Properties

Label 3450.2.a.u.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -4.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -4.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -4.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +8.00000 q^{29} +8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} -4.00000 q^{38} -4.00000 q^{39} +6.00000 q^{41} -4.00000 q^{42} -6.00000 q^{43} +2.00000 q^{44} +1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} +9.00000 q^{49} +6.00000 q^{51} -4.00000 q^{52} +14.0000 q^{53} +1.00000 q^{54} -4.00000 q^{56} -4.00000 q^{57} +8.00000 q^{58} +4.00000 q^{59} +6.00000 q^{61} +8.00000 q^{62} -4.00000 q^{63} +1.00000 q^{64} +2.00000 q^{66} -14.0000 q^{67} +6.00000 q^{68} +1.00000 q^{69} +10.0000 q^{71} +1.00000 q^{72} -14.0000 q^{73} +10.0000 q^{74} -4.00000 q^{76} -8.00000 q^{77} -4.00000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{84} -6.00000 q^{86} +8.00000 q^{87} +2.00000 q^{88} +16.0000 q^{91} +1.00000 q^{92} +8.00000 q^{93} +4.00000 q^{94} +1.00000 q^{96} +8.00000 q^{97} +9.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −4.00000 −1.51186 −0.755929 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 2.00000 0.426401
\(23\) 1.00000 0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 1.00000 0.176777
\(33\) 2.00000 0.348155
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −4.00000 −0.648886
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) −4.00000 −0.617213
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) −4.00000 −0.554700
\(53\) 14.0000 1.92305 0.961524 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) −4.00000 −0.529813
\(58\) 8.00000 1.05045
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 8.00000 1.01600
\(63\) −4.00000 −0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.00000 0.246183
\(67\) −14.0000 −1.71037 −0.855186 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) 6.00000 0.727607
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.0000 −1.63858 −0.819288 0.573382i \(-0.805631\pi\)
−0.819288 + 0.573382i \(0.805631\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −8.00000 −0.911685
\(78\) −4.00000 −0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −6.00000 −0.646997
\(87\) 8.00000 0.857690
\(88\) 2.00000 0.213201
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 16.0000 1.67726
\(92\) 1.00000 0.104257
\(93\) 8.00000 0.829561
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 9.00000 0.909137
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −8.00000 −0.796030 −0.398015 0.917379i \(-0.630301\pi\)
−0.398015 + 0.917379i \(0.630301\pi\)
\(102\) 6.00000 0.594089
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) −4.00000 −0.392232
\(105\) 0 0
\(106\) 14.0000 1.35980
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 1.00000 0.0962250
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 10.0000 0.949158
\(112\) −4.00000 −0.377964
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 8.00000 0.742781
\(117\) −4.00000 −0.369800
\(118\) 4.00000 0.368230
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 6.00000 0.543214
\(123\) 6.00000 0.541002
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) −4.00000 −0.356348
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 2.00000 0.174078
\(133\) 16.0000 1.38738
\(134\) −14.0000 −1.20942
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 1.00000 0.0851257
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) 10.0000 0.839181
\(143\) −8.00000 −0.668994
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −14.0000 −1.15865
\(147\) 9.00000 0.742307
\(148\) 10.0000 0.821995
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −4.00000 −0.324443
\(153\) 6.00000 0.485071
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) −4.00000 −0.320256
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) −8.00000 −0.636446
\(159\) 14.0000 1.11027
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 1.00000 0.0785674
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 8.00000 0.619059 0.309529 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) −4.00000 −0.308607
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) −6.00000 −0.457496
\(173\) −10.0000 −0.760286 −0.380143 0.924928i \(-0.624125\pi\)
−0.380143 + 0.924928i \(0.624125\pi\)
\(174\) 8.00000 0.606478
