Properties

Label 9075.2.a.bk.1.2
Level $9075$
Weight $2$
Character 9075.1
Self dual yes
Analytic conductor $72.464$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 9075 = 3 \cdot 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9075.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.4642398343\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 9075.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.23607 q^{2} -1.00000 q^{3} +3.00000 q^{4} -2.23607 q^{6} +2.23607 q^{7} +2.23607 q^{8} +1.00000 q^{9} -3.00000 q^{12} +4.47214 q^{13} +5.00000 q^{14} -1.00000 q^{16} +2.23607 q^{17} +2.23607 q^{18} -2.23607 q^{19} -2.23607 q^{21} -1.00000 q^{23} -2.23607 q^{24} +10.0000 q^{26} -1.00000 q^{27} +6.70820 q^{28} +4.47214 q^{29} +10.0000 q^{31} -6.70820 q^{32} +5.00000 q^{34} +3.00000 q^{36} -7.00000 q^{37} -5.00000 q^{38} -4.47214 q^{39} +6.70820 q^{41} -5.00000 q^{42} -2.23607 q^{46} -3.00000 q^{47} +1.00000 q^{48} -2.00000 q^{49} -2.23607 q^{51} +13.4164 q^{52} +14.0000 q^{53} -2.23607 q^{54} +5.00000 q^{56} +2.23607 q^{57} +10.0000 q^{58} -5.00000 q^{59} +4.47214 q^{61} +22.3607 q^{62} +2.23607 q^{63} -13.0000 q^{64} +2.00000 q^{67} +6.70820 q^{68} +1.00000 q^{69} -13.0000 q^{71} +2.23607 q^{72} +13.4164 q^{73} -15.6525 q^{74} -6.70820 q^{76} -10.0000 q^{78} -15.6525 q^{79} +1.00000 q^{81} +15.0000 q^{82} +4.47214 q^{83} -6.70820 q^{84} -4.47214 q^{87} +6.00000 q^{89} +10.0000 q^{91} -3.00000 q^{92} -10.0000 q^{93} -6.70820 q^{94} +6.70820 q^{96} -17.0000 q^{97} -4.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 6 q^{4} + 2 q^{9} - 6 q^{12} + 10 q^{14} - 2 q^{16} - 2 q^{23} + 20 q^{26} - 2 q^{27} + 20 q^{31} + 10 q^{34} + 6 q^{36} - 14 q^{37} - 10 q^{38} - 10 q^{42} - 6 q^{47} + 2 q^{48} - 4 q^{49} + 28 q^{53} + 10 q^{56} + 20 q^{58} - 10 q^{59} - 26 q^{64} + 4 q^{67} + 2 q^{69} - 26 q^{71} - 20 q^{78} + 2 q^{81} + 30 q^{82} + 12 q^{89} + 20 q^{91} - 6 q^{92} - 20 q^{93} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.00000 1.50000
\(5\) 0 0
\(6\) −2.23607 −0.912871
\(7\) 2.23607 0.845154 0.422577 0.906327i \(-0.361126\pi\)
0.422577 + 0.906327i \(0.361126\pi\)
\(8\) 2.23607 0.790569
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) −3.00000 −0.866025
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) 5.00000 1.33631
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.23607 0.542326 0.271163 0.962533i \(-0.412592\pi\)
0.271163 + 0.962533i \(0.412592\pi\)
\(18\) 2.23607 0.527046
\(19\) −2.23607 −0.512989 −0.256495 0.966546i \(-0.582568\pi\)
−0.256495 + 0.966546i \(0.582568\pi\)
\(20\) 0 0
\(21\) −2.23607 −0.487950
\(22\) 0 0
\(23\) −1.00000 −0.208514 −0.104257 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) −2.23607 −0.456435
\(25\) 0 0
\(26\) 10.0000 1.96116
\(27\) −1.00000 −0.192450
\(28\) 6.70820 1.26773
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 10.0000 1.79605 0.898027 0.439941i \(-0.145001\pi\)
0.898027 + 0.439941i \(0.145001\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) −5.00000 −0.811107
\(39\) −4.47214 −0.716115
\(40\) 0 0
\(41\) 6.70820 1.04765 0.523823 0.851827i \(-0.324505\pi\)
0.523823 + 0.851827i \(0.324505\pi\)
\(42\) −5.00000 −0.771517
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −2.23607 −0.329690
\(47\) −3.00000 −0.437595 −0.218797 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) 1.00000 0.144338
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) −2.23607 −0.313112
\(52\) 13.4164 1.86052
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) 5.00000 0.668153
\(57\) 2.23607 0.296174
\(58\) 10.0000 1.31306
\(59\) −5.00000 −0.650945 −0.325472 0.945552i \(-0.605523\pi\)
−0.325472 + 0.945552i \(0.605523\pi\)
\(60\) 0 0
\(61\) 4.47214 0.572598 0.286299 0.958140i \(-0.407575\pi\)
0.286299 + 0.958140i \(0.407575\pi\)
\(62\) 22.3607 2.83981
\(63\) 2.23607 0.281718
\(64\) −13.0000 −1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 6.70820 0.813489
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −13.0000 −1.54282 −0.771408 0.636341i \(-0.780447\pi\)
−0.771408 + 0.636341i \(0.780447\pi\)
\(72\) 2.23607 0.263523
