(322\) 4.00000 0.222911
\(323\) 0 0
\(324\) −9.00000 −0.500000
\(325\) −22.0000 −1.22034
\(326\) −18.0000 −0.996928
\(327\) −27.0000 −1.49310
\(328\) 0 0
\(329\) −3.00000 −0.165395
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 1.00000 0.0548821
\(333\) 60.0000 3.28798
\(334\) 1.00000 0.0547176
\(335\) −32.0000 −1.74835
\(336\) 3.00000 0.163663
\(337\) 13.0000 0.708155 0.354078 0.935216i \(-0.384795\pi\)
0.354078 + 0.935216i \(0.384795\pi\)
\(338\) 9.00000 0.489535
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 0 0
\(342\) 42.0000 2.27110
\(343\) −1.00000 −0.0539949
\(344\) −6.00000 −0.323498
\(345\) 48.0000 2.58423
\(346\) 20.0000 1.07521
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) −9.00000 −0.482451
\(349\) −36.0000 −1.92704 −0.963518 0.267644i \(-0.913755\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 11.0000 0.587975
\(351\) −18.0000 −0.960769
\(352\) 0 0
\(353\) 24.0000 1.27739 0.638696 0.769460i \(-0.279474\pi\)
0.638696 + 0.769460i \(0.279474\pi\)
\(354\) 27.0000 1.43503
\(355\) 8.00000 0.424596
\(356\) 8.00000 0.423999
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 72.0000 3.79473
\(361\) 30.0000 1.57895
\(362\) −16.0000 −0.840941
\(363\) −33.0000 −1.73205
\(364\) −2.00000 −0.104828
\(365\) 24.0000 1.25622
\(366\) −24.0000 −1.25450
\(367\) −23.0000 −1.20059 −0.600295 0.799779i \(-0.704950\pi\)
−0.600295 + 0.799779i \(0.704950\pi\)
\(368\) −4.00000 −0.208514
\(369\) 0 0
\(370\) −40.0000 −2.07950
\(371\) 3.00000 0.155752
\(372\) −21.0000 −1.08880
\(373\) −15.0000 −0.776671 −0.388335 0.921518i \(-0.626950\pi\)
−0.388335 + 0.921518i \(0.626950\pi\)
\(374\) 0 0
\(375\) 72.0000 3.71806
\(376\) 9.00000 0.464140
\(377\) −6.00000 −0.309016
\(378\) 9.00000 0.462910
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 28.0000 1.43637
\(381\) −12.0000 −0.614779
\(382\) 22.0000 1.12562
\(383\) 21.0000 1.07305 0.536525 0.843884i \(-0.319737\pi\)
0.536525 + 0.843884i \(0.319737\pi\)
\(384\) 9.00000 0.459279
\(385\) 0 0
\(386\) −23.0000 −1.17067
\(387\) −12.0000 −0.609994
\(388\) −2.00000 −0.101535
\(389\) 27.0000 1.36895 0.684477 0.729034i \(-0.260031\pi\)
0.684477 + 0.729034i \(0.260031\pi\)
\(390\) 24.0000 1.21529
\(391\) 0 0
\(392\) 3.00000 0.151523
\(393\) 57.0000 2.87527
\(394\) 15.0000 0.755689
\(395\) 24.0000 1.20757
\(396\) 0 0
\(397\) −24.0000 −1.20453 −0.602263 0.798298i \(-0.705734\pi\)
−0.602263 + 0.798298i \(0.705734\pi\)
\(398\) −16.0000 −0.802008
\(399\) 21.0000 1.05131
\(400\) −11.0000 −0.550000
\(401\) 1.00000 0.0499376 0.0249688 0.999688i \(-0.492051\pi\)
0.0249688 + 0.999688i \(0.492051\pi\)
\(402\) 24.0000 1.19701
\(403\) −14.0000 −0.697390
\(404\) −2.00000 −0.0995037
\(405\) 36.0000 1.78885
\(406\) 3.00000 0.148888
\(407\) 0 0
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) 27.0000 1.33181
\(412\) 7.00000 0.344865
\(413\) 9.00000 0.442861
\(414\) −24.0000 −1.17954
\(415\) −4.00000 −0.196352
\(416\) 10.0000 0.490290
\(417\) −60.0000 −2.93821
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 12.0000 0.585540
\(421\) 13.0000 0.633581 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(422\) 10.0000 0.486792