\(175\) 0 0
\(176\) 2.00000 0.150756
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 16.0000 1.18600
\(183\) 6.00000 0.443533
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) 12.0000 0.877527
\(188\) 4.00000 0.291730
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.0000 1.58359 0.791797 0.610784i \(-0.209146\pi\)
0.791797 + 0.610784i \(0.209146\pi\)
\(194\) 8.00000 0.574367
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 2.00000 0.142134
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) −14.0000 −0.987484
\(202\) −8.00000 −0.562878
\(203\) −32.0000 −2.24596
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −4.00000 −0.278693
\(207\) 1.00000 0.0695048
\(208\) −4.00000 −0.277350
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 14.0000 0.961524
\(213\) 10.0000 0.685189
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −32.0000 −2.17230
\(218\) −14.0000 −0.948200
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) −24.0000 −1.61441
\(222\) 10.0000 0.671156
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) −4.00000 −0.264906
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 8.00000 0.525226
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) −4.00000 −0.261488
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) −8.00000 −0.519656
\(238\) −24.0000 −1.55569
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −7.00000 −0.449977
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 16.0000 1.01806
\(248\) 8.00000 0.508001
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −2.00000 −0.126239 −0.0631194 0.998006i \(-0.520105\pi\)
−0.0631194 + 0.998006i \(0.520105\pi\)
\(252\) −4.00000 −0.251976
\(253\) 2.00000 0.125739
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) −6.00000 −0.373544
\(259\) −40.0000 −2.48548
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) 16.0000 0.988483
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 2.00000 0.123091
\(265\) 0 0
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) 6.00000 0.363803
\(273\) 16.0000 0.968364
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 12.0000 0.721010 0.360505 0.932757i \(-0.382604\pi\)
0.360505 + 0.932757i \(0.382604\pi\)
\(278\) −20.0000 −1.19952
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 8.00000 0.477240 0.238620 0.971113i \(-0.423305\pi\)
0.238620 + 0.971113i \(0.423305\pi\)
\(282\) 4.00000 0.238197
\(283\) 22.0000 1.30776 0.653882 0.756596i \(-0.273139\pi\)
0.653882 + 0.756596i \(0.273139\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) −8.00000 −0.473050
\(287\) −24.0000 −1.41668
\(288\) 1.00000 0.0589256
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 8.00000 0.468968
\(292\) −14.0000 −0.819288
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 10.0000 0.581238
\(297\) 2.00000 0.116052
\(298\) −10.0000 −0.579284
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 24.0000 1.38334
\(302\) 20.0000 1.15087
\(303\) −8.00000 −0.459588
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) −8.00000 −0.455842
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −14.0000 −0.793867 −0.396934 0.917847i \(-0.629926\pi\)
−0.396934 + 0.917847i \(0.629926\pi\)
\(312\) −4.00000 −0.226455
\(313\) 4.00000 0.226093 0.113047 0.993590i \(-0.463939\pi\)
0.113047 + 0.993590i \(0.463939\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 14.0000 0.785081
\(319\) 16.0000 0.895828
\(320\) 0 0
\(321\) −8.00000 −0.446516
\(322\) −4.00000 −0.222911
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 8.00000 0.443079
\(327\) −14.0000 −0.774202
\(328\) 6.00000 0.331295
\(329\) −16.0000 −0.882109
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 4.00000 0.219529
\(333\) 10.0000 0.547997
\(334\) 8.00000 0.437741
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 3.00000 0.163178
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 16.0000 0.866449
\(342\) −4.00000 −0.216295
\(343\) −8.00000 −0.431959
\(344\) −6.00000 −0.323498
\(345\) 0 0
\(346\) −10.0000 −0.537603
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 8.00000 0.428845
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 2.00000 0.106600
\(353\) −14.0000 −0.745145 −0.372572 0.928003i \(-0.621524\pi\)
−0.372572 + 0.928003i \(0.621524\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) 0 0
\(357\) −24.0000 −1.27021
\(358\) −4.00000 −0.211407