\(73\) 13.4164 1.57027 0.785136 0.619324i \(-0.212593\pi\)
0.785136 + 0.619324i \(0.212593\pi\)
\(74\) −15.6525 −1.81956
\(75\) 0 0
\(76\) −6.70820 −0.769484
\(77\) 0 0
\(78\) −10.0000 −1.13228
\(79\) −15.6525 −1.76104 −0.880521 0.474008i \(-0.842807\pi\)
−0.880521 + 0.474008i \(0.842807\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.0000 1.65647
\(83\) 4.47214 0.490881 0.245440 0.969412i \(-0.421067\pi\)
0.245440 + 0.969412i \(0.421067\pi\)
\(84\) −6.70820 −0.731925
\(85\) 0 0
\(86\) 0 0
\(87\) −4.47214 −0.479463
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 10.0000 1.04828
\(92\) −3.00000 −0.312772
\(93\) −10.0000 −1.03695
\(94\) −6.70820 −0.691898
\(95\) 0 0
\(96\) 6.70820 0.684653
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) −4.47214 −0.451754
\(99\) 0 0
\(100\) 0 0
\(101\) 11.1803 1.11249 0.556243 0.831020i \(-0.312242\pi\)
0.556243 + 0.831020i \(0.312242\pi\)
\(102\) −5.00000 −0.495074
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 10.0000 0.980581
\(105\) 0 0
\(106\) 31.3050 3.04061
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) −3.00000 −0.288675
\(109\) 17.8885 1.71341 0.856706 0.515805i \(-0.172507\pi\)
0.856706 + 0.515805i \(0.172507\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) −2.23607 −0.211289
\(113\) 14.0000 1.31701 0.658505 0.752577i \(-0.271189\pi\)
0.658505 + 0.752577i \(0.271189\pi\)
\(114\) 5.00000 0.468293
\(115\) 0 0
\(116\) 13.4164 1.24568
\(117\) 4.47214 0.413449
\(118\) −11.1803 −1.02923
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) 0 0
\(122\) 10.0000 0.905357
\(123\) −6.70820 −0.604858
\(124\) 30.0000 2.69408
\(125\) 0 0
\(126\) 5.00000 0.445435
\(127\) 15.6525 1.38893 0.694466 0.719525i \(-0.255641\pi\)
0.694466 + 0.719525i \(0.255641\pi\)
\(128\) −15.6525 −1.38350
\(129\) 0 0
\(130\) 0 0
\(131\) −13.4164 −1.17220 −0.586098 0.810240i \(-0.699337\pi\)
−0.586098 + 0.810240i \(0.699337\pi\)
\(132\) 0 0
\(133\) −5.00000 −0.433555
\(134\) 4.47214 0.386334
\(135\) 0 0
\(136\) 5.00000 0.428746
\(137\) 8.00000 0.683486 0.341743 0.939793i \(-0.388983\pi\)
0.341743 + 0.939793i \(0.388983\pi\)
\(138\) 2.23607 0.190347
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) −29.0689 −2.43941
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 30.0000 2.48282
\(147\) 2.00000 0.164957
\(148\) −21.0000 −1.72619
\(149\) −20.1246 −1.64867 −0.824336 0.566101i \(-0.808451\pi\)
−0.824336 + 0.566101i \(0.808451\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −5.00000 −0.405554
\(153\) 2.23607 0.180775
\(154\) 0 0
\(155\) 0 0
\(156\) −13.4164 −1.07417
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) −35.0000 −2.78445
\(159\) −14.0000 −1.11027
\(160\) 0 0
\(161\) −2.23607 −0.176227
\(162\) 2.23607 0.175682
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 20.1246 1.57147
\(165\) 0 0
\(166\) 10.0000 0.776151
\(167\) 4.47214 0.346064 0.173032 0.984916i \(-0.444644\pi\)
0.173032 + 0.984916i \(0.444644\pi\)
\(168\) −5.00000 −0.385758
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) −2.23607 −0.170996
\(172\) 0 0
\(173\) 20.1246 1.53005 0.765023 0.644003i \(-0.222728\pi\)
0.765023 + 0.644003i \(0.222728\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) 0 0
\(177\) 5.00000 0.375823
\(178\) 13.4164 1.00560
\(179\) 11.0000 0.822179 0.411089 0.911595i \(-0.365148\pi\)
0.411089 + 0.911595i \(0.365148\pi\)
\(180\) 0 0
\(181\) −5.00000 −0.371647 −0.185824 0.982583i \(-0.559495\pi\)
−0.185824 + 0.982583i \(0.559495\pi\)
\(182\) 22.3607 1.65748
\(183\) −4.47214 −0.330590
\(184\) −2.23607 −0.164845
\(185\) 0 0
\(186\) −22.3607 −1.63956
\(187\) 0 0
\(188\) −9.00000 −0.656392
\(189\) −2.23607 −0.162650
\(190\) 0 0
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 13.0000 0.938194
\(193\) −8.94427 −0.643823 −0.321911 0.946770i \(-0.604325\pi\)
−0.321911 + 0.946770i \(0.604325\pi\)
\(194\) −38.0132 −2.72919
\(195\) 0 0
\(196\) −6.00000 −0.428571
\(197\) −15.6525 −1.11519 −0.557596 0.830112i \(-0.688276\pi\)
−0.557596 + 0.830112i \(0.688276\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 25.0000 1.75899
