Properties

Label 6762.2.a.r.1.1
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{12} -4.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} -6.00000 q^{19} +2.00000 q^{20} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{30} +10.0000 q^{31} -1.00000 q^{32} +1.00000 q^{36} -6.00000 q^{37} +6.00000 q^{38} -4.00000 q^{39} -2.00000 q^{40} +2.00000 q^{41} +12.0000 q^{43} +2.00000 q^{45} +1.00000 q^{46} -10.0000 q^{47} +1.00000 q^{48} +1.00000 q^{50} -4.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} -6.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} +2.00000 q^{60} +14.0000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -8.00000 q^{65} -12.0000 q^{67} -1.00000 q^{69} -12.0000 q^{71} -1.00000 q^{72} -14.0000 q^{73} +6.00000 q^{74} -1.00000 q^{75} -6.00000 q^{76} +4.00000 q^{78} +2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -2.00000 q^{83} -12.0000 q^{86} -6.00000 q^{87} +12.0000 q^{89} -2.00000 q^{90} -1.00000 q^{92} +10.0000 q^{93} +10.0000 q^{94} -12.0000 q^{95} -1.00000 q^{96} -12.0000 q^{97} +O(q^{100})\)

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 \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 4.00000 0.784465
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) −2.00000 −0.365148
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 6.00000 0.973329
\(39\) −4.00000 −0.640513
\(40\) −2.00000 −0.316228
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 12.0000 1.82998 0.914991 0.403473i \(-0.132197\pi\)
0.914991 + 0.403473i \(0.132197\pi\)
\(44\) 0 0
\(45\) 2.00000 0.298142
\(46\) 1.00000 0.147442
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −6.00000 −0.794719
\(58\) 6.00000 0.787839
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 2.00000 0.258199
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −10.0000 −1.27000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 6.00000 0.697486
\(75\) −1.00000 −0.115470
\(76\) −6.00000 −0.688247
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −2.00000 −0.219529 −0.109764 0.993958i \(-0.535010\pi\)
−0.109764 + 0.993958i \(0.535010\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −12.0000 −1.29399
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 10.0000 1.03695
\(94\) 10.0000 1.03142
\(95\) −12.0000 −1.23117
\(96\) −1.00000 −0.102062
\(97\) −12.0000 −1.21842 −0.609208 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 6.00000 0.561951
\(115\) −2.00000 −0.186501
\(116\) −6.00000 −0.557086
\(117\) −4.00000 −0.369800
\(118\) −12.0000 −1.10469
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −11.0000 −1.00000
\(122\) −14.0000 −1.26750
\(123\) 2.00000 0.180334
\(124\) 10.0000 0.898027
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000 1.05654
\(130\) 8.00000 0.701646
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) 2.00000 0.172133
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 1.00000 0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −10.0000 −0.842152
\(142\) 12.0000 1.00702
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −12.0000 −0.996546
\(146\) 14.0000 1.15865
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) 18.0000 1.47462 0.737309 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 1.00000 0.0816497
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 6.00000 0.486664
\(153\) 0 0
\(154\) 0 0
\(155\) 20.0000 1.60644
\(156\) −4.00000 −0.320256
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) −10.0000 −0.793052
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 12.0000 0.914991
\(173\) 24.0000 1.82469 0.912343 0.409426i \(-0.134271\pi\)
0.912343 + 0.409426i \(0.134271\pi\)
\(174\) 6.00000 0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) −12.0000 −0.899438
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 2.00000 0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 1.00000 0.0737210
\(185\) −12.0000 −0.882258
\(186\) −10.0000 −0.733236
\(187\) 0 0
\(188\) −10.0000 −0.729325
\(189\) 0 0
\(190\) 12.0000 0.870572
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 12.0000 0.861550
\(195\) −8.00000 −0.572892
\(196\) 0 0
