Properties

Label 3450.2.a.bm.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
Defining polynomial: \(x^{2} - x - 14\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.27492\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.27492 q^{11} +1.00000 q^{12} +3.27492 q^{13} +3.00000 q^{14} +1.00000 q^{16} -2.27492 q^{17} +1.00000 q^{18} +3.27492 q^{19} +3.00000 q^{21} -1.27492 q^{22} -1.00000 q^{23} +1.00000 q^{24} +3.27492 q^{26} +1.00000 q^{27} +3.00000 q^{28} -9.54983 q^{29} +6.00000 q^{31} +1.00000 q^{32} -1.27492 q^{33} -2.27492 q^{34} +1.00000 q^{36} +10.8248 q^{37} +3.27492 q^{38} +3.27492 q^{39} -2.72508 q^{41} +3.00000 q^{42} +7.27492 q^{43} -1.27492 q^{44} -1.00000 q^{46} +8.27492 q^{47} +1.00000 q^{48} +2.00000 q^{49} -2.27492 q^{51} +3.27492 q^{52} -10.5498 q^{53} +1.00000 q^{54} +3.00000 q^{56} +3.27492 q^{57} -9.54983 q^{58} +12.5498 q^{59} +0.549834 q^{61} +6.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} -1.27492 q^{66} -4.54983 q^{67} -2.27492 q^{68} -1.00000 q^{69} -8.82475 q^{71} +1.00000 q^{72} -15.5498 q^{73} +10.8248 q^{74} +3.27492 q^{76} -3.82475 q^{77} +3.27492 q^{78} -11.8248 q^{79} +1.00000 q^{81} -2.72508 q^{82} -2.45017 q^{83} +3.00000 q^{84} +7.27492 q^{86} -9.54983 q^{87} -1.27492 q^{88} +8.27492 q^{89} +9.82475 q^{91} -1.00000 q^{92} +6.00000 q^{93} +8.27492 q^{94} +1.00000 q^{96} +14.0000 q^{97} +2.00000 q^{98} -1.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{7} + 2q^{8} + 2q^{9} + 5q^{11} + 2q^{12} - q^{13} + 6q^{14} + 2q^{16} + 3q^{17} + 2q^{18} - q^{19} + 6q^{21} + 5q^{22} - 2q^{23} + 2q^{24} - q^{26} + 2q^{27} + 6q^{28} - 4q^{29} + 12q^{31} + 2q^{32} + 5q^{33} + 3q^{34} + 2q^{36} - q^{37} - q^{38} - q^{39} - 13q^{41} + 6q^{42} + 7q^{43} + 5q^{44} - 2q^{46} + 9q^{47} + 2q^{48} + 4q^{49} + 3q^{51} - q^{52} - 6q^{53} + 2q^{54} + 6q^{56} - q^{57} - 4q^{58} + 10q^{59} - 14q^{61} + 12q^{62} + 6q^{63} + 2q^{64} + 5q^{66} + 6q^{67} + 3q^{68} - 2q^{69} + 5q^{71} + 2q^{72} - 16q^{73} - q^{74} - q^{76} + 15q^{77} - q^{78} - q^{79} + 2q^{81} - 13q^{82} - 20q^{83} + 6q^{84} + 7q^{86} - 4q^{87} + 5q^{88} + 9q^{89} - 3q^{91} - 2q^{92} + 12q^{93} + 9q^{94} + 2q^{96} + 28q^{97} + 4q^{98} + 5q^{99} + O(q^{100}) \)

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
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.27492 −0.384402 −0.192201 0.981356i \(-0.561563\pi\)
−0.192201 + 0.981356i \(0.561563\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.27492 0.908299 0.454149 0.890926i \(-0.349943\pi\)
0.454149 + 0.890926i \(0.349943\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.27492 −0.551748 −0.275874 0.961194i \(-0.588967\pi\)
−0.275874 + 0.961194i \(0.588967\pi\)
\(18\) 1.00000 0.235702
\(19\) 3.27492 0.751318 0.375659 0.926758i \(-0.377416\pi\)
0.375659 + 0.926758i \(0.377416\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) −1.27492 −0.271813
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 3.27492 0.642264
\(27\) 1.00000 0.192450
\(28\) 3.00000 0.566947
\(29\) −9.54983 −1.77336 −0.886680 0.462384i \(-0.846994\pi\)
−0.886680 + 0.462384i \(0.846994\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.27492 −0.221935
\(34\) −2.27492 −0.390145
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 10.8248 1.77958 0.889789 0.456372i \(-0.150851\pi\)
0.889789 + 0.456372i \(0.150851\pi\)
