Properties

Label 7623.2.a.bn.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of \(x^{2} - x - 1\)
Character \(\chi\) \(=\) 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} -2.23607 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-0.618034 q^{2} -1.61803 q^{4} -2.23607 q^{5} -1.00000 q^{7} +2.23607 q^{8} +1.38197 q^{10} +4.23607 q^{13} +0.618034 q^{14} +1.85410 q^{16} +2.00000 q^{17} -7.47214 q^{19} +3.61803 q^{20} +2.00000 q^{23} -2.61803 q^{26} +1.61803 q^{28} +5.00000 q^{29} +1.52786 q^{31} -5.61803 q^{32} -1.23607 q^{34} +2.23607 q^{35} +7.47214 q^{37} +4.61803 q^{38} -5.00000 q^{40} -6.47214 q^{41} -0.472136 q^{43} -1.23607 q^{46} -0.527864 q^{47} +1.00000 q^{49} -6.85410 q^{52} +2.47214 q^{53} -2.23607 q^{56} -3.09017 q^{58} +9.94427 q^{59} -3.52786 q^{61} -0.944272 q^{62} -0.236068 q^{64} -9.47214 q^{65} -13.1803 q^{67} -3.23607 q^{68} -1.38197 q^{70} -6.47214 q^{71} -2.70820 q^{73} -4.61803 q^{74} +12.0902 q^{76} -8.47214 q^{79} -4.14590 q^{80} +4.00000 q^{82} +16.4721 q^{83} -4.47214 q^{85} +0.291796 q^{86} -8.47214 q^{89} -4.23607 q^{91} -3.23607 q^{92} +0.326238 q^{94} +16.7082 q^{95} -17.4164 q^{97} -0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{4} - 2q^{7} + O(q^{10}) \) \( 2q + q^{2} - q^{4} - 2q^{7} + 5q^{10} + 4q^{13} - q^{14} - 3q^{16} + 4q^{17} - 6q^{19} + 5q^{20} + 4q^{23} - 3q^{26} + q^{28} + 10q^{29} + 12q^{31} - 9q^{32} + 2q^{34} + 6q^{37} + 7q^{38} - 10q^{40} - 4q^{41} + 8q^{43} + 2q^{46} - 10q^{47} + 2q^{49} - 7q^{52} - 4q^{53} + 5q^{58} + 2q^{59} - 16q^{61} + 16q^{62} + 4q^{64} - 10q^{65} - 4q^{67} - 2q^{68} - 5q^{70} - 4q^{71} + 8q^{73} - 7q^{74} + 13q^{76} - 8q^{79} - 15q^{80} + 8q^{82} + 24q^{83} + 14q^{86} - 8q^{89} - 4q^{91} - 2q^{92} - 15q^{94} + 20q^{95} - 8q^{97} + q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) −2.23607 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 1.38197 0.437016
\(11\) 0 0
\(12\) 0 0
\(13\) 4.23607 1.17487 0.587437 0.809270i \(-0.300137\pi\)
0.587437 + 0.809270i \(0.300137\pi\)
\(14\) 0.618034 0.165177
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −7.47214 −1.71423 −0.857113 0.515129i \(-0.827744\pi\)
−0.857113 + 0.515129i \(0.827744\pi\)
\(20\) 3.61803 0.809017
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.61803 −0.513439
\(27\) 0 0
\(28\) 1.61803 0.305780
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) 1.52786 0.274412 0.137206 0.990543i \(-0.456188\pi\)
0.137206 + 0.990543i \(0.456188\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) −1.23607 −0.211984
\(35\) 2.23607 0.377964
\(36\) 0 0
\(37\) 7.47214 1.22841 0.614206 0.789146i \(-0.289476\pi\)
0.614206 + 0.789146i \(0.289476\pi\)
\(38\) 4.61803 0.749144
\(39\) 0 0
\(40\) −5.00000 −0.790569
\(41\) −6.47214 −1.01078 −0.505389 0.862892i \(-0.668651\pi\)
−0.505389 + 0.862892i \(0.668651\pi\)
\(42\) 0 0
\(43\) −0.472136 −0.0720001 −0.0360000 0.999352i \(-0.511462\pi\)
−0.0360000 + 0.999352i \(0.511462\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.23607 −0.182248
\(47\) −0.527864 −0.0769969 −0.0384984 0.999259i \(-0.512257\pi\)
−0.0384984 + 0.999259i \(0.512257\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −6.85410 −0.950493
\(53\) 2.47214 0.339574 0.169787 0.985481i \(-0.445692\pi\)
0.169787 + 0.985481i \(0.445692\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) −3.09017 −0.405759
\(59\) 9.94427 1.29463 0.647317 0.762221i \(-0.275891\pi\)
0.647317 + 0.762221i \(0.275891\pi\)
\(60\) 0 0
\(61\) −3.52786 −0.451697 −0.225848 0.974162i \(-0.572515\pi\)
−0.225848 + 0.974162i \(0.572515\pi\)
\(62\) −0.944272 −0.119923
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) −9.47214 −1.17487
\(66\) 0 0
\(67\) −13.1803 −1.61023 −0.805117 0.593115i \(-0.797898\pi\)
−0.805117 + 0.593115i \(0.797898\pi\)
\(68\) −3.23607 −0.392431
\(69\) 0 0
\(70\) −1.38197 −0.165177
\(71\) −6.47214 −0.768101 −0.384051 0.923312i \(-0.625471\pi\)
−0.384051 + 0.923312i \(0.625471\pi\)
\(72\) 0 0
\(73\) −2.70820 −0.316971 −0.158486 0.987361i \(-0.550661\pi\)
−0.158486 + 0.987361i \(0.550661\pi\)
\(74\) −4.61803 −0.536836
\(75\) 0 0
\(76\) 12.0902 1.38684
\(77\) 0 0
\(78\) 0 0
\(79\) −8.47214 −0.953190 −0.476595 0.879123i \(-0.658129\pi\)
−0.476595 + 0.879123i \(0.658129\pi\)