\(197\) 6.00000 0.427482 0.213741 0.976890i \(-0.431435\pi\)
0.213741 + 0.976890i \(0.431435\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 1.00000 0.0707107
\(201\) −12.0000 −0.846415
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) −4.00000 −0.278693
\(207\) −1.00000 −0.0695048
\(208\) −4.00000 −0.277350
\(209\) 0 0
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) −10.0000 −0.686803
\(213\) −12.0000 −0.822226
\(214\) −8.00000 −0.546869
\(215\) 24.0000 1.63679
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 10.0000 0.677285
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 0 0
\(222\) 6.00000 0.402694
\(223\) −18.0000 −1.20537 −0.602685 0.797980i \(-0.705902\pi\)
−0.602685 + 0.797980i \(0.705902\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 14.0000 0.931266
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) −6.00000 −0.397360
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 2.00000 0.131876
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 4.00000 0.261488
\(235\) −20.0000 −1.30466
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 2.00000 0.129099
\(241\) −24.0000 −1.54598 −0.772988 0.634421i \(-0.781239\pi\)
−0.772988 + 0.634421i \(0.781239\pi\)
\(242\) 11.0000 0.707107
\(243\) 1.00000 0.0641500
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) 24.0000 1.52708
\(248\) −10.0000 −0.635001
\(249\) −2.00000 −0.126745
\(250\) 12.0000 0.758947
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 12.0000 0.752947
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) −12.0000 −0.747087
\(259\) 0 0
\(260\) −8.00000 −0.496139
\(261\) −6.00000 −0.371391
\(262\) 4.00000 0.247121
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) −20.0000 −1.22859
\(266\) 0 0
\(267\) 12.0000 0.734388
\(268\) −12.0000 −0.733017
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) −2.00000 −0.121716
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −2.00000 −0.119310 −0.0596550 0.998219i \(-0.519000\pi\)
−0.0596550 + 0.998219i \(0.519000\pi\)
\(282\) 10.0000 0.595491
\(283\) 18.0000 1.06999 0.534994 0.844856i \(-0.320314\pi\)
0.534994 + 0.844856i \(0.320314\pi\)
\(284\) −12.0000 −0.712069
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −17.0000 −1.00000
\(290\) 12.0000 0.704664
\(291\) −12.0000 −0.703452
\(292\) −14.0000 −0.819288
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 24.0000 1.39733
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 4.00000 0.231326
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) −12.0000 −0.690522
\(303\) 0 0
\(304\) −6.00000 −0.344124
\(305\) 28.0000 1.60328
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) −20.0000 −1.13592
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 4.00000 0.226455
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 10.0000 0.560772
\(319\) 0 0
\(320\) 2.00000 0.111803
\(321\) 8.00000 0.446516
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) 4.00000 0.221880
\(326\) 12.0000 0.664619
\(327\) −10.0000 −0.553001
\(328\) −2.00000 −0.110432
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) −2.00000 −0.109764
\(333\) −6.00000 −0.328798
\(334\) 14.0000 0.766046
\(335\) −24.0000 −1.31126
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) −3.00000 −0.163178
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) −12.0000 −0.646997
\(345\) −2.00000 −0.107676
\(346\) −24.0000 −1.29025
\(347\) 4.00000 0.214731 0.107366 0.994220i \(-0.465758\pi\)
0.107366 + 0.994220i \(0.465758\pi\)
\(348\) −6.00000 −0.321634
\(349\) 16.0000 0.856460 0.428230 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) −12.0000 −0.637793
\(355\) −24.0000 −1.27379
\(356\) 12.0000 0.635999
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −2.00000 −0.105409
\(361\) 17.0000 0.894737
\(362\) 2.00000 0.105118
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) −28.0000 −1.46559
\(366\) −14.0000 −0.731792
\(367\) 20.0000 1.04399 0.521996 0.852948i \(-0.325188\pi\)
0.521996 + 0.852948i \(0.325188\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 2.00000 0.104116
\(370\) 12.0000 0.623850
\(371\) 0 0
\(372\) 10.0000 0.518476