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −2.00000 −0.105118
\(363\) −7.00000 −0.367405
\(364\) 16.0000 0.838628
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) 12.0000 0.626395 0.313197 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) 1.00000 0.0521286
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −56.0000 −2.90738
\(372\) 8.00000 0.414781
\(373\) 18.0000 0.932005 0.466002 0.884783i \(-0.345694\pi\)
0.466002 + 0.884783i \(0.345694\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) −32.0000 −1.64808
\(378\) −4.00000 −0.205738
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) −16.0000 −0.818631
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) −6.00000 −0.304997
\(388\) 8.00000 0.406138
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) 9.00000 0.454569
\(393\) 16.0000 0.807093
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) 2.00000 0.100504
\(397\) 28.0000 1.40528 0.702640 0.711546i \(-0.252005\pi\)
0.702640 + 0.711546i \(0.252005\pi\)
\(398\) 16.0000 0.802008
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 20.0000 0.998752 0.499376 0.866385i \(-0.333563\pi\)
0.499376 + 0.866385i \(0.333563\pi\)
\(402\) −14.0000 −0.698257
\(403\) −32.0000 −1.59403
\(404\) −8.00000 −0.398015
\(405\) 0 0
\(406\) −32.0000 −1.58813
\(407\) 20.0000 0.991363
\(408\) 6.00000 0.297044
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) −4.00000 −0.197066
\(413\) −16.0000 −0.787309
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −4.00000 −0.196116
\(417\) −20.0000 −0.979404
\(418\) −8.00000 −0.391293
\(419\) −30.0000 −1.46560 −0.732798 0.680446i \(-0.761786\pi\)
−0.732798 + 0.680446i \(0.761786\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) −20.0000 −0.973585
\(423\) 4.00000 0.194487
\(424\) 14.0000 0.679900
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) −24.0000 −1.16144
\(428\) −8.00000 −0.386695
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000 0.0481125
\(433\) −32.0000 −1.53782 −0.768911 0.639356i \(-0.779201\pi\)
−0.768911 + 0.639356i \(0.779201\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −4.00000 −0.191346
\(438\) −14.0000 −0.668946
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −24.0000 −1.14156
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 10.0000 0.474579
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) −10.0000 −0.472984
\(448\) −4.00000 −0.188982
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 10.0000 0.470360
\(453\) 20.0000 0.939682
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) −16.0000 −0.748448 −0.374224 0.927338i \(-0.622091\pi\)
−0.374224 + 0.927338i \(0.622091\pi\)
\(458\) −22.0000 −1.02799
\(459\) 6.00000 0.280056
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) −8.00000 −0.372194
\(463\) 10.0000 0.464739 0.232370 0.972628i \(-0.425352\pi\)
0.232370 + 0.972628i \(0.425352\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −4.00000 −0.184900
\(469\) 56.0000 2.58584
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 4.00000 0.184115
\(473\) −12.0000 −0.551761
\(474\) −8.00000 −0.367452
\(475\) 0 0
\(476\) −24.0000 −1.10004
\(477\) 14.0000 0.641016
\(478\) 26.0000 1.18921
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) −40.0000 −1.82384
\(482\) 10.0000 0.455488
\(483\) −4.00000 −0.182006
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) 6.00000 0.271607
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 6.00000 0.270501
\(493\) 48.0000 2.16181
\(494\) 16.0000 0.719874
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) −40.0000 −1.79425
\(498\) 4.00000 0.179244
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 8.00000 0.357414
\(502\) −2.00000 −0.0892644
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 2.00000 0.0889108
\(507\) 3.00000 0.133235
\(508\) −2.00000 −0.0887357
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) 0 0
\(511\) 56.0000 2.47729
\(512\) 1.00000 0.0441942
\(513\) −4.00000 −0.176604
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −6.00000 −0.264135
\(517\) 8.00000 0.351840
\(518\) −40.0000 −1.75750
\(519\) −10.0000 −0.438951
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 8.00000 0.350150
\(523\) 14.0000 0.612177 0.306089 0.952003i \(-0.400980\pi\)
0.306089 + 0.952003i \(0.400980\pi\)
\(524\) 16.0000 0.698963
\(525\) 0 0
\(526\) −16.0000 −0.697633