\(80\) −4.14590 −0.463525
\(81\) 0 0
\(82\) 4.00000 0.441726
\(83\) 16.4721 1.80805 0.904026 0.427478i \(-0.140598\pi\)
0.904026 + 0.427478i \(0.140598\pi\)
\(84\) 0 0
\(85\) −4.47214 −0.485071
\(86\) 0.291796 0.0314652
\(87\) 0 0
\(88\) 0 0
\(89\) −8.47214 −0.898045 −0.449022 0.893521i \(-0.648228\pi\)
−0.449022 + 0.893521i \(0.648228\pi\)
\(90\) 0 0
\(91\) −4.23607 −0.444061
\(92\) −3.23607 −0.337383
\(93\) 0 0
\(94\) 0.326238 0.0336489
\(95\) 16.7082 1.71423
\(96\) 0 0
\(97\) −17.4164 −1.76837 −0.884184 0.467139i \(-0.845285\pi\)
−0.884184 + 0.467139i \(0.845285\pi\)
\(98\) −0.618034 −0.0624309
\(99\) 0 0
\(100\) 0 0
\(101\) 18.4721 1.83805 0.919023 0.394204i \(-0.128980\pi\)
0.919023 + 0.394204i \(0.128980\pi\)
\(102\) 0 0
\(103\) −1.52786 −0.150545 −0.0752725 0.997163i \(-0.523983\pi\)
−0.0752725 + 0.997163i \(0.523983\pi\)
\(104\) 9.47214 0.928819
\(105\) 0 0
\(106\) −1.52786 −0.148399
\(107\) −11.6525 −1.12649 −0.563244 0.826291i \(-0.690447\pi\)
−0.563244 + 0.826291i \(0.690447\pi\)
\(108\) 0 0
\(109\) 1.52786 0.146343 0.0731714 0.997319i \(-0.476688\pi\)
0.0731714 + 0.997319i \(0.476688\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.85410 −0.175196
\(113\) 14.4721 1.36142 0.680712 0.732551i \(-0.261671\pi\)
0.680712 + 0.732551i \(0.261671\pi\)
\(114\) 0 0
\(115\) −4.47214 −0.417029
\(116\) −8.09017 −0.751153
\(117\) 0 0
\(118\) −6.14590 −0.565776
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 0 0
\(122\) 2.18034 0.197399
\(123\) 0 0
\(124\) −2.47214 −0.222004
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) −17.4164 −1.54546 −0.772728 0.634737i \(-0.781108\pi\)
−0.772728 + 0.634737i \(0.781108\pi\)
\(128\) 11.3820 1.00603
\(129\) 0 0
\(130\) 5.85410 0.513439
\(131\) −7.41641 −0.647975 −0.323987 0.946061i \(-0.605024\pi\)
−0.323987 + 0.946061i \(0.605024\pi\)
\(132\) 0 0
\(133\) 7.47214 0.647916
\(134\) 8.14590 0.703698
\(135\) 0 0
\(136\) 4.47214 0.383482
\(137\) −17.4164 −1.48798 −0.743992 0.668188i \(-0.767070\pi\)
−0.743992 + 0.668188i \(0.767070\pi\)
\(138\) 0 0
\(139\) −8.94427 −0.758643 −0.379322 0.925265i \(-0.623843\pi\)
−0.379322 + 0.925265i \(0.623843\pi\)
\(140\) −3.61803 −0.305780
\(141\) 0 0
\(142\) 4.00000 0.335673
\(143\) 0 0
\(144\) 0 0
\(145\) −11.1803 −0.928477
\(146\) 1.67376 0.138522
\(147\) 0 0
\(148\) −12.0902 −0.993806
\(149\) 21.0000 1.72039 0.860194 0.509968i \(-0.170343\pi\)
0.860194 + 0.509968i \(0.170343\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) −16.7082 −1.35521
\(153\) 0 0
\(154\) 0 0
\(155\) −3.41641 −0.274412
\(156\) 0 0
\(157\) 12.4721 0.995385 0.497692 0.867354i \(-0.334181\pi\)
0.497692 + 0.867354i \(0.334181\pi\)
\(158\) 5.23607 0.416559
\(159\) 0 0
\(160\) 12.5623 0.993137
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) −8.23607 −0.645099 −0.322549 0.946553i \(-0.604540\pi\)
−0.322549 + 0.946553i \(0.604540\pi\)
\(164\) 10.4721 0.817736
\(165\) 0 0
\(166\) −10.1803 −0.790148
\(167\) 21.4164 1.65725 0.828626 0.559803i \(-0.189123\pi\)
0.828626 + 0.559803i \(0.189123\pi\)
\(168\) 0 0
\(169\) 4.94427 0.380329
\(170\) 2.76393 0.211984
\(171\) 0 0
\(172\) 0.763932 0.0582493
\(173\) −16.4721 −1.25235 −0.626177 0.779681i \(-0.715381\pi\)
−0.626177 + 0.779681i \(0.715381\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 5.23607 0.392460
\(179\) −18.3607 −1.37234 −0.686171 0.727440i \(-0.740710\pi\)
−0.686171 + 0.727440i \(0.740710\pi\)
\(180\) 0 0
\(181\) −4.94427 −0.367505 −0.183752 0.982973i \(-0.558824\pi\)
−0.183752 + 0.982973i \(0.558824\pi\)
\(182\) 2.61803 0.194062
\(183\) 0 0
\(184\) 4.47214 0.329690
\(185\) −16.7082 −1.22841
\(186\) 0 0
\(187\) 0 0
\(188\) 0.854102 0.0622918
\(189\) 0 0
\(190\) −10.3262 −0.749144
\(191\) −24.4721 −1.77074 −0.885371 0.464886i \(-0.846095\pi\)
−0.885371 + 0.464886i \(0.846095\pi\)
\(192\) 0 0
\(193\) −10.4721 −0.753801 −0.376900 0.926254i \(-0.623010\pi\)
−0.376900 + 0.926254i \(0.623010\pi\)
\(194\) 10.7639 0.772805
\(195\) 0 0
\(196\) −1.61803 −0.115574
\(197\) 11.8885 0.847024 0.423512 0.905891i \(-0.360797\pi\)
0.423512 + 0.905891i \(0.360797\pi\)
\(198\) 0 0
\(199\) 23.8885 1.69341 0.846707 0.532059i \(-0.178582\pi\)
0.846707 + 0.532059i \(0.178582\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −11.4164 −0.803256