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) −12.0000 −0.619677
\(376\) 10.0000 0.515711
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) −12.0000 −0.615587
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) −16.0000 −0.817562 −0.408781 0.912633i \(-0.634046\pi\)
−0.408781 + 0.912633i \(0.634046\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 12.0000 0.609994
\(388\) −12.0000 −0.609208
\(389\) −18.0000 −0.912636 −0.456318 0.889817i \(-0.650832\pi\)
−0.456318 + 0.889817i \(0.650832\pi\)
\(390\) 8.00000 0.405096
\(391\) 0 0
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) 12.0000 0.602263 0.301131 0.953583i \(-0.402636\pi\)
0.301131 + 0.953583i \(0.402636\pi\)
\(398\) 20.0000 1.00251
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 12.0000 0.598506
\(403\) −40.0000 −1.99254
\(404\) 0 0
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) −4.00000 −0.197546
\(411\) −18.0000 −0.887875
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) −4.00000 −0.196352
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 0 0
\(419\) 14.0000 0.683945 0.341972 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) 28.0000 1.36302
\(423\) −10.0000 −0.486217
\(424\) 10.0000 0.485643
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 0 0
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 1.00000 0.0481125
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) −12.0000 −0.575356
\(436\) −10.0000 −0.478913
\(437\) 6.00000 0.287019
\(438\) 14.0000 0.668946
\(439\) 22.0000 1.05000 0.525001 0.851101i \(-0.324065\pi\)
0.525001 + 0.851101i \(0.324065\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 36.0000 1.71041 0.855206 0.518289i \(-0.173431\pi\)
0.855206 + 0.518289i \(0.173431\pi\)
\(444\) −6.00000 −0.284747
\(445\) 24.0000 1.13771
\(446\) 18.0000 0.852325
\(447\) 18.0000 0.851371
\(448\) 0 0
\(449\) 14.0000 0.660701 0.330350 0.943858i \(-0.392833\pi\)
0.330350 + 0.943858i \(0.392833\pi\)
\(450\) 1.00000 0.0471405
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) 12.0000 0.563809
\(454\) 14.0000 0.657053
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) 26.0000 1.21623 0.608114 0.793849i \(-0.291926\pi\)
0.608114 + 0.793849i \(0.291926\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) −2.00000 −0.0932505
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −6.00000 −0.278543
\(465\) 20.0000 0.927478
\(466\) −22.0000 −1.01913
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) −4.00000 −0.184900
\(469\) 0 0
\(470\) 20.0000 0.922531
\(471\) 2.00000 0.0921551
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) −20.0000 −0.914779
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 24.0000 1.09431
\(482\) 24.0000 1.09317
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −24.0000 −1.08978
\(486\) −1.00000 −0.0453609
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) −14.0000 −0.633750
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) 2.00000 0.0901670
\(493\) 0 0
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 0 0
\(498\) 2.00000 0.0896221
\(499\) −44.0000 −1.96971 −0.984855 0.173379i \(-0.944532\pi\)
−0.984855 + 0.173379i \(0.944532\pi\)
\(500\) −12.0000 −0.536656
\(501\) −14.0000 −0.625474
\(502\) 18.0000 0.803379
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) −12.0000 −0.532414
\(509\) 16.0000 0.709188 0.354594 0.935020i \(-0.384619\pi\)
0.354594 + 0.935020i \(0.384619\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) −6.00000 −0.264906
\(514\) −6.00000 −0.264649
\(515\) 8.00000 0.352522
\(516\) 12.0000 0.528271
\(517\) 0 0
\(518\) 0 0
\(519\) 24.0000 1.05348
\(520\) 8.00000 0.350823
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 6.00000 0.262613
\(523\) −38.0000 −1.66162 −0.830812 0.556553i \(-0.812124\pi\)
−0.830812 + 0.556553i \(0.812124\pi\)
\(524\) −4.00000 −0.174741
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 20.0000 0.868744
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) −12.0000 −0.519291
\(535\) 16.0000 0.691740
\(536\) 12.0000 0.518321
\(537\) −4.00000 −0.172613
\(538\) −24.0000 −1.03471
\(539\) 0 0
\(540\) 2.00000 0.0860663