\(203\) 10.0000 0.701862
\(204\) −6.70820 −0.469668
\(205\) 0 0
\(206\) 8.94427 0.623177
\(207\) −1.00000 −0.0695048
\(208\) −4.47214 −0.310087
\(209\) 0 0
\(210\) 0 0
\(211\) 26.8328 1.84725 0.923624 0.383301i \(-0.125213\pi\)
0.923624 + 0.383301i \(0.125213\pi\)
\(212\) 42.0000 2.88457
\(213\) 13.0000 0.890745
\(214\) 0 0
\(215\) 0 0
\(216\) −2.23607 −0.152145
\(217\) 22.3607 1.51794
\(218\) 40.0000 2.70914
\(219\) −13.4164 −0.906597
\(220\) 0 0
\(221\) 10.0000 0.672673
\(222\) 15.6525 1.05053
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −15.0000 −1.00223
\(225\) 0 0
\(226\) 31.3050 2.08237
\(227\) 13.4164 0.890478 0.445239 0.895412i \(-0.353119\pi\)
0.445239 + 0.895412i \(0.353119\pi\)
\(228\) 6.70820 0.444262
\(229\) −5.00000 −0.330409 −0.165205 0.986259i \(-0.552828\pi\)
−0.165205 + 0.986259i \(0.552828\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.0000 0.656532
\(233\) −20.1246 −1.31841 −0.659204 0.751965i \(-0.729106\pi\)
−0.659204 + 0.751965i \(0.729106\pi\)
\(234\) 10.0000 0.653720
\(235\) 0 0
\(236\) −15.0000 −0.976417
\(237\) 15.6525 1.01674
\(238\) 11.1803 0.724714
\(239\) 8.94427 0.578557 0.289278 0.957245i \(-0.406585\pi\)
0.289278 + 0.957245i \(0.406585\pi\)
\(240\) 0 0
\(241\) 22.3607 1.44038 0.720189 0.693778i \(-0.244055\pi\)
0.720189 + 0.693778i \(0.244055\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 13.4164 0.858898
\(245\) 0 0
\(246\) −15.0000 −0.956365
\(247\) −10.0000 −0.636285
\(248\) 22.3607 1.41990
\(249\) −4.47214 −0.283410
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 6.70820 0.422577
\(253\) 0 0
\(254\) 35.0000 2.19610
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −15.6525 −0.972598
\(260\) 0 0
\(261\) 4.47214 0.276818
\(262\) −30.0000 −1.85341
\(263\) −17.8885 −1.10305 −0.551527 0.834157i \(-0.685955\pi\)
−0.551527 + 0.834157i \(0.685955\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −11.1803 −0.685511
\(267\) −6.00000 −0.367194
\(268\) 6.00000 0.366508
\(269\) 30.0000 1.82913 0.914566 0.404436i \(-0.132532\pi\)
0.914566 + 0.404436i \(0.132532\pi\)
\(270\) 0 0
\(271\) 20.1246 1.22248 0.611242 0.791444i \(-0.290670\pi\)
0.611242 + 0.791444i \(0.290670\pi\)
\(272\) −2.23607 −0.135582
\(273\) −10.0000 −0.605228
\(274\) 17.8885 1.08069
\(275\) 0 0
\(276\) 3.00000 0.180579
\(277\) −13.4164 −0.806114 −0.403057 0.915175i \(-0.632052\pi\)
−0.403057 + 0.915175i \(0.632052\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 24.5967 1.46732 0.733659 0.679517i \(-0.237811\pi\)
0.733659 + 0.679517i \(0.237811\pi\)
\(282\) 6.70820 0.399468
\(283\) −6.70820 −0.398761 −0.199381 0.979922i \(-0.563893\pi\)
−0.199381 + 0.979922i \(0.563893\pi\)
\(284\) −39.0000 −2.31422
\(285\) 0 0
\(286\) 0 0
\(287\) 15.0000 0.885422
\(288\) −6.70820 −0.395285
\(289\) −12.0000 −0.705882
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) 40.2492 2.35541
\(293\) 15.6525 0.914427 0.457214 0.889357i \(-0.348847\pi\)
0.457214 + 0.889357i \(0.348847\pi\)
\(294\) 4.47214 0.260820
\(295\) 0 0
\(296\) −15.6525 −0.909782
\(297\) 0 0
\(298\) −45.0000 −2.60678
\(299\) −4.47214 −0.258630
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −11.1803 −0.642294
\(304\) 2.23607 0.128247
\(305\) 0 0
\(306\) 5.00000 0.285831
\(307\) −17.8885 −1.02095 −0.510477 0.859892i \(-0.670531\pi\)
−0.510477 + 0.859892i \(0.670531\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −32.0000 −1.81455 −0.907277 0.420534i \(-0.861843\pi\)
−0.907277 + 0.420534i \(0.861843\pi\)
\(312\) −10.0000 −0.566139
\(313\) −9.00000 −0.508710 −0.254355 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) −4.47214 −0.252377
\(315\) 0 0
\(316\) −46.9574 −2.64156
\(317\) 22.0000 1.23564 0.617822 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) −31.3050 −1.75549
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) −5.00000 −0.278639
\(323\) −5.00000 −0.278207
\(324\) 3.00000 0.166667
\(325\) 0 0
\(326\) −31.3050 −1.73382
\(327\) −17.8885 −0.989239
\(328\) 15.0000 0.828236
\(329\) −6.70820 −0.369835
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 13.4164 0.736321
\(333\) −7.00000 −0.383598