\(423\) 18.0000 0.875190
\(424\) −9.00000 −0.437079
\(425\) 0 0
\(426\) −6.00000 −0.290701
\(427\) −8.00000 −0.387147
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) −9.00000 −0.433013
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 7.00000 0.336011
\(435\) 36.0000 1.72607
\(436\) 9.00000 0.431022
\(437\) −28.0000 −1.33942
\(438\) −18.0000 −0.860073
\(439\) 1.00000 0.0477274 0.0238637 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) 0 0
\(443\) −26.0000 −1.23530 −0.617649 0.786454i \(-0.711915\pi\)
−0.617649 + 0.786454i \(0.711915\pi\)
\(444\) −30.0000 −1.42374
\(445\) −32.0000 −1.51695
\(446\) 15.0000 0.710271
\(447\) −21.0000 −0.993266
\(448\) −7.00000 −0.330719
\(449\) 7.00000 0.330350 0.165175 0.986264i \(-0.447181\pi\)
0.165175 + 0.986264i \(0.447181\pi\)
\(450\) −66.0000 −3.11127
\(451\) 0 0
\(452\) 2.00000 0.0940721
\(453\) 6.00000 0.281905
\(454\) 9.00000 0.422391
\(455\) 8.00000 0.375046
\(456\) −63.0000 −2.95025
\(457\) 3.00000 0.140334 0.0701670 0.997535i \(-0.477647\pi\)
0.0701670 + 0.997535i \(0.477647\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) −16.0000 −0.746004
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) −3.00000 −0.139272
\(465\) 84.0000 3.89541
\(466\) −9.00000 −0.416917
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 12.0000 0.554700
\(469\) 8.00000 0.369406
\(470\) −12.0000 −0.553519
\(471\) −42.0000 −1.93526
\(472\) −27.0000 −1.24278
\(473\) 0 0
\(474\) −18.0000 −0.826767
\(475\) −77.0000 −3.53300
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) −2.00000 −0.0914779
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) −60.0000 −2.73861
\(481\) −20.0000 −0.911922
\(482\) 12.0000 0.546585
\(483\) −12.0000 −0.546019
\(484\) 11.0000 0.500000
\(485\) 8.00000 0.363261
\(486\) 0 0
\(487\) −10.0000 −0.453143 −0.226572 0.973995i \(-0.572752\pi\)
−0.226572 + 0.973995i \(0.572752\pi\)
\(488\) 24.0000 1.08643
\(489\) 54.0000 2.44196
\(490\) −4.00000 −0.180702
\(491\) −22.0000 −0.992846 −0.496423 0.868081i \(-0.665354\pi\)
−0.496423 + 0.868081i \(0.665354\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −14.0000 −0.629890
\(495\) 0 0
\(496\) −7.00000 −0.314309
\(497\) −2.00000 −0.0897123
\(498\) 3.00000 0.134433
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) −24.0000 −1.07331
\(501\) −3.00000 −0.134030
\(502\) 24.0000 1.07117
\(503\) 9.00000 0.401290 0.200645 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) −18.0000 −0.801784
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) −27.0000 −1.19911
\(508\) 4.00000 0.177471
\(509\) −20.0000 −0.886484 −0.443242 0.896402i \(-0.646172\pi\)
−0.443242 + 0.896402i \(0.646172\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) 11.0000 0.486136
\(513\) −63.0000 −2.78152
\(514\) −30.0000 −1.32324
\(515\) −28.0000 −1.23383
\(516\) 6.00000 0.264135
\(517\) 0 0
\(518\) 10.0000 0.439375
\(519\) −60.0000 −2.63371
\(520\) −24.0000 −1.05247
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) −18.0000 −0.787839
\(523\) −12.0000 −0.524723 −0.262362 0.964970i \(-0.584501\pi\)