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) −2.00000 −0.0859074
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) −20.0000 −0.856706
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −18.0000 −0.768922
\(549\) 14.0000 0.597505
\(550\) 0 0
\(551\) 36.0000 1.53365
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) −12.0000 −0.509372
\(556\) 0 0
\(557\) 18.0000 0.762684 0.381342 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) −10.0000 −0.423334
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) 0 0
\(562\) 2.00000 0.0843649
\(563\) 22.0000 0.927189 0.463595 0.886047i \(-0.346559\pi\)
0.463595 + 0.886047i \(0.346559\pi\)
\(564\) −10.0000 −0.421076
\(565\) −28.0000 −1.17797
\(566\) −18.0000 −0.756596
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 46.0000 1.92842 0.964210 0.265139i \(-0.0854179\pi\)
0.964210 + 0.265139i \(0.0854179\pi\)
\(570\) 12.0000 0.502625
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 1.00000 0.0416667
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 17.0000 0.707107
\(579\) −10.0000 −0.415586
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) 12.0000 0.497416
\(583\) 0 0
\(584\) 14.0000 0.579324
\(585\) −8.00000 −0.330759
\(586\) −18.0000 −0.743573
\(587\) −16.0000 −0.660391 −0.330195 0.943913i \(-0.607115\pi\)
−0.330195 + 0.943913i \(0.607115\pi\)
\(588\) 0 0
\(589\) −60.0000 −2.47226
\(590\) −24.0000 −0.988064
\(591\) 6.00000 0.246807
\(592\) −6.00000 −0.246598
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 18.0000 0.737309
\(597\) −20.0000 −0.818546
\(598\) −4.00000 −0.163572
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 1.00000 0.0408248
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 12.0000 0.488273
\(605\) −22.0000 −0.894427
\(606\) 0 0
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) 6.00000 0.243332
\(609\) 0 0
\(610\) −28.0000 −1.13369
\(611\) 40.0000 1.61823
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 16.0000 0.645707
\(615\) 4.00000 0.161296
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −4.00000 −0.160904
\(619\) 6.00000 0.241160 0.120580 0.992704i \(-0.461525\pi\)
0.120580 + 0.992704i \(0.461525\pi\)
\(620\) 20.0000 0.803219
\(621\) −1.00000 −0.0401286
\(622\) 10.0000 0.400963
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) −19.0000 −0.760000
\(626\) −8.00000 −0.319744
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 0 0
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) −28.0000 −1.11290
\(634\) 2.00000 0.0794301
\(635\) −24.0000 −0.952411
\(636\) −10.0000 −0.396526
\(637\) 0 0
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) −2.00000 −0.0790569
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) −8.00000 −0.315735
\(643\) 46.0000 1.81406 0.907031 0.421063i \(-0.138343\pi\)
0.907031 + 0.421063i \(0.138343\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) −38.0000 −1.49393 −0.746967 0.664861i \(-0.768491\pi\)
−0.746967 + 0.664861i \(0.768491\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 0 0
\(650\) −4.00000 −0.156893
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −46.0000 −1.80012 −0.900060 0.435767i \(-0.856477\pi\)
−0.900060 + 0.435767i \(0.856477\pi\)
\(654\) 10.0000 0.391031
\(655\) −8.00000 −0.312586
\(656\) 2.00000 0.0780869
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 20.0000 0.777322
\(663\) 0 0
\(664\) 2.00000 0.0776151
\(665\) 0 0
\(666\) 6.00000 0.232495
\(667\) 6.00000 0.232321
\(668\) −14.0000 −0.541676
\(669\) −18.0000 −0.695920
\(670\) 24.0000 0.927201
\(671\) 0 0
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) −14.0000 −0.539260
\(675\) −1.00000 −0.0384900
\(676\) 3.00000 0.115385
\(677\) −30.0000 −1.15299 −0.576497 0.817099i \(-0.695581\pi\)
−0.576497 + 0.817099i \(0.695581\pi\)
\(678\) 14.0000 0.537667
\(679\) 0 0
\(680\) 0 0
\(681\) −14.0000 −0.536481
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −6.00000 −0.229416
\(685\) −36.0000 −1.37549
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 12.0000 0.457496
\(689\) 40.0000 1.52388
\(690\) 2.00000 0.0761387
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 24.0000 0.912343