\(203\) −5.00000 −0.350931
\(204\) 0 0
\(205\) 14.4721 1.01078
\(206\) 0.944272 0.0657905
\(207\) 0 0
\(208\) 7.85410 0.544584
\(209\) 0 0
\(210\) 0 0
\(211\) 18.3607 1.26400 0.632001 0.774968i \(-0.282234\pi\)
0.632001 + 0.774968i \(0.282234\pi\)
\(212\) −4.00000 −0.274721
\(213\) 0 0
\(214\) 7.20163 0.492293
\(215\) 1.05573 0.0720001
\(216\) 0 0
\(217\) −1.52786 −0.103718
\(218\) −0.944272 −0.0639542
\(219\) 0 0
\(220\) 0 0
\(221\) 8.47214 0.569898
\(222\) 0 0
\(223\) 11.8885 0.796116 0.398058 0.917360i \(-0.369684\pi\)
0.398058 + 0.917360i \(0.369684\pi\)
\(224\) 5.61803 0.375371
\(225\) 0 0
\(226\) −8.94427 −0.594964
\(227\) 12.4721 0.827805 0.413902 0.910321i \(-0.364165\pi\)
0.413902 + 0.910321i \(0.364165\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 2.76393 0.182248
\(231\) 0 0
\(232\) 11.1803 0.734025
\(233\) −2.94427 −0.192886 −0.0964428 0.995339i \(-0.530746\pi\)
−0.0964428 + 0.995339i \(0.530746\pi\)
\(234\) 0 0
\(235\) 1.18034 0.0769969
\(236\) −16.0902 −1.04738
\(237\) 0 0
\(238\) 1.23607 0.0801224
\(239\) 14.1246 0.913645 0.456823 0.889558i \(-0.348987\pi\)
0.456823 + 0.889558i \(0.348987\pi\)
\(240\) 0 0
\(241\) −3.18034 −0.204864 −0.102432 0.994740i \(-0.532662\pi\)
−0.102432 + 0.994740i \(0.532662\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 5.70820 0.365430
\(245\) −2.23607 −0.142857
\(246\) 0 0
\(247\) −31.6525 −2.01400
\(248\) 3.41641 0.216942
\(249\) 0 0
\(250\) −6.90983 −0.437016
\(251\) 11.0000 0.694314 0.347157 0.937807i \(-0.387147\pi\)
0.347157 + 0.937807i \(0.387147\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 10.7639 0.675389
\(255\) 0 0
\(256\) −6.56231 −0.410144
\(257\) −17.6525 −1.10113 −0.550566 0.834792i \(-0.685588\pi\)
−0.550566 + 0.834792i \(0.685588\pi\)
\(258\) 0 0
\(259\) −7.47214 −0.464296
\(260\) 15.3262 0.950493
\(261\) 0 0
\(262\) 4.58359 0.283175
\(263\) 24.7082 1.52357 0.761787 0.647828i \(-0.224322\pi\)
0.761787 + 0.647828i \(0.224322\pi\)
\(264\) 0 0
\(265\) −5.52786 −0.339574
\(266\) −4.61803 −0.283150
\(267\) 0 0
\(268\) 21.3262 1.30271
\(269\) 13.4164 0.818013 0.409006 0.912532i \(-0.365875\pi\)
0.409006 + 0.912532i \(0.365875\pi\)
\(270\) 0 0
\(271\) 28.8885 1.75485 0.877427 0.479710i \(-0.159258\pi\)
0.877427 + 0.479710i \(0.159258\pi\)
\(272\) 3.70820 0.224843
\(273\) 0 0
\(274\) 10.7639 0.650273
\(275\) 0 0
\(276\) 0 0
\(277\) 14.3607 0.862850 0.431425 0.902149i \(-0.358011\pi\)
0.431425 + 0.902149i \(0.358011\pi\)
\(278\) 5.52786 0.331539
\(279\) 0 0
\(280\) 5.00000 0.298807
\(281\) 2.52786 0.150800 0.0753999 0.997153i \(-0.475977\pi\)
0.0753999 + 0.997153i \(0.475977\pi\)
\(282\) 0 0
\(283\) −8.41641 −0.500304 −0.250152 0.968207i \(-0.580481\pi\)
−0.250152 + 0.968207i \(0.580481\pi\)
\(284\) 10.4721 0.621407
\(285\) 0 0
\(286\) 0 0
\(287\) 6.47214 0.382038
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 6.90983 0.405759
\(291\) 0 0
\(292\) 4.38197 0.256435
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −22.2361 −1.29463
\(296\) 16.7082 0.971145
\(297\) 0 0
\(298\) −12.9787 −0.751837
\(299\) 8.47214 0.489956
\(300\) 0 0
\(301\) 0.472136 0.0272135
\(302\) 6.18034 0.355639
\(303\) 0 0
\(304\) −13.8541 −0.794587
\(305\) 7.88854 0.451697
\(306\) 0 0
\(307\) 5.88854 0.336077 0.168038 0.985780i \(-0.446257\pi\)
0.168038 + 0.985780i \(0.446257\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.11146 0.119923
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −2.94427 −0.166420 −0.0832100 0.996532i \(-0.526517\pi\)
−0.0832100 + 0.996532i \(0.526517\pi\)
\(314\) −7.70820 −0.434999
\(315\) 0 0
\(316\) 13.7082 0.771147
\(317\) 20.9443 1.17635 0.588174 0.808735i \(-0.299847\pi\)
0.588174 + 0.808735i \(0.299847\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.527864 0.0295085
\(321\) 0 0
\(322\) 1.23607 0.0688834
\(323\) −14.9443 −0.831522
\(324\) 0 0
\(325\) 0 0
\(326\) 5.09017 0.281918
\(327\) 0 0
\(328\) −14.4721 −0.799090
\(329\) 0.527864 0.0291021
\(330\) 0 0
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) −26.6525 −1.46274
\(333\) 0 0
\(334\) −13.2361 −0.724245
\(335\) 29.4721 1.61023
\(336\) 0 0
\(337\) 36.4721 1.98676 0.993382 0.114858i \(-0.0366413\pi\)
0.993382 + 0.114858i \(0.0366413\pi\)
\(338\) −3.05573 −0.166210
\(339\) 0 0