\(527\) 48.0000 2.09091
\(528\) 2.00000 0.0870388
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 4.00000 0.173585
\(532\) 16.0000 0.693688
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) −14.0000 −0.604708
\(537\) −4.00000 −0.172613
\(538\) 12.0000 0.517357
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −8.00000 −0.343629
\(543\) −2.00000 −0.0858282
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 16.0000 0.684737
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −6.00000 −0.256307
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) 1.00000 0.0425628
\(553\) 32.0000 1.36078
\(554\) 12.0000 0.509831
\(555\) 0 0
\(556\) −20.0000 −0.848189
\(557\) 2.00000 0.0847427 0.0423714 0.999102i \(-0.486509\pi\)
0.0423714 + 0.999102i \(0.486509\pi\)
\(558\) 8.00000 0.338667
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 12.0000 0.506640
\(562\) 8.00000 0.337460
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) 22.0000 0.924729
\(567\) −4.00000 −0.167984
\(568\) 10.0000 0.419591
\(569\) −44.0000 −1.84458 −0.922288 0.386503i \(-0.873683\pi\)
−0.922288 + 0.386503i \(0.873683\pi\)
\(570\) 0 0
\(571\) −24.0000 −1.00437 −0.502184 0.864761i \(-0.667470\pi\)
−0.502184 + 0.864761i \(0.667470\pi\)
\(572\) −8.00000 −0.334497
\(573\) −16.0000 −0.668410
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 19.0000 0.790296
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) −16.0000 −0.663792
\(582\) 8.00000 0.331611
\(583\) 28.0000 1.15964
\(584\) −14.0000 −0.579324
\(585\) 0 0
\(586\) −26.0000 −1.07405
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 9.00000 0.371154
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 22.0000 0.904959
\(592\) 10.0000 0.410997
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 2.00000 0.0820610
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 16.0000 0.654836
\(598\) −4.00000 −0.163572
\(599\) 14.0000 0.572024 0.286012 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) −26.0000 −1.06056 −0.530281 0.847822i \(-0.677914\pi\)
−0.530281 + 0.847822i \(0.677914\pi\)
\(602\) 24.0000 0.978167
\(603\) −14.0000 −0.570124
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) −8.00000 −0.324978
\(607\) −30.0000 −1.21766 −0.608831 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(608\) −4.00000 −0.162221
\(609\) −32.0000 −1.29671
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 6.00000 0.242536
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) −8.00000 −0.322329
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) −4.00000 −0.160904
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −14.0000 −0.561349
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) 4.00000 0.159872
\(627\) −8.00000 −0.319489
\(628\) 2.00000 0.0798087
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) −8.00000 −0.318223
\(633\) −20.0000 −0.794929
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 14.0000 0.555136
\(637\) −36.0000 −1.42637
\(638\) 16.0000 0.633446
\(639\) 10.0000 0.395594
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) −8.00000 −0.315735
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 1.00000 0.0392837
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) 8.00000 0.313304
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) −14.0000 −0.546192
\(658\) −16.0000 −0.623745
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 12.0000 0.466393
\(663\) −24.0000 −0.932083
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) 8.00000 0.309761
\(668\) 8.00000 0.309529
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) −4.00000 −0.154303
\(673\) 34.0000 1.31060 0.655302 0.755367i \(-0.272541\pi\)
0.655302 + 0.755367i \(0.272541\pi\)
\(674\) 20.0000 0.770371
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 26.0000 0.999261 0.499631 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 10.0000 0.384048
\(679\) −32.0000 −1.22805
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) 16.0000 0.612672
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) −8.00000 −0.305441
\(687\) −22.0000 −0.839352
\(688\) −6.00000 −0.228748
\(689\) −56.0000 −2.13343
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) −10.0000 −0.380143
\(693\) −8.00000 −0.303895
\(694\) −12.0000 −0.455514
\(695\) 0 0
\(696\) 8.00000 0.303239
\(697\) 36.0000 1.36360