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) 0 0
\(698\) −16.0000 −0.605609
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 4.00000 0.150970
\(703\) 36.0000 1.35777
\(704\) 0 0
\(705\) −20.0000 −0.753244
\(706\) −26.0000 −0.978523
\(707\) 0 0
\(708\) 12.0000 0.450988
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 24.0000 0.900704
\(711\) 0 0
\(712\) −12.0000 −0.449719
\(713\) −10.0000 −0.374503
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 20.0000 0.746914
\(718\) 0 0
\(719\) 18.0000 0.671287 0.335643 0.941989i \(-0.391046\pi\)
0.335643 + 0.941989i \(0.391046\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) −17.0000 −0.632674
\(723\) −24.0000 −0.892570
\(724\) −2.00000 −0.0743294
\(725\) 6.00000 0.222834
\(726\) 11.0000 0.408248
\(727\) −36.0000 −1.33517 −0.667583 0.744535i \(-0.732671\pi\)
−0.667583 + 0.744535i \(0.732671\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 28.0000 1.03633
\(731\) 0 0
\(732\) 14.0000 0.517455
\(733\) −6.00000 −0.221615 −0.110808 0.993842i \(-0.535344\pi\)
−0.110808 + 0.993842i \(0.535344\pi\)
\(734\) −20.0000 −0.738213
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) −2.00000 −0.0736210
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) −12.0000 −0.441129
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −10.0000 −0.366618
\(745\) 36.0000 1.31894
\(746\) 14.0000 0.512576
\(747\) −2.00000 −0.0731762
\(748\) 0 0
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) −10.0000 −0.364662
\(753\) −18.0000 −0.655956
\(754\) −24.0000 −0.874028
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) −24.0000 −0.871719
\(759\) 0 0
\(760\) 12.0000 0.435286
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 12.0000 0.434714
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) −48.0000 −1.73318
\(768\) 1.00000 0.0360844
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) −10.0000 −0.359908
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) −12.0000 −0.431331
\(775\) −10.0000 −0.359211
\(776\) 12.0000 0.430775
\(777\) 0 0
\(778\) 18.0000 0.645331
\(779\) −12.0000 −0.429945
\(780\) −8.00000 −0.286446
\(781\) 0 0
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 4.00000 0.142675
\(787\) −2.00000 −0.0712923 −0.0356462 0.999364i \(-0.511349\pi\)
−0.0356462 + 0.999364i \(0.511349\pi\)
\(788\) 6.00000 0.213741
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −56.0000 −1.98862
\(794\) −12.0000 −0.425864
\(795\) −20.0000 −0.709327
\(796\) −20.0000 −0.708881
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 1.00000 0.0353553
\(801\) 12.0000 0.423999
\(802\) −22.0000 −0.776847
\(803\) 0 0
\(804\) −12.0000 −0.423207
\(805\) 0 0
\(806\) 40.0000 1.40894
\(807\) 24.0000 0.844840
\(808\) 0 0
\(809\) −2.00000 −0.0703163 −0.0351581 0.999382i \(-0.511193\pi\)
−0.0351581 + 0.999382i \(0.511193\pi\)
\(810\) −2.00000 −0.0702728
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 0 0
\(813\) 2.00000 0.0701431
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −72.0000 −2.51896
\(818\) −22.0000 −0.769212
\(819\) 0 0
\(820\) 4.00000 0.139686
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 18.0000 0.627822
\(823\) 36.0000 1.25488 0.627441 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 12.0000 0.416777 0.208389 0.978046i \(-0.433178\pi\)
0.208389 + 0.978046i \(0.433178\pi\)
\(830\) 4.00000 0.138842
\(831\) −10.0000 −0.346896
\(832\) −4.00000 −0.138675
\(833\) 0 0
\(834\) 0 0
\(835\) −28.0000 −0.968980
\(836\) 0 0
\(837\) 10.0000 0.345651
\(838\) −14.0000 −0.483622
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) −2.00000 −0.0688837
\(844\) −28.0000 −0.963800
\(845\) 6.00000 0.206406
\(846\) 10.0000 0.343807
\(847\) 0 0
\(848\) −10.0000 −0.343401
\(849\) 18.0000 0.617758
\(850\) 0 0
\(851\) 6.00000 0.205677
\(852\) −12.0000 −0.411113
\(853\) 12.0000 0.410872 0.205436 0.978671i \(-0.434139\pi\)
0.205436 + 0.978671i \(0.434139\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) −8.00000 −0.273434
\(857\) −38.0000 −1.29806 −0.649028 0.760765i \(-0.724824\pi\)
−0.649028 + 0.760765i \(0.724824\pi\)
\(858\) 0 0
\(859\) −56.0000 −1.91070 −0.955348 0.295484i \(-0.904519\pi\)