\(340\) 7.23607 0.392431
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −1.05573 −0.0569210
\(345\) 0 0
\(346\) 10.1803 0.547298
\(347\) 12.9443 0.694885 0.347442 0.937701i \(-0.387050\pi\)
0.347442 + 0.937701i \(0.387050\pi\)
\(348\) 0 0
\(349\) 22.1246 1.18430 0.592152 0.805827i \(-0.298279\pi\)
0.592152 + 0.805827i \(0.298279\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −17.1803 −0.914417 −0.457209 0.889359i \(-0.651151\pi\)
−0.457209 + 0.889359i \(0.651151\pi\)
\(354\) 0 0
\(355\) 14.4721 0.768101
\(356\) 13.7082 0.726533
\(357\) 0 0
\(358\) 11.3475 0.599735
\(359\) 11.0557 0.583499 0.291750 0.956495i \(-0.405763\pi\)
0.291750 + 0.956495i \(0.405763\pi\)
\(360\) 0 0
\(361\) 36.8328 1.93857
\(362\) 3.05573 0.160606
\(363\) 0 0
\(364\) 6.85410 0.359253
\(365\) 6.05573 0.316971
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 3.70820 0.193303
\(369\) 0 0
\(370\) 10.3262 0.536836
\(371\) −2.47214 −0.128347
\(372\) 0 0
\(373\) −6.00000 −0.310668 −0.155334 0.987862i \(-0.549645\pi\)
−0.155334 + 0.987862i \(0.549645\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.18034 −0.0608714
\(377\) 21.1803 1.09084
\(378\) 0 0
\(379\) 16.2361 0.833991 0.416995 0.908909i \(-0.363083\pi\)
0.416995 + 0.908909i \(0.363083\pi\)
\(380\) −27.0344 −1.38684
\(381\) 0 0
\(382\) 15.1246 0.773842
\(383\) −16.9443 −0.865812 −0.432906 0.901439i \(-0.642512\pi\)
−0.432906 + 0.901439i \(0.642512\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 6.47214 0.329423
\(387\) 0 0
\(388\) 28.1803 1.43064
\(389\) 34.4721 1.74781 0.873903 0.486100i \(-0.161581\pi\)
0.873903 + 0.486100i \(0.161581\pi\)
\(390\) 0 0
\(391\) 4.00000 0.202289
\(392\) 2.23607 0.112938
\(393\) 0 0
\(394\) −7.34752 −0.370163
\(395\) 18.9443 0.953190
\(396\) 0 0
\(397\) 9.05573 0.454494 0.227247 0.973837i \(-0.427028\pi\)
0.227247 + 0.973837i \(0.427028\pi\)
\(398\) −14.7639 −0.740049
\(399\) 0 0
\(400\) 0 0
\(401\) 26.0000 1.29838 0.649189 0.760627i \(-0.275108\pi\)
0.649189 + 0.760627i \(0.275108\pi\)
\(402\) 0 0
\(403\) 6.47214 0.322400
\(404\) −29.8885 −1.48701
\(405\) 0 0
\(406\) 3.09017 0.153363
\(407\) 0 0
\(408\) 0 0
\(409\) 16.4721 0.814495 0.407247 0.913318i \(-0.366489\pi\)
0.407247 + 0.913318i \(0.366489\pi\)
\(410\) −8.94427 −0.441726
\(411\) 0 0
\(412\) 2.47214 0.121793
\(413\) −9.94427 −0.489326
\(414\) 0 0
\(415\) −36.8328 −1.80805
\(416\) −23.7984 −1.16681
\(417\) 0 0
\(418\) 0 0
\(419\) −17.0000 −0.830504 −0.415252 0.909706i \(-0.636307\pi\)
−0.415252 + 0.909706i \(0.636307\pi\)
\(420\) 0 0
\(421\) 7.36068 0.358738 0.179369 0.983782i \(-0.442594\pi\)
0.179369 + 0.983782i \(0.442594\pi\)
\(422\) −11.3475 −0.552389
\(423\) 0 0
\(424\) 5.52786 0.268457
\(425\) 0 0
\(426\) 0 0
\(427\) 3.52786 0.170725
\(428\) 18.8541 0.911347
\(429\) 0 0
\(430\) −0.652476 −0.0314652
\(431\) 15.7639 0.759322 0.379661 0.925126i \(-0.376041\pi\)
0.379661 + 0.925126i \(0.376041\pi\)
\(432\) 0 0
\(433\) 33.4164 1.60589 0.802945 0.596053i \(-0.203265\pi\)
0.802945 + 0.596053i \(0.203265\pi\)
\(434\) 0.944272 0.0453265
\(435\) 0 0
\(436\) −2.47214 −0.118394
\(437\) −14.9443 −0.714881
\(438\) 0 0
\(439\) 8.88854 0.424227 0.212114 0.977245i \(-0.431965\pi\)
0.212114 + 0.977245i \(0.431965\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −5.23607 −0.249054
\(443\) −14.0000 −0.665160 −0.332580 0.943075i \(-0.607919\pi\)
−0.332580 + 0.943075i \(0.607919\pi\)
\(444\) 0 0
\(445\) 18.9443 0.898045
\(446\) −7.34752 −0.347915
\(447\) 0 0
\(448\) 0.236068 0.0111532
\(449\) 8.47214 0.399825 0.199912 0.979814i \(-0.435934\pi\)
0.199912 + 0.979814i \(0.435934\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −23.4164 −1.10142
\(453\) 0 0
\(454\) −7.70820 −0.361764
\(455\) 9.47214 0.444061
\(456\) 0 0
\(457\) 38.3607 1.79444 0.897218 0.441587i \(-0.145584\pi\)
0.897218 + 0.441587i \(0.145584\pi\)
\(458\) −8.65248 −0.404304
\(459\) 0 0
\(460\) 7.23607 0.337383
\(461\) 1.05573 0.0491702 0.0245851 0.999698i \(-0.492174\pi\)
0.0245851 + 0.999698i \(0.492174\pi\)
\(462\) 0 0
\(463\) −6.70820 −0.311757 −0.155878 0.987776i \(-0.549821\pi\)
−0.155878 + 0.987776i \(0.549821\pi\)
\(464\) 9.27051 0.430373
\(465\) 0 0
\(466\) 1.81966 0.0842941
\(467\) −27.9443 −1.29311 −0.646553 0.762869i \(-0.723790\pi\)