\(334\) 10.0000 0.547176
\(335\) 0 0
\(336\) 2.23607 0.121988
\(337\) −13.4164 −0.730838 −0.365419 0.930843i \(-0.619074\pi\)
−0.365419 + 0.930843i \(0.619074\pi\)
\(338\) 15.6525 0.851382
\(339\) −14.0000 −0.760376
\(340\) 0 0
\(341\) 0 0
\(342\) −5.00000 −0.270369
\(343\) −20.1246 −1.08663
\(344\) 0 0
\(345\) 0 0
\(346\) 45.0000 2.41921
\(347\) 13.4164 0.720231 0.360115 0.932908i \(-0.382737\pi\)
0.360115 + 0.932908i \(0.382737\pi\)
\(348\) −13.4164 −0.719195
\(349\) −13.4164 −0.718164 −0.359082 0.933306i \(-0.616910\pi\)
−0.359082 + 0.933306i \(0.616910\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) 34.0000 1.80964 0.904819 0.425797i \(-0.140006\pi\)
0.904819 + 0.425797i \(0.140006\pi\)
\(354\) 11.1803 0.594228
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) −5.00000 −0.264628
\(358\) 24.5967 1.29998
\(359\) 4.47214 0.236030 0.118015 0.993012i \(-0.462347\pi\)
0.118015 + 0.993012i \(0.462347\pi\)
\(360\) 0 0
\(361\) −14.0000 −0.736842
\(362\) −11.1803 −0.587626
\(363\) 0 0
\(364\) 30.0000 1.57243
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.70820 0.349215
\(370\) 0 0
\(371\) 31.3050 1.62527
\(372\) −30.0000 −1.55543
\(373\) −8.94427 −0.463117 −0.231558 0.972821i \(-0.574382\pi\)
−0.231558 + 0.972821i \(0.574382\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −6.70820 −0.345949
\(377\) 20.0000 1.03005
\(378\) −5.00000 −0.257172
\(379\) 10.0000 0.513665 0.256833 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) −15.6525 −0.801901
\(382\) 33.5410 1.71611
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 15.6525 0.798762
\(385\) 0 0
\(386\) −20.0000 −1.01797
\(387\) 0 0
\(388\) −51.0000 −2.58913
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −2.23607 −0.113083
\(392\) −4.47214 −0.225877
\(393\) 13.4164 0.676768
\(394\) −35.0000 −1.76327
\(395\) 0 0
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 22.3607 1.12084
\(399\) 5.00000 0.250313
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −4.47214 −0.223050
\(403\) 44.7214 2.22773
\(404\) 33.5410 1.66873
\(405\) 0 0
\(406\) 22.3607 1.10974
\(407\) 0 0
\(408\) −5.00000 −0.247537
\(409\) 13.4164 0.663399 0.331699 0.943385i \(-0.392378\pi\)
0.331699 + 0.943385i \(0.392378\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) 12.0000 0.591198
\(413\) −11.1803 −0.550149
\(414\) −2.23607 −0.109897
\(415\) 0 0
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) 0 0
\(419\) 9.00000 0.439679 0.219839 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) −15.0000 −0.731055 −0.365528 0.930800i \(-0.619111\pi\)
−0.365528 + 0.930800i \(0.619111\pi\)
\(422\) 60.0000 2.92075
\(423\) −3.00000 −0.145865
\(424\) 31.3050 1.52030
\(425\) 0 0
\(426\) 29.0689 1.40839
\(427\) 10.0000 0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 22.3607 1.07708 0.538538 0.842601i \(-0.318977\pi\)
0.538538 + 0.842601i \(0.318977\pi\)
\(432\) 1.00000 0.0481125
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 50.0000 2.40008
\(435\) 0 0
\(436\) 53.6656 2.57012
\(437\) 2.23607 0.106966
\(438\) −30.0000 −1.43346
\(439\) 20.1246 0.960495 0.480248 0.877133i \(-0.340547\pi\)
0.480248 + 0.877133i \(0.340547\pi\)
\(440\) 0 0
\(441\) −2.00000 −0.0952381
\(442\) 22.3607 1.06359
\(443\) −21.0000 −0.997740 −0.498870 0.866677i \(-0.666252\pi\)
−0.498870 + 0.866677i \(0.666252\pi\)
\(444\) 21.0000 0.996616
\(445\) 0 0
\(446\) −31.3050 −1.48233
\(447\) 20.1246 0.951861
\(448\) −29.0689 −1.37338
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 42.0000 1.97551
\(453\) 0 0
\(454\) 30.0000 1.40797
\(455\) 0 0
\(456\) 5.00000 0.234146
\(457\) −22.3607 −1.04599 −0.522994 0.852336i \(-0.675185\pi\)
−0.522994 + 0.852336i \(0.675185\pi\)
\(458\) −11.1803 −0.522423
\(459\) −2.23607 −0.104371
\(460\) 0 0
\(461\) −22.3607 −1.04144 −0.520720 0.853727i \(-0.674337\pi\)
−0.520720 + 0.853727i \(0.674337\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) −4.47214 −0.207614
\(465\) 0 0
\(466\) −45.0000 −2.08458
\(467\) −32.0000 −1.48078 −0.740392 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 13.4164 0.620174
\(469\) 4.47214 0.206504