−0.955348 + 0.295484i \(0.904519\pi\)
\(860\) 24.0000 0.818393
\(861\) 0 0
\(862\) 24.0000 0.817443
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 48.0000 1.63205
\(866\) 4.00000 0.135926
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) 0 0
\(870\) 12.0000 0.406838
\(871\) 48.0000 1.62642
\(872\) 10.0000 0.338643
\(873\) −12.0000 −0.406138
\(874\) −6.00000 −0.202953
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) 10.0000 0.337676 0.168838 0.985644i \(-0.445999\pi\)
0.168838 + 0.985644i \(0.445999\pi\)
\(878\) −22.0000 −0.742464
\(879\) 18.0000 0.607125
\(880\) 0 0
\(881\) 32.0000 1.07811 0.539054 0.842271i \(-0.318782\pi\)
0.539054 + 0.842271i \(0.318782\pi\)
\(882\) 0 0
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) 0 0
\(885\) 24.0000 0.806751
\(886\) −36.0000 −1.20944
\(887\) 10.0000 0.335767 0.167884 0.985807i \(-0.446307\pi\)
0.167884 + 0.985807i \(0.446307\pi\)
\(888\) 6.00000 0.201347
\(889\) 0 0
\(890\) −24.0000 −0.804482
\(891\) 0 0
\(892\) −18.0000 −0.602685
\(893\) 60.0000 2.00782
\(894\) −18.0000 −0.602010
\(895\) −8.00000 −0.267411
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) −14.0000 −0.467186
\(899\) −60.0000 −2.00111
\(900\) −1.00000 −0.0333333
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 14.0000 0.465633
\(905\) −4.00000 −0.132964
\(906\) −12.0000 −0.398673
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) −14.0000 −0.464606
\(909\) 0 0
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) −6.00000 −0.198680
\(913\) 0 0
\(914\) −26.0000 −0.860004
\(915\) 28.0000 0.925651
\(916\) 6.00000 0.198246
\(917\) 0 0
\(918\) 0 0
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 2.00000 0.0659380
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) 48.0000 1.57994
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 16.0000 0.525793
\(927\) 4.00000 0.131377
\(928\) 6.00000 0.196960
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) −20.0000 −0.655826
\(931\) 0 0
\(932\) 22.0000 0.720634
\(933\) −10.0000 −0.327385
\(934\) 6.00000 0.196326
\(935\) 0 0
\(936\) 4.00000 0.130744
\(937\) −56.0000 −1.82944 −0.914720 0.404088i \(-0.867589\pi\)
−0.914720 + 0.404088i \(0.867589\pi\)
\(938\) 0 0
\(939\) 8.00000 0.261070
\(940\) −20.0000 −0.652328
\(941\) −6.00000 −0.195594 −0.0977972 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) −2.00000 −0.0651635
\(943\) −2.00000 −0.0651290
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 0 0
\(947\) 20.0000 0.649913 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) 0 0
\(949\) 56.0000 1.81784
\(950\) −6.00000 −0.194666
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) 36.0000 1.16311
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) 69.0000 2.22581
\(962\) −24.0000 −0.773791
\(963\) 8.00000 0.257796
\(964\) −24.0000 −0.772988
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 11.0000 0.353553
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) 22.0000 0.706014 0.353007 0.935621i \(-0.385159\pi\)
0.353007 + 0.935621i \(0.385159\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 4.00000 0.128103
\(976\) 14.0000 0.448129
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 12.0000 0.383718
\(979\) 0 0
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 20.0000 0.638226
\(983\) 32.0000 1.02064 0.510321 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 12.0000 0.382352
\(986\) 0 0
\(987\) 0 0
\(988\) 24.0000 0.763542
\(989\) −12.0000 −0.381578
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) −10.0000 −0.317500
\(993\) −20.0000 −0.634681
\(994\) 0 0
\(995\) −40.0000 −1.26809
\(996\) −2.00000 −0.0633724
\(997\) −28.0000 −0.886769 −0.443384 0.896332i \(-0.646222\pi\)
−0.443384 + 0.896332i \(0.646222\pi\)
\(998\) 44.0000 1.39280
\(999\) −6.00000 −0.189832
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 6762.2.a.r.1.1 1
7.6 odd 2 966.2.a.b.1.1 1
21.20 even 2 2898.2.a.r.1.1 1
28.27 even 2 7728.2.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.a.b.1.1 1 7.6 odd 2
2898.2.a.r.1.1 1 21.20 even 2
6762.2.a.r.1.1 1 1.1 even 1 trivial
7728.2.a.p.1.1 1 28.27 even 2