−0.262362 + 0.964970i \(0.584501\pi\)
\(524\) −19.0000 −0.830019
\(525\) −33.0000 −1.44024
\(526\) 16.0000 0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 12.0000 0.521247
\(531\) −54.0000 −2.34340
\(532\) −7.00000 −0.303488
\(533\) 0 0
\(534\) 24.0000 1.03858
\(535\) −64.0000 −2.76696
\(536\) −24.0000 −1.03664
\(537\) −18.0000 −0.776757
\(538\) −10.0000 −0.431131
\(539\) 0 0
\(540\) −36.0000 −1.54919
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) −17.0000 −0.730213
\(543\) 48.0000 2.05988
\(544\) 0 0
\(545\) −36.0000 −1.54207
\(546\) −6.00000 −0.256776
\(547\) 4.00000 0.171028 0.0855138 0.996337i \(-0.472747\pi\)
0.0855138 + 0.996337i \(0.472747\pi\)
\(548\) −9.00000 −0.384461
\(549\) 48.0000 2.04859
\(550\) 0 0
\(551\) −21.0000 −0.894630
\(552\) 36.0000 1.53226
\(553\) −6.00000 −0.255146
\(554\) −5.00000 −0.212430
\(555\) 120.000 5.09372
\(556\) 20.0000 0.848189
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −42.0000 −1.77800
\(559\) 4.00000 0.169182
\(560\) 4.00000 0.169031
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) 11.0000 0.463595 0.231797 0.972764i \(-0.425539\pi\)
0.231797 + 0.972764i \(0.425539\pi\)
\(564\) −9.00000 −0.378968
\(565\) −8.00000 −0.336563
\(566\) −5.00000 −0.210166
\(567\) −9.00000 −0.377964
\(568\) 6.00000 0.251754
\(569\) −23.0000 −0.964210 −0.482105 0.876113i \(-0.660128\pi\)
−0.482105 + 0.876113i \(0.660128\pi\)
\(570\) 84.0000 3.51837
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 0 0
\(573\) −66.0000 −2.75719
\(574\) 0 0
\(575\) 44.0000 1.83493
\(576\) 42.0000 1.75000
\(577\) 16.0000 0.666089 0.333044 0.942911i \(-0.391924\pi\)
0.333044 + 0.942911i \(0.391924\pi\)
\(578\) 0 0
\(579\) 69.0000 2.86754
\(580\) −12.0000 −0.498273
\(581\) 1.00000 0.0414870
\(582\) −6.00000 −0.248708
\(583\) 0 0
\(584\) 18.0000 0.744845
\(585\) −48.0000 −1.98456
\(586\) −12.0000 −0.495715
\(587\) −17.0000 −0.701665 −0.350833 0.936438i \(-0.614101\pi\)
−0.350833 + 0.936438i \(0.614101\pi\)
\(588\) −3.00000 −0.123718
\(589\) −49.0000 −2.01901
\(590\) 36.0000 1.48210
\(591\) −45.0000 −1.85105
\(592\) −10.0000 −0.410997
\(593\) 42.0000 1.72473 0.862367 0.506284i \(-0.168981\pi\)
0.862367 + 0.506284i \(0.168981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.00000 0.286731
\(597\) 48.0000 1.96451
\(598\) 8.00000 0.327144
\(599\) 18.0000 0.735460 0.367730 0.929933i \(-0.380135\pi\)
0.367730 + 0.929933i \(0.380135\pi\)
\(600\) 99.0000 4.04166
\(601\) 16.0000 0.652654 0.326327 0.945257i \(-0.394189\pi\)
0.326327 + 0.945257i \(0.394189\pi\)
\(602\) −2.00000 −0.0815139
\(603\) −48.0000 −1.95471
\(604\) −2.00000 −0.0813788
\(605\) −44.0000 −1.78885
\(606\) −6.00000 −0.243733
\(607\) −7.00000 −0.284121 −0.142061 0.989858i \(-0.545373\pi\)
−0.142061 + 0.989858i \(0.545373\pi\)
\(608\) 35.0000 1.41944
\(609\) −9.00000 −0.364698
\(610\) −32.0000 −1.29564
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) −15.0000 −0.605351
\(615\) 0 0
\(616\) 0 0
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 21.0000 0.844744
\(619\) 15.0000 0.602901 0.301450 0.953482i \(-0.402529\pi\)
0.301450 + 0.953482i \(0.402529\pi\)