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) −11.1803 −0.514617
\(473\) 0 0
\(474\) 35.0000 1.60760
\(475\) 0 0
\(476\) 15.0000 0.687524
\(477\) 14.0000 0.641016
\(478\) 20.0000 0.914779
\(479\) 22.3607 1.02169 0.510843 0.859674i \(-0.329333\pi\)
0.510843 + 0.859674i \(0.329333\pi\)
\(480\) 0 0
\(481\) −31.3050 −1.42738
\(482\) 50.0000 2.27744
\(483\) 2.23607 0.101745
\(484\) 0 0
\(485\) 0 0
\(486\) −2.23607 −0.101430
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 10.0000 0.452679
\(489\) 14.0000 0.633102
\(490\) 0 0
\(491\) −4.47214 −0.201825 −0.100912 0.994895i \(-0.532176\pi\)
−0.100912 + 0.994895i \(0.532176\pi\)
\(492\) −20.1246 −0.907288
\(493\) 10.0000 0.450377
\(494\) −22.3607 −1.00605
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) −29.0689 −1.30392
\(498\) −10.0000 −0.448111
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) −4.47214 −0.199800
\(502\) −26.8328 −1.19761
\(503\) −35.7771 −1.59522 −0.797611 0.603173i \(-0.793903\pi\)
−0.797611 + 0.603173i \(0.793903\pi\)
\(504\) 5.00000 0.222718
\(505\) 0 0
\(506\) 0 0
\(507\) −7.00000 −0.310881
\(508\) 46.9574 2.08340
\(509\) 10.0000 0.443242 0.221621 0.975133i \(-0.428865\pi\)
0.221621 + 0.975133i \(0.428865\pi\)
\(510\) 0 0
\(511\) 30.0000 1.32712
\(512\) 11.1803 0.494106
\(513\) 2.23607 0.0987248
\(514\) −40.2492 −1.77532
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −35.0000 −1.53781
\(519\) −20.1246 −0.883372
\(520\) 0 0
\(521\) 20.0000 0.876216 0.438108 0.898922i \(-0.355649\pi\)
0.438108 + 0.898922i \(0.355649\pi\)
\(522\) 10.0000 0.437688
\(523\) 6.70820 0.293329 0.146665 0.989186i \(-0.453146\pi\)
0.146665 + 0.989186i \(0.453146\pi\)
\(524\) −40.2492 −1.75830
\(525\) 0 0
\(526\) −40.0000 −1.74408
\(527\) 22.3607 0.974047
\(528\) 0 0
\(529\) −22.0000 −0.956522
\(530\) 0 0
\(531\) −5.00000 −0.216982
\(532\) −15.0000 −0.650332
\(533\) 30.0000 1.29944
\(534\) −13.4164 −0.580585
\(535\) 0 0
\(536\) 4.47214 0.193167
\(537\) −11.0000 −0.474685
\(538\) 67.0820 2.89211
\(539\) 0 0
\(540\) 0 0
\(541\) 17.8885 0.769089 0.384544 0.923107i \(-0.374359\pi\)
0.384544 + 0.923107i \(0.374359\pi\)
\(542\) 45.0000 1.93292
\(543\) 5.00000 0.214571
\(544\) −15.0000 −0.643120
\(545\) 0 0
\(546\) −22.3607 −0.956949
\(547\) −20.1246 −0.860466 −0.430233 0.902718i \(-0.641569\pi\)
−0.430233 + 0.902718i \(0.641569\pi\)
\(548\) 24.0000 1.02523
\(549\) 4.47214 0.190866
\(550\) 0 0
\(551\) −10.0000 −0.426014
\(552\) 2.23607 0.0951734
\(553\) −35.0000 −1.48835
\(554\) −30.0000 −1.27458
\(555\) 0 0
\(556\) 0 0
\(557\) 22.3607 0.947452 0.473726 0.880672i \(-0.342909\pi\)
0.473726 + 0.880672i \(0.342909\pi\)
\(558\) 22.3607 0.946603
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 55.0000 2.32003
\(563\) −35.7771 −1.50782 −0.753912 0.656975i \(-0.771836\pi\)
−0.753912 + 0.656975i \(0.771836\pi\)
\(564\) 9.00000 0.378968
\(565\) 0 0
\(566\) −15.0000 −0.630497
\(567\) 2.23607 0.0939060
\(568\) −29.0689 −1.21970
\(569\) 11.1803 0.468704 0.234352 0.972152i \(-0.424703\pi\)
0.234352 + 0.972152i \(0.424703\pi\)
\(570\) 0 0
\(571\) 35.7771 1.49722 0.748612 0.663008i \(-0.230720\pi\)
0.748612 + 0.663008i \(0.230720\pi\)
\(572\) 0 0
\(573\) −15.0000 −0.626634
\(574\) 33.5410 1.39998
\(575\) 0 0
\(576\) −13.0000 −0.541667
\(577\) −3.00000 −0.124892 −0.0624458 0.998048i \(-0.519890\pi\)
−0.0624458 + 0.998048i \(0.519890\pi\)
\(578\) −26.8328 −1.11610
\(579\) 8.94427 0.371711
\(580\) 0 0
\(581\) 10.0000 0.414870
\(582\) 38.0132 1.57570
\(583\) 0 0
\(584\) 30.0000 1.24141
\(585\) 0 0
\(586\) 35.0000 1.44584
\(587\) 23.0000 0.949312 0.474656 0.880172i \(-0.342573\pi\)
0.474656 + 0.880172i \(0.342573\pi\)
\(588\) 6.00000 0.247436
\(589\) −22.3607 −0.921356
\(590\) 0 0
\(591\) 15.6525 0.643857
\(592\) 7.00000 0.287698
\(593\) 4.47214 0.183649 0.0918243 0.995775i \(-0.470730\pi\)
0.0918243 + 0.995775i \(0.470730\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −60.3738 −2.47301
\(597\) −10.0000 −0.409273
\(598\) −10.0000 −0.408930
\(599\) 9.00000 0.367730 0.183865 0.982952i \(-0.441139\pi\)
0.183865 + 0.982952i \(0.441139\pi\)