−0.646553 + 0.762869i \(0.723790\pi\)
\(468\) 0 0
\(469\) 13.1803 0.608612
\(470\) −0.729490 −0.0336489
\(471\) 0 0
\(472\) 22.2361 1.02350
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 3.23607 0.148325
\(477\) 0 0
\(478\) −8.72949 −0.399278
\(479\) −0.111456 −0.00509256 −0.00254628 0.999997i \(-0.500811\pi\)
−0.00254628 + 0.999997i \(0.500811\pi\)
\(480\) 0 0
\(481\) 31.6525 1.44323
\(482\) 1.96556 0.0895287
\(483\) 0 0
\(484\) 0 0
\(485\) 38.9443 1.76837
\(486\) 0 0
\(487\) −29.8885 −1.35438 −0.677190 0.735809i \(-0.736802\pi\)
−0.677190 + 0.735809i \(0.736802\pi\)
\(488\) −7.88854 −0.357098
\(489\) 0 0
\(490\) 1.38197 0.0624309
\(491\) 28.1246 1.26925 0.634623 0.772822i \(-0.281155\pi\)
0.634623 + 0.772822i \(0.281155\pi\)
\(492\) 0 0
\(493\) 10.0000 0.450377
\(494\) 19.5623 0.880150
\(495\) 0 0
\(496\) 2.83282 0.127197
\(497\) 6.47214 0.290315
\(498\) 0 0
\(499\) 24.2361 1.08496 0.542478 0.840070i \(-0.317486\pi\)
0.542478 + 0.840070i \(0.317486\pi\)
\(500\) −18.0902 −0.809017
\(501\) 0 0
\(502\) −6.79837 −0.303426
\(503\) −25.8885 −1.15431 −0.577157 0.816634i \(-0.695838\pi\)
−0.577157 + 0.816634i \(0.695838\pi\)
\(504\) 0 0
\(505\) −41.3050 −1.83805
\(506\) 0 0
\(507\) 0 0
\(508\) 28.1803 1.25030
\(509\) −6.58359 −0.291813 −0.145906 0.989298i \(-0.546610\pi\)
−0.145906 + 0.989298i \(0.546610\pi\)
\(510\) 0 0
\(511\) 2.70820 0.119804
\(512\) −18.7082 −0.826794
\(513\) 0 0
\(514\) 10.9098 0.481212
\(515\) 3.41641 0.150545
\(516\) 0 0
\(517\) 0 0
\(518\) 4.61803 0.202905
\(519\) 0 0
\(520\) −21.1803 −0.928819
\(521\) 30.1246 1.31978 0.659892 0.751361i \(-0.270602\pi\)
0.659892 + 0.751361i \(0.270602\pi\)
\(522\) 0 0
\(523\) 28.4164 1.24256 0.621281 0.783588i \(-0.286612\pi\)
0.621281 + 0.783588i \(0.286612\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −15.2705 −0.665826
\(527\) 3.05573 0.133110
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 3.41641 0.148399
\(531\) 0 0
\(532\) −12.0902 −0.524175
\(533\) −27.4164 −1.18754
\(534\) 0 0
\(535\) 26.0557 1.12649
\(536\) −29.4721 −1.27300
\(537\) 0 0
\(538\) −8.29180 −0.357485
\(539\) 0 0
\(540\) 0 0
\(541\) −20.3607 −0.875374 −0.437687 0.899127i \(-0.644202\pi\)
−0.437687 + 0.899127i \(0.644202\pi\)
\(542\) −17.8541 −0.766899
\(543\) 0 0
\(544\) −11.2361 −0.481742
\(545\) −3.41641 −0.146343
\(546\) 0 0
\(547\) −26.0000 −1.11168 −0.555840 0.831289i \(-0.687603\pi\)
−0.555840 + 0.831289i \(0.687603\pi\)
\(548\) 28.1803 1.20380
\(549\) 0 0
\(550\) 0 0
\(551\) −37.3607 −1.59162
\(552\) 0 0
\(553\) 8.47214 0.360272
\(554\) −8.87539 −0.377079
\(555\) 0 0
\(556\) 14.4721 0.613755
\(557\) −37.8328 −1.60303 −0.801514 0.597976i \(-0.795972\pi\)
−0.801514 + 0.597976i \(0.795972\pi\)
\(558\) 0 0
\(559\) −2.00000 −0.0845910
\(560\) 4.14590 0.175196
\(561\) 0 0
\(562\) −1.56231 −0.0659019
\(563\) −13.8885 −0.585332 −0.292666 0.956215i \(-0.594542\pi\)
−0.292666 + 0.956215i \(0.594542\pi\)
\(564\) 0 0
\(565\) −32.3607 −1.36142
\(566\) 5.20163 0.218641
\(567\) 0 0
\(568\) −14.4721 −0.607237
\(569\) −2.00000 −0.0838444 −0.0419222 0.999121i \(-0.513348\pi\)
−0.0419222 + 0.999121i \(0.513348\pi\)
\(570\) 0 0
\(571\) 8.83282 0.369642 0.184821 0.982772i \(-0.440829\pi\)
0.184821 + 0.982772i \(0.440829\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) 0 0
\(577\) 17.5279 0.729695 0.364847 0.931067i \(-0.381121\pi\)
0.364847 + 0.931067i \(0.381121\pi\)
\(578\) 8.03444 0.334189
\(579\) 0 0
\(580\) 18.0902 0.751153
\(581\) −16.4721 −0.683379
\(582\) 0 0
\(583\) 0 0
\(584\) −6.05573 −0.250588
\(585\) 0 0
\(586\) 0 0
\(587\) −36.7771 −1.51795 −0.758976 0.651118i \(-0.774300\pi\)
−0.758976 + 0.651118i \(0.774300\pi\)
\(588\) 0 0
\(589\) −11.4164 −0.470405
\(590\) 13.7426 0.565776
\(591\) 0 0
\(592\) 13.8541 0.569400
\(593\) 39.8885 1.63803 0.819013 0.573775i \(-0.194522\pi\)
0.819013 + 0.573775i \(0.194522\pi\)
\(594\) 0 0
\(595\) 4.47214 0.183340
\(596\) −33.9787 −1.39182
\(597\) 0 0
\(598\) −5.23607 −0.214119
\(599\) 4.47214 0.182727 0.0913633 0.995818i \(-0.470878\pi\)
0.0913633 + 0.995818i \(0.470878\pi\)
\(600\) 0 0
\(601\) 5.29180 0.215857 0.107928 0.994159i \(-0.465578\pi\)
0.107928 + 0.994159i \(0.465578\pi\)
\(602\) −0.291796 −0.0118927
\(603\) 0 0
\(604\) 16.1803 0.658369
\(605\) 0 0