\(620\) −28.0000 −1.12451
\(621\) 36.0000 1.44463
\(622\) −31.0000 −1.24299
\(623\) 8.00000 0.320513
\(624\) 6.00000 0.240192
\(625\) 41.0000 1.64000
\(626\) 10.0000 0.399680
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 0 0
\(630\) 24.0000 0.956183
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) 18.0000 0.716002
\(633\) −30.0000 −1.19239
\(634\) 31.0000 1.23117
\(635\) −16.0000 −0.634941
\(636\) 9.00000 0.356873
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) 12.0000 0.474713
\(640\) 12.0000 0.474342
\(641\) 45.0000 1.77739 0.888697 0.458496i \(-0.151612\pi\)
0.888697 + 0.458496i \(0.151612\pi\)
\(642\) 48.0000 1.89441
\(643\) −28.0000 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(644\) 4.00000 0.157622
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) −28.0000 −1.10079 −0.550397 0.834903i \(-0.685524\pi\)
−0.550397 + 0.834903i \(0.685524\pi\)
\(648\) 27.0000 1.06066
\(649\) 0 0
\(650\) 22.0000 0.862911
\(651\) −21.0000 −0.823055
\(652\) −18.0000 −0.704934
\(653\) 3.00000 0.117399 0.0586995 0.998276i \(-0.481305\pi\)
0.0586995 + 0.998276i \(0.481305\pi\)
\(654\) 27.0000 1.05578
\(655\) 76.0000 2.96957
\(656\) 0 0
\(657\) 36.0000 1.40449
\(658\) 3.00000 0.116952
\(659\) 28.0000 1.09073 0.545363 0.838200i \(-0.316392\pi\)
0.545363 + 0.838200i \(0.316392\pi\)
\(660\) 0 0
\(661\) −4.00000 −0.155582 −0.0777910 0.996970i \(-0.524787\pi\)
−0.0777910 + 0.996970i \(0.524787\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) −3.00000 −0.116423
\(665\) 28.0000 1.08579
\(666\) −60.0000 −2.32495
\(667\) 12.0000 0.464642
\(668\) 1.00000 0.0386912
\(669\) −45.0000 −1.73980
\(670\) 32.0000 1.23627
\(671\) 0 0
\(672\) 15.0000 0.578638
\(673\) −19.0000 −0.732396 −0.366198 0.930537i \(-0.619341\pi\)
−0.366198 + 0.930537i \(0.619341\pi\)
\(674\) −13.0000 −0.500741
\(675\) 99.0000 3.81051
\(676\) 9.00000 0.346154
\(677\) −42.0000 −1.61419 −0.807096 0.590421i \(-0.798962\pi\)
−0.807096 + 0.590421i \(0.798962\pi\)
\(678\) 6.00000 0.230429
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −27.0000 −1.03464
\(682\) 0 0
\(683\) 8.00000 0.306111 0.153056 0.988218i \(-0.451089\pi\)
0.153056 + 0.988218i \(0.451089\pi\)
\(684\) 42.0000 1.60591
\(685\) 36.0000 1.37549
\(686\) 1.00000 0.0381802
\(687\) −18.0000 −0.686743
\(688\) 2.00000 0.0762493
\(689\) 6.00000 0.228582
\(690\) −48.0000 −1.82733
\(691\) 1.00000 0.0380418 0.0190209 0.999819i \(-0.493945\pi\)
0.0190209 + 0.999819i \(0.493945\pi\)
\(692\) 20.0000 0.760286
\(693\) 0 0
\(694\) 16.0000 0.607352
\(695\) −80.0000 −3.03457
\(696\) 27.0000 1.02343
\(697\) 0 0
\(698\) 36.0000 1.36262
\(699\) 27.0000 1.02123
\(700\) 11.0000 0.415761
\(701\) −9.00000 −0.339925 −0.169963 0.985451i \(-0.554365\pi\)
−0.169963 + 0.985451i \(0.554365\pi\)
\(702\) 18.0000 0.679366
\(703\) −70.0000 −2.64010
\(704\) 0 0
\(705\) 36.0000 1.35584
\(706\) −24.0000 −0.903252
\(707\) −2.00000 −0.0752177
\(708\) 27.0000 1.01472
\(709\) −46.0000 −1.72757 −0.863783 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) −8.00000 −0.300235
\(711\) 36.0000 1.35011
\(712\) −24.0000 −0.899438
\(713\) 28.0000 1.04861