\(600\) 0 0
\(601\) −13.4164 −0.547267 −0.273633 0.961834i \(-0.588225\pi\)
−0.273633 + 0.961834i \(0.588225\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) 0 0
\(606\) −25.0000 −1.01556
\(607\) −26.8328 −1.08911 −0.544555 0.838725i \(-0.683302\pi\)
−0.544555 + 0.838725i \(0.683302\pi\)
\(608\) 15.0000 0.608330
\(609\) −10.0000 −0.405220
\(610\) 0 0
\(611\) −13.4164 −0.542770
\(612\) 6.70820 0.271163
\(613\) −8.94427 −0.361256 −0.180628 0.983552i \(-0.557813\pi\)
−0.180628 + 0.983552i \(0.557813\pi\)
\(614\) −40.0000 −1.61427
\(615\) 0 0
\(616\) 0 0
\(617\) 22.0000 0.885687 0.442843 0.896599i \(-0.353970\pi\)
0.442843 + 0.896599i \(0.353970\pi\)
\(618\) −8.94427 −0.359791
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −71.5542 −2.86906
\(623\) 13.4164 0.537517
\(624\) 4.47214 0.179029
\(625\) 0 0
\(626\) −20.1246 −0.804341
\(627\) 0 0
\(628\) −6.00000 −0.239426
\(629\) −15.6525 −0.624105
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) −35.0000 −1.39223
\(633\) −26.8328 −1.06651
\(634\) 49.1935 1.95372
\(635\) 0 0
\(636\) −42.0000 −1.66541
\(637\) −8.94427 −0.354385
\(638\) 0 0
\(639\) −13.0000 −0.514272
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) −6.70820 −0.264340
\(645\) 0 0
\(646\) −11.1803 −0.439885
\(647\) −43.0000 −1.69050 −0.845252 0.534368i \(-0.820550\pi\)
−0.845252 + 0.534368i \(0.820550\pi\)
\(648\) 2.23607 0.0878410
\(649\) 0 0
\(650\) 0 0
\(651\) −22.3607 −0.876384
\(652\) −42.0000 −1.64485
\(653\) −26.0000 −1.01746 −0.508729 0.860927i \(-0.669885\pi\)
−0.508729 + 0.860927i \(0.669885\pi\)
\(654\) −40.0000 −1.56412
\(655\) 0 0
\(656\) −6.70820 −0.261911
\(657\) 13.4164 0.523424
\(658\) −15.0000 −0.584761
\(659\) −31.3050 −1.21947 −0.609734 0.792606i \(-0.708724\pi\)
−0.609734 + 0.792606i \(0.708724\pi\)
\(660\) 0 0
\(661\) 3.00000 0.116686 0.0583432 0.998297i \(-0.481418\pi\)
0.0583432 + 0.998297i \(0.481418\pi\)
\(662\) 44.7214 1.73814
\(663\) −10.0000 −0.388368
\(664\) 10.0000 0.388075
\(665\) 0 0
\(666\) −15.6525 −0.606521
\(667\) −4.47214 −0.173162
\(668\) 13.4164 0.519096
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 0 0
\(672\) 15.0000 0.578638
\(673\) −8.94427 −0.344776 −0.172388 0.985029i \(-0.555148\pi\)
−0.172388 + 0.985029i \(0.555148\pi\)
\(674\) −30.0000 −1.15556
\(675\) 0 0
\(676\) 21.0000 0.807692
\(677\) −49.1935 −1.89066 −0.945330 0.326116i \(-0.894260\pi\)
−0.945330 + 0.326116i \(0.894260\pi\)
\(678\) −31.3050 −1.20226
\(679\) −38.0132 −1.45881
\(680\) 0 0
\(681\) −13.4164 −0.514118
\(682\) 0 0
\(683\) 21.0000 0.803543 0.401771 0.915740i \(-0.368395\pi\)
0.401771 + 0.915740i \(0.368395\pi\)
\(684\) −6.70820 −0.256495
\(685\) 0 0
\(686\) −45.0000 −1.71811
\(687\) 5.00000 0.190762
\(688\) 0 0
\(689\) 62.6099 2.38525
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 60.3738 2.29507
\(693\) 0 0
\(694\) 30.0000 1.13878
\(695\) 0 0
\(696\) −10.0000 −0.379049
\(697\) 15.0000 0.568166
\(698\) −30.0000 −1.13552
\(699\) 20.1246 0.761183
\(700\) 0 0
\(701\) 42.4853 1.60465 0.802324 0.596889i \(-0.203597\pi\)
0.802324 + 0.596889i \(0.203597\pi\)
\(702\) −10.0000 −0.377426
\(703\) 15.6525 0.590344
\(704\) 0 0
\(705\) 0 0
\(706\) 76.0263 2.86129
\(707\) 25.0000 0.940222
\(708\) 15.0000 0.563735
\(709\) 19.0000 0.713560 0.356780 0.934188i \(-0.383875\pi\)
0.356780 + 0.934188i \(0.383875\pi\)
\(710\) 0 0
\(711\) −15.6525 −0.587014
\(712\) 13.4164 0.502801
\(713\) −10.0000 −0.374503
\(714\) −11.1803 −0.418414
\(715\) 0 0
\(716\) 33.0000 1.23327
\(717\) −8.94427 −0.334030
\(718\) 10.0000 0.373197
\(719\) −4.00000 −0.149175 −0.0745874 0.997214i \(-0.523764\pi\)
−0.0745874 + 0.997214i \(0.523764\pi\)
\(720\) 0 0
\(721\) 8.94427 0.333102
\(722\) −31.3050 −1.16505
\(723\) −22.3607 −0.831603
\(724\) −15.0000 −0.557471
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 22.3607 0.828742
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −13.4164 −0.495885
\(733\) 40.2492 1.48664 0.743319 0.668937i \(-0.233250\pi\)
0.743319 + 0.668937i \(0.233250\pi\)
\(734\) −40.2492 −1.48563
\(735\) 0 0
\(736\) 6.70820 0.247268
\(737\) 0 0