\(606\) 0 0
\(607\) 10.0557 0.408149 0.204075 0.978955i \(-0.434581\pi\)
0.204075 + 0.978955i \(0.434581\pi\)
\(608\) 41.9787 1.70246
\(609\) 0 0
\(610\) −4.87539 −0.197399
\(611\) −2.23607 −0.0904616
\(612\) 0 0
\(613\) 24.0000 0.969351 0.484675 0.874694i \(-0.338938\pi\)
0.484675 + 0.874694i \(0.338938\pi\)
\(614\) −3.63932 −0.146871
\(615\) 0 0
\(616\) 0 0
\(617\) 26.8328 1.08025 0.540124 0.841585i \(-0.318377\pi\)
0.540124 + 0.841585i \(0.318377\pi\)
\(618\) 0 0
\(619\) −29.8885 −1.20132 −0.600661 0.799504i \(-0.705096\pi\)
−0.600661 + 0.799504i \(0.705096\pi\)
\(620\) 5.52786 0.222004
\(621\) 0 0
\(622\) 0 0
\(623\) 8.47214 0.339429
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 1.81966 0.0727282
\(627\) 0 0
\(628\) −20.1803 −0.805283
\(629\) 14.9443 0.595867
\(630\) 0 0
\(631\) −26.8328 −1.06820 −0.534099 0.845422i \(-0.679349\pi\)
−0.534099 + 0.845422i \(0.679349\pi\)
\(632\) −18.9443 −0.753563
\(633\) 0 0
\(634\) −12.9443 −0.514083
\(635\) 38.9443 1.54546
\(636\) 0 0
\(637\) 4.23607 0.167839
\(638\) 0 0
\(639\) 0 0
\(640\) −25.4508 −1.00603
\(641\) 11.4164 0.450921 0.225460 0.974252i \(-0.427611\pi\)
0.225460 + 0.974252i \(0.427611\pi\)
\(642\) 0 0
\(643\) 33.7771 1.33204 0.666019 0.745935i \(-0.267997\pi\)
0.666019 + 0.745935i \(0.267997\pi\)
\(644\) 3.23607 0.127519
\(645\) 0 0
\(646\) 9.23607 0.363388
\(647\) 12.3050 0.483758 0.241879 0.970306i \(-0.422236\pi\)
0.241879 + 0.970306i \(0.422236\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 13.3262 0.521896
\(653\) −19.0557 −0.745708 −0.372854 0.927890i \(-0.621621\pi\)
−0.372854 + 0.927890i \(0.621621\pi\)
\(654\) 0 0
\(655\) 16.5836 0.647975
\(656\) −12.0000 −0.468521
\(657\) 0 0
\(658\) −0.326238 −0.0127181
\(659\) 16.1246 0.628126 0.314063 0.949402i \(-0.398310\pi\)
0.314063 + 0.949402i \(0.398310\pi\)
\(660\) 0 0
\(661\) 37.3050 1.45099 0.725497 0.688225i \(-0.241610\pi\)
0.725497 + 0.688225i \(0.241610\pi\)
\(662\) 9.88854 0.384329
\(663\) 0 0
\(664\) 36.8328 1.42939
\(665\) −16.7082 −0.647916
\(666\) 0 0
\(667\) 10.0000 0.387202
\(668\) −34.6525 −1.34074
\(669\) 0 0
\(670\) −18.2148 −0.703698
\(671\) 0 0
\(672\) 0 0
\(673\) −7.05573 −0.271978 −0.135989 0.990710i \(-0.543421\pi\)
−0.135989 + 0.990710i \(0.543421\pi\)
\(674\) −22.5410 −0.868248
\(675\) 0 0
\(676\) −8.00000 −0.307692
\(677\) 26.4721 1.01741 0.508703 0.860942i \(-0.330125\pi\)
0.508703 + 0.860942i \(0.330125\pi\)
\(678\) 0 0
\(679\) 17.4164 0.668380
\(680\) −10.0000 −0.383482
\(681\) 0 0
\(682\) 0 0
\(683\) −27.0557 −1.03526 −0.517629 0.855605i \(-0.673185\pi\)
−0.517629 + 0.855605i \(0.673185\pi\)
\(684\) 0 0
\(685\) 38.9443 1.48798
\(686\) 0.618034 0.0235966
\(687\) 0 0
\(688\) −0.875388 −0.0333739
\(689\) 10.4721 0.398957
\(690\) 0 0
\(691\) 38.8328 1.47727 0.738635 0.674106i \(-0.235471\pi\)
0.738635 + 0.674106i \(0.235471\pi\)
\(692\) 26.6525 1.01318
\(693\) 0 0
\(694\) −8.00000 −0.303676
\(695\) 20.0000 0.758643
\(696\) 0 0
\(697\) −12.9443 −0.490299
\(698\) −13.6738 −0.517560
\(699\) 0 0
\(700\) 0 0
\(701\) −47.8885 −1.80873 −0.904363 0.426765i \(-0.859653\pi\)
−0.904363 + 0.426765i \(0.859653\pi\)
\(702\) 0 0
\(703\) −55.8328 −2.10577
\(704\) 0 0
\(705\) 0 0
\(706\) 10.6180 0.399615
\(707\) −18.4721 −0.694716
\(708\) 0 0
\(709\) −43.4721 −1.63263 −0.816315 0.577607i \(-0.803987\pi\)
−0.816315 + 0.577607i \(0.803987\pi\)
\(710\) −8.94427 −0.335673
\(711\) 0 0
\(712\) −18.9443 −0.709967
\(713\) 3.05573 0.114438
\(714\) 0 0
\(715\) 0 0
\(716\) 29.7082 1.11025
\(717\) 0 0
\(718\) −6.83282 −0.254998
\(719\) 23.3607 0.871206 0.435603 0.900139i \(-0.356535\pi\)
0.435603 + 0.900139i \(0.356535\pi\)
\(720\) 0 0
\(721\) 1.52786 0.0569006
\(722\) −22.7639 −0.847186
\(723\) 0 0
\(724\) 8.00000 0.297318
\(725\) 0 0
\(726\) 0 0
\(727\) −12.4721 −0.462566 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(728\) −9.47214 −0.351061
\(729\) 0 0
\(730\) −3.74265 −0.138522
\(731\) −0.944272 −0.0349252
\(732\) 0 0
\(733\) −18.3607 −0.678167 −0.339084 0.940756i \(-0.610117\pi\)
−0.339084 + 0.940756i \(0.610117\pi\)
\(734\) 9.88854 0.364993
\(735\) 0 0
\(736\) −11.2361 −0.414167
\(737\) 0 0
\(738\) 0 0
\(739\) −16.1115 −0.592669 −0.296335 0.955084i \(-0.595764\pi\)
−0.296335 + 0.955084i \(0.595764\pi\)