\(714\) 0 0
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 6.00000 0.224074
\(718\) 12.0000 0.447836
\(719\) −8.00000 −0.298350 −0.149175 0.988811i \(-0.547662\pi\)
−0.149175 + 0.988811i \(0.547662\pi\)
\(720\) −24.0000 −0.894427
\(721\) 7.00000 0.260694
\(722\) −30.0000 −1.11648
\(723\) −36.0000 −1.33885
\(724\) −16.0000 −0.594635
\(725\) 33.0000 1.22559
\(726\) 33.0000 1.22474
\(727\) −20.0000 −0.741759 −0.370879 0.928681i \(-0.620944\pi\)
−0.370879 + 0.928681i \(0.620944\pi\)
\(728\) 6.00000 0.222375
\(729\) −27.0000 −1.00000
\(730\) −24.0000 −0.888280
\(731\) 0 0
\(732\) −24.0000 −0.887066
\(733\) −8.00000 −0.295487 −0.147743 0.989026i \(-0.547201\pi\)
−0.147743 + 0.989026i \(0.547201\pi\)
\(734\) 23.0000 0.848945
\(735\) 12.0000 0.442627
\(736\) −20.0000 −0.737210
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000 1.25071 0.625355 0.780340i \(-0.284954\pi\)
0.625355 + 0.780340i \(0.284954\pi\)
\(740\) −40.0000 −1.47043
\(741\) 42.0000 1.54291
\(742\) −3.00000 −0.110133
\(743\) 36.0000 1.32071 0.660356 0.750953i \(-0.270405\pi\)
0.660356 + 0.750953i \(0.270405\pi\)
\(744\) 63.0000 2.30969
\(745\) −28.0000 −1.02584
\(746\) 15.0000 0.549189
\(747\) −6.00000 −0.219529
\(748\) 0 0
\(749\) 16.0000 0.584627
\(750\) −72.0000 −2.62907
\(751\) 38.0000 1.38664 0.693320 0.720630i \(-0.256147\pi\)
0.693320 + 0.720630i \(0.256147\pi\)
\(752\) −3.00000 −0.109399
\(753\) −72.0000 −2.62383
\(754\) 6.00000 0.218507
\(755\) 8.00000 0.291150
\(756\) 9.00000 0.327327
\(757\) −5.00000 −0.181728 −0.0908640 0.995863i \(-0.528963\pi\)
−0.0908640 + 0.995863i \(0.528963\pi\)
\(758\) −4.00000 −0.145287
\(759\) 0 0
\(760\) −84.0000 −3.04700
\(761\) 38.0000 1.37750 0.688749 0.724999i \(-0.258160\pi\)
0.688749 + 0.724999i \(0.258160\pi\)
\(762\) 12.0000 0.434714
\(763\) 9.00000 0.325822
\(764\) 22.0000 0.795932
\(765\) 0 0
\(766\) −21.0000 −0.758761
\(767\) 18.0000 0.649942
\(768\) −51.0000 −1.84030
\(769\) 4.00000 0.144244 0.0721218 0.997396i \(-0.477023\pi\)
0.0721218 + 0.997396i \(0.477023\pi\)
\(770\) 0 0
\(771\) 90.0000 3.24127
\(772\) −23.0000 −0.827788
\(773\) 40.0000 1.43870 0.719350 0.694648i \(-0.244440\pi\)
0.719350 + 0.694648i \(0.244440\pi\)
\(774\) 12.0000 0.431331
\(775\) 77.0000 2.76592
\(776\) 6.00000 0.215387
\(777\) −30.0000 −1.07624
\(778\) −27.0000 −0.967997
\(779\) 0 0
\(780\) 24.0000 0.859338
\(781\) 0 0
\(782\) 0 0
\(783\) 27.0000 0.964901
\(784\) −1.00000 −0.0357143
\(785\) −56.0000 −1.99873
\(786\) −57.0000 −2.03312
\(787\) −20.0000 −0.712923 −0.356462 0.934310i \(-0.616017\pi\)
−0.356462 + 0.934310i \(0.616017\pi\)
\(788\) 15.0000 0.534353
\(789\) −48.0000 −1.70885
\(790\) −24.0000 −0.853882
\(791\) 2.00000 0.0711118
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) 24.0000 0.851728
\(795\) −36.0000 −1.27679
\(796\) −16.0000 −0.567105
\(797\) −8.00000 −0.283375 −0.141687 0.989911i \(-0.545253\pi\)
−0.141687 + 0.989911i \(0.545253\pi\)
\(798\) −21.0000 −0.743392
\(799\) 0 0
\(800\) −55.0000 −1.94454
\(801\) −48.0000 −1.69600
\(802\) −1.00000 −0.0353112
\(803\) 0 0
\(804\) 24.0000 0.846415
\(805\) −16.0000 −0.563926