\(738\) 15.0000 0.552158
\(739\) −24.5967 −0.904806 −0.452403 0.891814i \(-0.649433\pi\)
−0.452403 + 0.891814i \(0.649433\pi\)
\(740\) 0 0
\(741\) 10.0000 0.367359
\(742\) 70.0000 2.56978
\(743\) −40.2492 −1.47660 −0.738300 0.674472i \(-0.764371\pi\)
−0.738300 + 0.674472i \(0.764371\pi\)
\(744\) −22.3607 −0.819782
\(745\) 0 0
\(746\) −20.0000 −0.732252
\(747\) 4.47214 0.163627
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 3.00000 0.109399
\(753\) 12.0000 0.437304
\(754\) 44.7214 1.62866
\(755\) 0 0
\(756\) −6.70820 −0.243975
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 22.3607 0.812176
\(759\) 0 0
\(760\) 0 0
\(761\) −40.2492 −1.45903 −0.729517 0.683963i \(-0.760255\pi\)
−0.729517 + 0.683963i \(0.760255\pi\)
\(762\) −35.0000 −1.26792
\(763\) 40.0000 1.44810
\(764\) 45.0000 1.62804
\(765\) 0 0
\(766\) 35.7771 1.29268
\(767\) −22.3607 −0.807397
\(768\) 9.00000 0.324760
\(769\) −35.7771 −1.29015 −0.645077 0.764117i \(-0.723175\pi\)
−0.645077 + 0.764117i \(0.723175\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −26.8328 −0.965734
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −38.0132 −1.36459
\(777\) 15.6525 0.561529
\(778\) 0 0
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) 0 0
\(782\) −5.00000 −0.178800
\(783\) −4.47214 −0.159821
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 30.0000 1.07006
\(787\) −46.9574 −1.67385 −0.836926 0.547316i \(-0.815649\pi\)
−0.836926 + 0.547316i \(0.815649\pi\)
\(788\) −46.9574 −1.67279
\(789\) 17.8885 0.636849
\(790\) 0 0
\(791\) 31.3050 1.11308
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) −49.1935 −1.74581
\(795\) 0 0
\(796\) 30.0000 1.06332
\(797\) 8.00000 0.283375 0.141687 0.989911i \(-0.454747\pi\)
0.141687 + 0.989911i \(0.454747\pi\)
\(798\) 11.1803 0.395780
\(799\) −6.70820 −0.237319
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 22.3607 0.789583
\(803\) 0 0
\(804\) −6.00000 −0.211604
\(805\) 0 0
\(806\) 100.000 3.52235
\(807\) −30.0000 −1.05605
\(808\) 25.0000 0.879497
\(809\) −24.5967 −0.864776 −0.432388 0.901688i \(-0.642329\pi\)
−0.432388 + 0.901688i \(0.642329\pi\)
\(810\) 0 0
\(811\) −46.9574 −1.64890 −0.824449 0.565936i \(-0.808515\pi\)
−0.824449 + 0.565936i \(0.808515\pi\)
\(812\) 30.0000 1.05279
\(813\) −20.1246 −0.705801
\(814\) 0 0
\(815\) 0 0
\(816\) 2.23607 0.0782780
\(817\) 0 0
\(818\) 30.0000 1.04893
\(819\) 10.0000 0.349428
\(820\) 0 0
\(821\) −31.3050 −1.09255 −0.546275 0.837606i \(-0.683955\pi\)
−0.546275 + 0.837606i \(0.683955\pi\)
\(822\) −17.8885 −0.623935
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) 8.94427 0.311588
\(825\) 0 0
\(826\) −25.0000 −0.869861
\(827\) 17.8885 0.622046 0.311023 0.950402i \(-0.399328\pi\)
0.311023 + 0.950402i \(0.399328\pi\)
\(828\) −3.00000 −0.104257
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 13.4164 0.465410
\(832\) −58.1378 −2.01556
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −10.0000 −0.345651
\(838\) 20.1246 0.695193
\(839\) −40.0000 −1.38095 −0.690477 0.723355i \(-0.742599\pi\)
−0.690477 + 0.723355i \(0.742599\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) −33.5410 −1.15590
\(843\) −24.5967 −0.847157
\(844\) 80.4984 2.77087
\(845\) 0 0
\(846\) −6.70820 −0.230633
\(847\) 0 0
\(848\) −14.0000 −0.480762
\(849\) 6.70820 0.230225
\(850\) 0 0
\(851\) 7.00000 0.239957
\(852\) 39.0000 1.33612
\(853\) −22.3607 −0.765615 −0.382808 0.923828i \(-0.625043\pi\)
−0.382808 + 0.923828i \(0.625043\pi\)
\(854\) 22.3607 0.765167
\(855\) 0 0
\(856\) 0 0
\(857\) 11.1803 0.381913 0.190957 0.981598i \(-0.438841\pi\)
0.190957 + 0.981598i \(0.438841\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −15.0000 −0.511199
\(862\) 50.0000 1.70301
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 6.70820 0.228218
\(865\) 0 0
\(866\) −31.3050 −1.06379
\(867\) 12.0000 0.407541
\(868\) 67.0820 2.27691
\(869\) 0 0
\(870\) 0 0
\(871\) 8.94427 0.303065
\(872\) 40.0000 1.35457
\(873\) −17.0000 −0.575363
\(874\) 5.00000 0.169128
\(875\) 0 0
\(876\) −40.2492 −1.35990
\(877\) −31.3050 −1.05709 −0.528547 0.848904i \(-0.677263\pi\)
−0.528547 + 0.848904i \(0.677263\pi\)