\(740\) 27.0344 0.993806
\(741\) 0 0
\(742\) 1.52786 0.0560897
\(743\) 18.5967 0.682249 0.341124 0.940018i \(-0.389192\pi\)
0.341124 + 0.940018i \(0.389192\pi\)
\(744\) 0 0
\(745\) −46.9574 −1.72039
\(746\) 3.70820 0.135767
\(747\) 0 0
\(748\) 0 0
\(749\) 11.6525 0.425772
\(750\) 0 0
\(751\) 15.1803 0.553938 0.276969 0.960879i \(-0.410670\pi\)
0.276969 + 0.960879i \(0.410670\pi\)
\(752\) −0.978714 −0.0356900
\(753\) 0 0
\(754\) −13.0902 −0.476716
\(755\) 22.3607 0.813788
\(756\) 0 0
\(757\) −42.4164 −1.54165 −0.770825 0.637047i \(-0.780156\pi\)
−0.770825 + 0.637047i \(0.780156\pi\)
\(758\) −10.0344 −0.364467
\(759\) 0 0
\(760\) 37.3607 1.35521
\(761\) 15.5279 0.562885 0.281442 0.959578i \(-0.409187\pi\)
0.281442 + 0.959578i \(0.409187\pi\)
\(762\) 0 0
\(763\) −1.52786 −0.0553124
\(764\) 39.5967 1.43256
\(765\) 0 0
\(766\) 10.4721 0.378374
\(767\) 42.1246 1.52103
\(768\) 0 0
\(769\) −7.18034 −0.258930 −0.129465 0.991584i \(-0.541326\pi\)
−0.129465 + 0.991584i \(0.541326\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 16.9443 0.609838
\(773\) −3.18034 −0.114389 −0.0571944 0.998363i \(-0.518215\pi\)
−0.0571944 + 0.998363i \(0.518215\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −38.9443 −1.39802
\(777\) 0 0
\(778\) −21.3050 −0.763820
\(779\) 48.3607 1.73270
\(780\) 0 0
\(781\) 0 0
\(782\) −2.47214 −0.0884034
\(783\) 0 0
\(784\) 1.85410 0.0662179
\(785\) −27.8885 −0.995385
\(786\) 0 0
\(787\) 35.2492 1.25650 0.628250 0.778012i \(-0.283772\pi\)
0.628250 + 0.778012i \(0.283772\pi\)
\(788\) −19.2361 −0.685257
\(789\) 0 0
\(790\) −11.7082 −0.416559
\(791\) −14.4721 −0.514570
\(792\) 0 0
\(793\) −14.9443 −0.530687
\(794\) −5.59675 −0.198621
\(795\) 0 0
\(796\) −38.6525 −1.37000
\(797\) 42.2361 1.49608 0.748039 0.663655i \(-0.230996\pi\)
0.748039 + 0.663655i \(0.230996\pi\)
\(798\) 0 0
\(799\) −1.05573 −0.0373490
\(800\) 0 0
\(801\) 0 0
\(802\) −16.0689 −0.567412
\(803\) 0 0
\(804\) 0 0
\(805\) 4.47214 0.157622
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 41.3050 1.45310
\(809\) 4.63932 0.163110 0.0815549 0.996669i \(-0.474011\pi\)
0.0815549 + 0.996669i \(0.474011\pi\)
\(810\) 0 0
\(811\) −49.4721 −1.73720 −0.868601 0.495512i \(-0.834980\pi\)
−0.868601 + 0.495512i \(0.834980\pi\)
\(812\) 8.09017 0.283909
\(813\) 0 0
\(814\) 0 0
\(815\) 18.4164 0.645099
\(816\) 0 0
\(817\) 3.52786 0.123424
\(818\) −10.1803 −0.355947
\(819\) 0 0
\(820\) −23.4164 −0.817736
\(821\) −51.8328 −1.80898 −0.904489 0.426497i \(-0.859747\pi\)
−0.904489 + 0.426497i \(0.859747\pi\)
\(822\) 0 0
\(823\) 25.0689 0.873846 0.436923 0.899499i \(-0.356068\pi\)
0.436923 + 0.899499i \(0.356068\pi\)
\(824\) −3.41641 −0.119016
\(825\) 0 0
\(826\) 6.14590 0.213843
\(827\) −21.7639 −0.756806 −0.378403 0.925641i \(-0.623527\pi\)
−0.378403 + 0.925641i \(0.623527\pi\)
\(828\) 0 0
\(829\) −3.05573 −0.106130 −0.0530649 0.998591i \(-0.516899\pi\)
−0.0530649 + 0.998591i \(0.516899\pi\)
\(830\) 22.7639 0.790148
\(831\) 0 0
\(832\) −1.00000 −0.0346688
\(833\) 2.00000 0.0692959
\(834\) 0 0
\(835\) −47.8885 −1.65725
\(836\) 0 0
\(837\) 0 0
\(838\) 10.5066 0.362944
\(839\) 51.2492 1.76932 0.884660 0.466237i \(-0.154391\pi\)
0.884660 + 0.466237i \(0.154391\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) −4.54915 −0.156774
\(843\) 0 0
\(844\) −29.7082 −1.02260
\(845\) −11.0557 −0.380329
\(846\) 0 0
\(847\) 0 0
\(848\) 4.58359 0.157401
\(849\) 0 0
\(850\) 0 0
\(851\) 14.9443 0.512283
\(852\) 0 0
\(853\) 2.58359 0.0884605 0.0442303 0.999021i \(-0.485916\pi\)
0.0442303 + 0.999021i \(0.485916\pi\)
\(854\) −2.18034 −0.0746097
\(855\) 0 0
\(856\) −26.0557 −0.890566
\(857\) −41.4164 −1.41476 −0.707379 0.706835i \(-0.750122\pi\)
−0.707379 + 0.706835i \(0.750122\pi\)
\(858\) 0 0
\(859\) 26.9443 0.919327 0.459663 0.888093i \(-0.347970\pi\)
0.459663 + 0.888093i \(0.347970\pi\)
\(860\) −1.70820 −0.0582493
\(861\) 0 0
\(862\) −9.74265 −0.331836
\(863\) 9.88854 0.336610 0.168305 0.985735i \(-0.446171\pi\)
0.168305 + 0.985735i \(0.446171\pi\)
\(864\) 0 0
\(865\) 36.8328 1.25235
\(866\) −20.6525 −0.701800
\(867\) 0 0
\(868\) 2.47214 0.0839098
\(869\) 0 0
\(870\) 0 0
\(871\) −55.8328 −1.89182
\(872\) 3.41641 0.115694
\(873\) 0 0
\(874\) 9.23607 0.312415
\(875\) −11.1803 −0.377964
\(876\) 0 0