\(806\) 14.0000 0.493129
\(807\) 30.0000 1.05605
\(808\) 6.00000 0.211079
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) −36.0000 −1.26491
\(811\) 49.0000 1.72062 0.860311 0.509769i \(-0.170269\pi\)
0.860311 + 0.509769i \(0.170269\pi\)
\(812\) 3.00000 0.105279
\(813\) 51.0000 1.78865
\(814\) 0 0
\(815\) 72.0000 2.52205
\(816\) 0 0
\(817\) 14.0000 0.489798
\(818\) −32.0000 −1.11885
\(819\) 12.0000 0.419314
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) −27.0000 −0.941733
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −21.0000 −0.731570
\(825\) 0 0
\(826\) −9.00000 −0.313150
\(827\) −36.0000 −1.25184 −0.625921 0.779886i \(-0.715277\pi\)
−0.625921 + 0.779886i \(0.715277\pi\)
\(828\) −24.0000 −0.834058
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 4.00000 0.138842
\(831\) 15.0000 0.520344
\(832\) −14.0000 −0.485363
\(833\) 0 0
\(834\) 60.0000 2.07763
\(835\) −4.00000 −0.138426
\(836\) 0 0
\(837\) 63.0000 2.17760
\(838\) 28.0000 0.967244
\(839\) 29.0000 1.00119 0.500596 0.865681i \(-0.333114\pi\)
0.500596 + 0.865681i \(0.333114\pi\)
\(840\) −36.0000 −1.24212
\(841\) −20.0000 −0.689655
\(842\) −13.0000 −0.448010
\(843\) 54.0000 1.85986
\(844\) 10.0000 0.344214
\(845\) −36.0000 −1.23844
\(846\) −18.0000 −0.618853
\(847\) 11.0000 0.377964
\(848\) 3.00000 0.103020
\(849\) 15.0000 0.514799
\(850\) 0 0
\(851\) 40.0000 1.37118
\(852\) −6.00000 −0.205557
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 8.00000 0.273754
\(855\) −168.000 −5.74548
\(856\) −48.0000 −1.64061
\(857\) 16.0000 0.546550 0.273275 0.961936i \(-0.411893\pi\)
0.273275 + 0.961936i \(0.411893\pi\)
\(858\) 0 0
\(859\) 43.0000 1.46714 0.733571 0.679613i \(-0.237852\pi\)
0.733571 + 0.679613i \(0.237852\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) −20.0000 −0.680808 −0.340404 0.940279i \(-0.610564\pi\)
−0.340404 + 0.940279i \(0.610564\pi\)
\(864\) −45.0000 −1.53093
\(865\) −80.0000 −2.72008
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 7.00000 0.237595
\(869\) 0 0
\(870\) −36.0000 −1.22051
\(871\) 16.0000 0.542139
\(872\) −27.0000 −0.914335
\(873\) 12.0000 0.406138
\(874\) 28.0000 0.947114
\(875\) −24.0000 −0.811348
\(876\) −18.0000 −0.608164
\(877\) −30.0000 −1.01303 −0.506514 0.862232i \(-0.669066\pi\)
−0.506514 + 0.862232i \(0.669066\pi\)
\(878\) −1.00000 −0.0337484
\(879\) 36.0000 1.21425
\(880\) 0 0
\(881\) 50.0000 1.68454 0.842271 0.539054i \(-0.181218\pi\)
0.842271 + 0.539054i \(0.181218\pi\)
\(882\) −6.00000 −0.202031
\(883\) −24.0000 −0.807664 −0.403832 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(884\) 0 0
\(885\) −108.000 −3.63038
\(886\) 26.0000 0.873487
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 90.0000 3.02020
\(889\) 4.00000 0.134156
\(890\) 32.0000 1.07264
\(891\) 0 0
\(892\) 15.0000 0.502237
\(893\) −21.0000 −0.702738
\(894\) 21.0000 0.702345
\(895\) −24.0000 −0.802232
\(896\) −3.00000 −0.100223
\(897\) −24.0000 −0.801337
\(898\) −7.00000 −0.233593
\(899\) 21.0000 0.700389
\(900\) −66.0000 −2.20000
\(901\) 0 0
\(902\) 0 0
\(903\) 6.00000 0.199667
\(904\) −6.00000 −0.199557
\(905\) 64.0000 2.12743