\(698\) −30.0000 −1.13552
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) −4.00000 −0.150970
\(703\) −40.0000 −1.50863
\(704\) 2.00000 0.0753778
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 32.0000 1.20348
\(708\) 4.00000 0.150329
\(709\) 34.0000 1.27690 0.638448 0.769665i \(-0.279577\pi\)
0.638448 + 0.769665i \(0.279577\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 8.00000 0.299602
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 26.0000 0.970988
\(718\) −16.0000 −0.597115
\(719\) 46.0000 1.71551 0.857755 0.514058i \(-0.171858\pi\)
0.857755 + 0.514058i \(0.171858\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) −3.00000 −0.111648
\(723\) 10.0000 0.371904
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) −7.00000 −0.259794
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 16.0000 0.592999
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) 6.00000 0.221766
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 12.0000 0.442928
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −28.0000 −1.03139
\(738\) 6.00000 0.220863
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) −56.0000 −2.05582
\(743\) −32.0000 −1.17397 −0.586983 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) 18.0000 0.659027
\(747\) 4.00000 0.146352
\(748\) 12.0000 0.438763
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 4.00000 0.145865
\(753\) −2.00000 −0.0728841
\(754\) −32.0000 −1.16537
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) −16.0000 −0.581146
\(759\) 2.00000 0.0725954
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) −2.00000 −0.0724524
\(763\) 56.0000 2.02734
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) −16.0000 −0.577727
\(768\) 1.00000 0.0360844
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 22.0000 0.791797
\(773\) −42.0000 −1.51064 −0.755318 0.655359i \(-0.772517\pi\)
−0.755318 + 0.655359i \(0.772517\pi\)
\(774\) −6.00000 −0.215666
\(775\) 0 0
\(776\) 8.00000 0.287183
\(777\) −40.0000 −1.43499
\(778\) −14.0000 −0.501924
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 20.0000 0.715656
\(782\) 6.00000 0.214560
\(783\) 8.00000 0.285897
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 16.0000 0.570701
\(787\) −50.0000 −1.78231 −0.891154 0.453701i \(-0.850103\pi\)
−0.891154 + 0.453701i \(0.850103\pi\)
\(788\) 22.0000 0.783718
\(789\) −16.0000 −0.569615
\(790\) 0 0
\(791\) −40.0000 −1.42224
\(792\) 2.00000 0.0710669
\(793\) −24.0000 −0.852265
\(794\) 28.0000 0.993683
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) −54.0000 −1.91278 −0.956389 0.292096i \(-0.905647\pi\)
−0.956389 + 0.292096i \(0.905647\pi\)
\(798\) 16.0000 0.566394
\(799\) 24.0000 0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 20.0000 0.706225
\(803\) −28.0000 −0.988099
\(804\) −14.0000 −0.493742
\(805\) 0 0
\(806\) −32.0000 −1.12715
\(807\) 12.0000 0.422420
\(808\) −8.00000 −0.281439
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −12.0000 −0.421377 −0.210688 0.977553i \(-0.567571\pi\)
−0.210688 + 0.977553i \(0.567571\pi\)
\(812\) −32.0000 −1.12298
\(813\) −8.00000 −0.280572
\(814\) 20.0000 0.701000
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 24.0000 0.839654
\(818\) −10.0000 −0.349642
\(819\) 16.0000 0.559085
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) −6.00000 −0.209274
\(823\) −42.0000 −1.46403 −0.732014 0.681290i \(-0.761419\pi\)
−0.732014 + 0.681290i \(0.761419\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 1.00000 0.0347524
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 12.0000 0.416275
\(832\) −4.00000 −0.138675
\(833\) 54.0000 1.87099
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) −8.00000 −0.276686
\(837\) 8.00000 0.276520
\(838\) −30.0000 −1.03633
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 6.00000 0.206774
\(843\) 8.00000 0.275535
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 28.0000 0.962091
\(848\) 14.0000 0.480762
\(849\) 22.0000 0.755038
\(850\) 0 0
\(851\) 10.0000 0.342796
\(852\) 10.0000 0.342594
\(853\) 4.00000 0.136957 0.0684787 0.997653i \(-0.478185\pi\)
0.0684787 + 0.997653i \(0.478185\pi\)
\(854\) −24.0000 −0.821263
\(855\) 0 0
\(856\) −8.00000 −0.273434
\(857\) −6.00000 −0.204956 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(858\) −8.00000 −0.273115
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) 8.00000 0.272481