\(878\) 45.0000 1.51868
\(879\) −15.6525 −0.527945
\(880\) 0 0
\(881\) 12.0000 0.404290 0.202145 0.979356i \(-0.435209\pi\)
0.202145 + 0.979356i \(0.435209\pi\)
\(882\) −4.47214 −0.150585
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 30.0000 1.00901
\(885\) 0 0
\(886\) −46.9574 −1.57757
\(887\) 4.47214 0.150160 0.0750798 0.997178i \(-0.476079\pi\)
0.0750798 + 0.997178i \(0.476079\pi\)
\(888\) 15.6525 0.525263
\(889\) 35.0000 1.17386
\(890\) 0 0
\(891\) 0 0
\(892\) −42.0000 −1.40626
\(893\) 6.70820 0.224481
\(894\) 45.0000 1.50503
\(895\) 0 0
\(896\) −35.0000 −1.16927
\(897\) 4.47214 0.149320
\(898\) 8.94427 0.298474
\(899\) 44.7214 1.49154
\(900\) 0 0
\(901\) 31.3050 1.04292
\(902\) 0 0
\(903\) 0 0
\(904\) 31.3050 1.04119
\(905\) 0 0
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 40.2492 1.33572
\(909\) 11.1803 0.370828
\(910\) 0 0
\(911\) 55.0000 1.82223 0.911116 0.412151i \(-0.135222\pi\)
0.911116 + 0.412151i \(0.135222\pi\)
\(912\) −2.23607 −0.0740436
\(913\) 0 0
\(914\) −50.0000 −1.65385
\(915\) 0 0
\(916\) −15.0000 −0.495614
\(917\) −30.0000 −0.990687
\(918\) −5.00000 −0.165025
\(919\) −24.5967 −0.811372 −0.405686 0.914013i \(-0.632967\pi\)
−0.405686 + 0.914013i \(0.632967\pi\)
\(920\) 0 0
\(921\) 17.8885 0.589448
\(922\) −50.0000 −1.64666
\(923\) −58.1378 −1.91363
\(924\) 0 0
\(925\) 0 0
\(926\) −31.3050 −1.02874
\(927\) 4.00000 0.131377
\(928\) −30.0000 −0.984798
\(929\) −20.0000 −0.656179 −0.328089 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) 4.47214 0.146568
\(932\) −60.3738 −1.97761
\(933\) 32.0000 1.04763
\(934\) −71.5542 −2.34132
\(935\) 0 0
\(936\) 10.0000 0.326860
\(937\) −26.8328 −0.876590 −0.438295 0.898831i \(-0.644417\pi\)
−0.438295 + 0.898831i \(0.644417\pi\)
\(938\) 10.0000 0.326512
\(939\) 9.00000 0.293704
\(940\) 0 0
\(941\) −24.5967 −0.801831 −0.400916 0.916115i \(-0.631308\pi\)
−0.400916 + 0.916115i \(0.631308\pi\)
\(942\) 4.47214 0.145710
\(943\) −6.70820 −0.218449
\(944\) 5.00000 0.162736
\(945\) 0 0
\(946\) 0 0
\(947\) −17.0000 −0.552426 −0.276213 0.961096i \(-0.589079\pi\)
−0.276213 + 0.961096i \(0.589079\pi\)
\(948\) 46.9574 1.52511
\(949\) 60.0000 1.94768
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) 11.1803 0.362357
\(953\) −42.4853 −1.37623 −0.688117 0.725600i \(-0.741562\pi\)
−0.688117 + 0.725600i \(0.741562\pi\)
\(954\) 31.3050 1.01354
\(955\) 0 0
\(956\) 26.8328 0.867835
\(957\) 0 0
\(958\) 50.0000 1.61543
\(959\) 17.8885 0.577651
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) −70.0000 −2.25689
\(963\) 0 0
\(964\) 67.0820 2.16057
\(965\) 0 0
\(966\) 5.00000 0.160872
\(967\) 8.94427 0.287628 0.143814 0.989605i \(-0.454063\pi\)
0.143814 + 0.989605i \(0.454063\pi\)
\(968\) 0 0
\(969\) 5.00000 0.160623
\(970\) 0 0
\(971\) 5.00000 0.160458 0.0802288 0.996776i \(-0.474435\pi\)
0.0802288 + 0.996776i \(0.474435\pi\)
\(972\) −3.00000 −0.0962250
\(973\) 0 0
\(974\) 71.5542 2.29274
\(975\) 0 0
\(976\) −4.47214 −0.143150
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 31.3050 1.00102
\(979\) 0 0
\(980\) 0 0
\(981\) 17.8885 0.571137
\(982\) −10.0000 −0.319113
\(983\) 49.0000 1.56286 0.781429 0.623995i \(-0.214491\pi\)
0.781429 + 0.623995i \(0.214491\pi\)
\(984\) −15.0000 −0.478183
\(985\) 0 0
\(986\) 22.3607 0.712109
\(987\) 6.70820 0.213524
\(988\) −30.0000 −0.954427
\(989\) 0 0
\(990\) 0 0
\(991\) 10.0000 0.317660 0.158830 0.987306i \(-0.449228\pi\)
0.158830 + 0.987306i \(0.449228\pi\)
\(992\) −67.0820 −2.12986
\(993\) −20.0000 −0.634681
\(994\) −65.0000 −2.06167
\(995\) 0 0
\(996\) −13.4164 −0.425115
\(997\) −62.6099 −1.98288 −0.991438 0.130580i \(-0.958316\pi\)
−0.991438 + 0.130580i \(0.958316\pi\)
\(998\) 44.7214 1.41563
\(999\) 7.00000 0.221470
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 9075.2.a.bk.1.2 yes 2
5.4 even 2 9075.2.a.br.1.1 yes 2
11.10 odd 2 inner 9075.2.a.bk.1.1 2
55.54 odd 2 9075.2.a.br.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9075.2.a.bk.1.1 2 11.10 odd 2 inner
9075.2.a.bk.1.2 yes 2 1.1 even 1 trivial
9075.2.a.br.1.1 yes 2 5.4 even 2
9075.2.a.br.1.2 yes 2 55.54 odd 2