\(877\) 28.0000 0.945493 0.472746 0.881199i \(-0.343263\pi\)
0.472746 + 0.881199i \(0.343263\pi\)
\(878\) −5.49342 −0.185394
\(879\) 0 0
\(880\) 0 0
\(881\) −35.7639 −1.20492 −0.602459 0.798150i \(-0.705812\pi\)
−0.602459 + 0.798150i \(0.705812\pi\)
\(882\) 0 0
\(883\) −2.81966 −0.0948891 −0.0474446 0.998874i \(-0.515108\pi\)
−0.0474446 + 0.998874i \(0.515108\pi\)
\(884\) −13.7082 −0.461057
\(885\) 0 0
\(886\) 8.65248 0.290686
\(887\) −0.944272 −0.0317055 −0.0158528 0.999874i \(-0.505046\pi\)
−0.0158528 + 0.999874i \(0.505046\pi\)
\(888\) 0 0
\(889\) 17.4164 0.584128
\(890\) −11.7082 −0.392460
\(891\) 0 0
\(892\) −19.2361 −0.644071
\(893\) 3.94427 0.131990
\(894\) 0 0
\(895\) 41.0557 1.37234
\(896\) −11.3820 −0.380245
\(897\) 0 0
\(898\) −5.23607 −0.174730
\(899\) 7.63932 0.254786
\(900\) 0 0
\(901\) 4.94427 0.164718
\(902\) 0 0
\(903\) 0 0
\(904\) 32.3607 1.07630
\(905\) 11.0557 0.367505
\(906\) 0 0
\(907\) −15.7771 −0.523870 −0.261935 0.965086i \(-0.584361\pi\)
−0.261935 + 0.965086i \(0.584361\pi\)
\(908\) −20.1803 −0.669708
\(909\) 0 0
\(910\) −5.85410 −0.194062
\(911\) −26.2492 −0.869676 −0.434838 0.900509i \(-0.643194\pi\)
−0.434838 + 0.900509i \(0.643194\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −23.7082 −0.784198
\(915\) 0 0
\(916\) −22.6525 −0.748459
\(917\) 7.41641 0.244911
\(918\) 0 0
\(919\) −13.5279 −0.446243 −0.223122 0.974791i \(-0.571625\pi\)
−0.223122 + 0.974791i \(0.571625\pi\)
\(920\) −10.0000 −0.329690
\(921\) 0 0
\(922\) −0.652476 −0.0214881
\(923\) −27.4164 −0.902422
\(924\) 0 0
\(925\) 0 0
\(926\) 4.14590 0.136243
\(927\) 0 0
\(928\) −28.0902 −0.922105
\(929\) 0.708204 0.0232354 0.0116177 0.999933i \(-0.496302\pi\)
0.0116177 + 0.999933i \(0.496302\pi\)
\(930\) 0 0
\(931\) −7.47214 −0.244889
\(932\) 4.76393 0.156048
\(933\) 0 0
\(934\) 17.2705 0.565108
\(935\) 0 0
\(936\) 0 0
\(937\) −6.36068 −0.207794 −0.103897 0.994588i \(-0.533131\pi\)
−0.103897 + 0.994588i \(0.533131\pi\)
\(938\) −8.14590 −0.265973
\(939\) 0 0
\(940\) −1.90983 −0.0622918
\(941\) 53.4164 1.74133 0.870663 0.491881i \(-0.163690\pi\)
0.870663 + 0.491881i \(0.163690\pi\)
\(942\) 0 0
\(943\) −12.9443 −0.421523
\(944\) 18.4377 0.600096
\(945\) 0 0
\(946\) 0 0
\(947\) 44.0000 1.42981 0.714904 0.699223i \(-0.246470\pi\)
0.714904 + 0.699223i \(0.246470\pi\)
\(948\) 0 0
\(949\) −11.4721 −0.372401
\(950\) 0 0
\(951\) 0 0
\(952\) −4.47214 −0.144943
\(953\) −18.4164 −0.596566 −0.298283 0.954477i \(-0.596414\pi\)
−0.298283 + 0.954477i \(0.596414\pi\)
\(954\) 0 0
\(955\) 54.7214 1.77074
\(956\) −22.8541 −0.739154
\(957\) 0 0
\(958\) 0.0688837 0.00222553
\(959\) 17.4164 0.562405
\(960\) 0 0
\(961\) −28.6656 −0.924698
\(962\) −19.5623 −0.630714
\(963\) 0 0
\(964\) 5.14590 0.165738
\(965\) 23.4164 0.753801
\(966\) 0 0
\(967\) 49.3050 1.58554 0.792770 0.609521i \(-0.208638\pi\)
0.792770 + 0.609521i \(0.208638\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −24.0689 −0.772805
\(971\) −29.9443 −0.960957 −0.480479 0.877006i \(-0.659537\pi\)
−0.480479 + 0.877006i \(0.659537\pi\)
\(972\) 0 0
\(973\) 8.94427 0.286740
\(974\) 18.4721 0.591885
\(975\) 0 0
\(976\) −6.54102 −0.209373
\(977\) −7.52786 −0.240838 −0.120419 0.992723i \(-0.538424\pi\)
−0.120419 + 0.992723i \(0.538424\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 3.61803 0.115574
\(981\) 0 0
\(982\) −17.3820 −0.554681
\(983\) 40.0000 1.27580 0.637901 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 0 0
\(985\) −26.5836 −0.847024
\(986\) −6.18034 −0.196822
\(987\) 0 0
\(988\) 51.2148 1.62936
\(989\) −0.944272 −0.0300261
\(990\) 0 0
\(991\) 11.1803 0.355155 0.177578 0.984107i \(-0.443174\pi\)
0.177578 + 0.984107i \(0.443174\pi\)
\(992\) −8.58359 −0.272529
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) −53.4164 −1.69341
\(996\) 0 0
\(997\) −41.1935 −1.30461 −0.652306 0.757956i \(-0.726198\pi\)
−0.652306 + 0.757956i \(0.726198\pi\)
\(998\) −14.9787 −0.474143
\(999\) 0 0
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 7623.2.a.bn.1.1 2
3.2 odd 2 2541.2.a.q.1.2 2
11.10 odd 2 7623.2.a.y.1.2 2
33.32 even 2 2541.2.a.y.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.q.1.2 2 3.2 odd 2
2541.2.a.y.1.1 yes 2 33.32 even 2
7623.2.a.y.1.2 2 11.10 odd 2
7623.2.a.bn.1.1 2 1.1 even 1 trivial