\(906\) −6.00000 −0.199337
\(907\) 26.0000 0.863316 0.431658 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) 9.00000 0.298675
\(909\) 12.0000 0.398015
\(910\) −8.00000 −0.265197
\(911\) −34.0000 −1.12647 −0.563235 0.826297i \(-0.690443\pi\)
−0.563235 + 0.826297i \(0.690443\pi\)
\(912\) 21.0000 0.695379
\(913\) 0 0
\(914\) −3.00000 −0.0992312
\(915\) 96.0000 3.17366
\(916\) 6.00000 0.198246
\(917\) −19.0000 −0.627435
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 48.0000 1.58251
\(921\) 45.0000 1.48280
\(922\) 14.0000 0.461065
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) 110.000 3.61678
\(926\) 40.0000 1.31448
\(927\) −42.0000 −1.37946
\(928\) −15.0000 −0.492399
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) −84.0000 −2.75447
\(931\) −7.00000 −0.229416
\(932\) −9.00000 −0.294805
\(933\) 93.0000 3.04468
\(934\) 12.0000 0.392652
\(935\) 0 0
\(936\) −36.0000 −1.17670
\(937\) 28.0000 0.914720 0.457360 0.889282i \(-0.348795\pi\)
0.457360 + 0.889282i \(0.348795\pi\)
\(938\) −8.00000 −0.261209
\(939\) −30.0000 −0.979013
\(940\) −12.0000 −0.391397
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 42.0000 1.36843
\(943\) 0 0
\(944\) 9.00000 0.292925
\(945\) −36.0000 −1.17108
\(946\) 0 0
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) −18.0000 −0.584613
\(949\) −12.0000 −0.389536
\(950\) 77.0000 2.49821
\(951\) −93.0000 −3.01573
\(952\) 0 0
\(953\) 11.0000 0.356325 0.178162 0.984001i \(-0.442985\pi\)
0.178162 + 0.984001i \(0.442985\pi\)
\(954\) 18.0000 0.582772
\(955\) −88.0000 −2.84761
\(956\) −2.00000 −0.0646846
\(957\) 0 0
\(958\) −36.0000 −1.16311
\(959\) −9.00000 −0.290625
\(960\) 84.0000 2.71109
\(961\) 18.0000 0.580645
\(962\) 20.0000 0.644826
\(963\) −96.0000 −3.09356
\(964\) 12.0000 0.386494
\(965\) 92.0000 2.96158
\(966\) 12.0000 0.386094
\(967\) 54.0000 1.73652 0.868261 0.496107i \(-0.165238\pi\)
0.868261 + 0.496107i \(0.165238\pi\)
\(968\) −33.0000 −1.06066
\(969\) 0 0
\(970\) −8.00000 −0.256865
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 10.0000 0.320421
\(975\) −66.0000 −2.11369
\(976\) −8.00000 −0.256074
\(977\) 21.0000 0.671850 0.335925 0.941889i \(-0.390951\pi\)
0.335925 + 0.941889i \(0.390951\pi\)
\(978\) −54.0000 −1.72673
\(979\) 0 0
\(980\) −4.00000 −0.127775
\(981\) −54.0000 −1.72409
\(982\) 22.0000 0.702048
\(983\) −9.00000 −0.287055 −0.143528 0.989646i \(-0.545845\pi\)
−0.143528 + 0.989646i \(0.545845\pi\)
\(984\) 0 0
\(985\) −60.0000 −1.91176
\(986\) 0 0
\(987\) −9.00000 −0.286473
\(988\) −14.0000 −0.445399
\(989\) −8.00000 −0.254385
\(990\) 0 0
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) −35.0000 −1.11125
\(993\) 12.0000 0.380808
\(994\) 2.00000 0.0634361
\(995\) 64.0000 2.02894
\(996\) 3.00000 0.0950586
\(997\) 46.0000 1.45683 0.728417 0.685134i \(-0.240256\pi\)
0.728417 + 0.685134i \(0.240256\pi\)
\(998\) −2.00000 −0.0633089
\(999\) 90.0000 2.84747
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 2023.2.a.b.1.1 yes 1
17.16 even 2 2023.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2023.2.a.a.1.1 1 17.16 even 2
2023.2.a.b.1.1 yes 1 1.1 even 1 trivial