\(863\) 36.0000 1.22545 0.612727 0.790295i \(-0.290072\pi\)
0.612727 + 0.790295i \(0.290072\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −32.0000 −1.08740
\(867\) 19.0000 0.645274
\(868\) −32.0000 −1.08615
\(869\) −16.0000 −0.542763
\(870\) 0 0
\(871\) 56.0000 1.89749
\(872\) −14.0000 −0.474100
\(873\) 8.00000 0.270759
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) −14.0000 −0.473016
\(877\) −20.0000 −0.675352 −0.337676 0.941262i \(-0.609641\pi\)
−0.337676 + 0.941262i \(0.609641\pi\)
\(878\) −20.0000 −0.674967
\(879\) −26.0000 −0.876958
\(880\) 0 0
\(881\) −48.0000 −1.61716 −0.808581 0.588386i \(-0.799764\pi\)
−0.808581 + 0.588386i \(0.799764\pi\)
\(882\) 9.00000 0.303046
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) −24.0000 −0.807207
\(885\) 0 0
\(886\) 28.0000 0.940678
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 10.0000 0.335578
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) −14.0000 −0.468755
\(893\) −16.0000 −0.535420
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) −4.00000 −0.133631
\(897\) −4.00000 −0.133556
\(898\) 30.0000 1.00111
\(899\) 64.0000 2.13452
\(900\) 0 0
\(901\) 84.0000 2.79845
\(902\) 12.0000 0.399556
\(903\) 24.0000 0.798670
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) −46.0000 −1.52740 −0.763702 0.645568i \(-0.776621\pi\)
−0.763702 + 0.645568i \(0.776621\pi\)
\(908\) −8.00000 −0.265489
\(909\) −8.00000 −0.265343
\(910\) 0 0
\(911\) −36.0000 −1.19273 −0.596367 0.802712i \(-0.703390\pi\)
−0.596367 + 0.802712i \(0.703390\pi\)
\(912\) −4.00000 −0.132453
\(913\) 8.00000 0.264761
\(914\) −16.0000 −0.529233
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) −64.0000 −2.11347
\(918\) 6.00000 0.198030
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) 0 0
\(923\) −40.0000 −1.31662
\(924\) −8.00000 −0.263181
\(925\) 0 0
\(926\) 10.0000 0.328620
\(927\) −4.00000 −0.131377
\(928\) 8.00000 0.262613
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) 6.00000 0.196537
\(933\) −14.0000 −0.458339
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −4.00000 −0.130744
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 56.0000 1.82846
\(939\) 4.00000 0.130535
\(940\) 0 0
\(941\) 38.0000 1.23876 0.619382 0.785090i \(-0.287383\pi\)
0.619382 + 0.785090i \(0.287383\pi\)
\(942\) 2.00000 0.0651635
\(943\) 6.00000 0.195387
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 20.0000 0.649913 0.324956 0.945729i \(-0.394650\pi\)
0.324956 + 0.945729i \(0.394650\pi\)
\(948\) −8.00000 −0.259828
\(949\) 56.0000 1.81784
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) −24.0000 −0.777844
\(953\) −38.0000 −1.23094 −0.615470 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 14.0000 0.453267
\(955\) 0 0
\(956\) 26.0000 0.840900
\(957\) 16.0000 0.517207
\(958\) 24.0000 0.775405
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) −40.0000 −1.28965
\(963\) −8.00000 −0.257796
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) −22.0000 −0.707472 −0.353736 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) −7.00000 −0.224989
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 1.00000 0.0320750
\(973\) 80.0000 2.56468
\(974\) 10.0000 0.320421
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) −10.0000 −0.319928 −0.159964 0.987123i \(-0.551138\pi\)
−0.159964 + 0.987123i \(0.551138\pi\)
\(978\) 8.00000 0.255812
\(979\) 0 0
\(980\) 0 0
\(981\) −14.0000 −0.446986
\(982\) −24.0000 −0.765871
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) 48.0000 1.52863
\(987\) −16.0000 −0.509286
\(988\) 16.0000 0.509028
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 8.00000 0.254000
\(993\) 12.0000 0.380808
\(994\) −40.0000 −1.26872
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) −4.00000 −0.126681 −0.0633406 0.997992i \(-0.520175\pi\)
−0.0633406 + 0.997992i \(0.520175\pi\)
\(998\) 20.0000 0.633089
\(999\) 10.0000 0.316386
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3450.2.a.u.1.1 1
5.2 odd 4 3450.2.d.s.2899.2 2
5.3 odd 4 3450.2.d.s.2899.1 2
5.4 even 2 690.2.a.d.1.1 1
15.14 odd 2 2070.2.a.o.1.1 1
20.19 odd 2 5520.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.d.1.1 1 5.4 even 2
2070.2.a.o.1.1 1 15.14 odd 2
3450.2.a.u.1.1 1 1.1 even 1 trivial
3450.2.d.s.2899.1 2 5.3 odd 4
3450.2.d.s.2899.2 2 5.2 odd 4
5520.2.a.bb.1.1 1 20.19 odd 2