\(38\) 3.27492 0.531262
\(39\) 3.27492 0.524406
\(40\) 0 0
\(41\) −2.72508 −0.425586 −0.212793 0.977097i \(-0.568256\pi\)
−0.212793 + 0.977097i \(0.568256\pi\)
\(42\) 3.00000 0.462910
\(43\) 7.27492 1.10941 0.554707 0.832046i \(-0.312830\pi\)
0.554707 + 0.832046i \(0.312830\pi\)
\(44\) −1.27492 −0.192201
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 8.27492 1.20702 0.603510 0.797355i \(-0.293768\pi\)
0.603510 + 0.797355i \(0.293768\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.00000 0.285714
\(50\) 0 0
\(51\) −2.27492 −0.318552
\(52\) 3.27492 0.454149
\(53\) −10.5498 −1.44913 −0.724566 0.689206i \(-0.757960\pi\)
−0.724566 + 0.689206i \(0.757960\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 3.00000 0.400892
\(57\) 3.27492 0.433773
\(58\) −9.54983 −1.25395
\(59\) 12.5498 1.63385 0.816925 0.576744i \(-0.195677\pi\)
0.816925 + 0.576744i \(0.195677\pi\)
\(60\) 0 0
\(61\) 0.549834 0.0703991 0.0351995 0.999380i \(-0.488793\pi\)
0.0351995 + 0.999380i \(0.488793\pi\)
\(62\) 6.00000 0.762001
\(63\) 3.00000 0.377964
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.27492 −0.156931
\(67\) −4.54983 −0.555851 −0.277925 0.960603i \(-0.589647\pi\)
−0.277925 + 0.960603i \(0.589647\pi\)
\(68\) −2.27492 −0.275874
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.82475 −1.04731 −0.523653 0.851932i \(-0.675431\pi\)
−0.523653 + 0.851932i \(0.675431\pi\)
\(72\) 1.00000 0.117851
\(73\) −15.5498 −1.81997 −0.909985 0.414641i \(-0.863907\pi\)
−0.909985 + 0.414641i \(0.863907\pi\)
\(74\) 10.8248 1.25835
\(75\) 0 0
\(76\) 3.27492 0.375659
\(77\) −3.82475 −0.435871
\(78\) 3.27492 0.370811
\(79\) −11.8248 −1.33039 −0.665194 0.746670i \(-0.731651\pi\)
−0.665194 + 0.746670i \(0.731651\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.72508 −0.300935
\(83\) −2.45017 −0.268941 −0.134470 0.990918i \(-0.542933\pi\)
−0.134470 + 0.990918i \(0.542933\pi\)
\(84\) 3.00000 0.327327
\(85\) 0 0
\(86\) 7.27492 0.784474
\(87\) −9.54983 −1.02385
\(88\) −1.27492 −0.135907
\(89\) 8.27492 0.877139 0.438570 0.898697i \(-0.355485\pi\)
0.438570 + 0.898697i \(0.355485\pi\)
\(90\) 0 0
\(91\) 9.82475 1.02991
\(92\) −1.00000 −0.104257
\(93\) 6.00000 0.622171
\(94\) 8.27492 0.853493
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 14.0000 1.42148 0.710742 0.703452i \(-0.248359\pi\)
0.710742 + 0.703452i \(0.248359\pi\)
\(98\) 2.00000 0.202031
\(99\) −1.27492 −0.128134
\(100\) 0 0
\(101\) 8.27492 0.823385 0.411693 0.911323i \(-0.364938\pi\)
0.411693 + 0.911323i \(0.364938\pi\)
\(102\) −2.27492 −0.225250
\(103\) −3.00000 −0.295599 −0.147799 0.989017i \(-0.547219\pi\)
−0.147799 + 0.989017i \(0.547219\pi\)
\(104\) 3.27492 0.321132
\(105\) 0 0
\(106\) −10.5498 −1.02469
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.82475 0.270562 0.135281 0.990807i \(-0.456806\pi\)
0.135281 + 0.990807i \(0.456806\pi\)
\(110\) 0 0
\(111\) 10.8248 1.02744
\(112\) 3.00000 0.283473
\(113\) −3.72508 −0.350426 −0.175213 0.984531i \(-0.556061\pi\)
−0.175213 + 0.984531i \(0.556061\pi\)
\(114\) 3.27492 0.306724
\(115\) 0 0
\(116\) −9.54983 −0.886680
\(117\) 3.27492 0.302766
\(118\) 12.5498 1.15531
\(119\) −6.82475 −0.625624
\(120\) 0 0
\(121\) −9.37459 −0.852235
\(122\) 0.549834 0.0497797
\(123\) −2.72508 −0.245712
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) 3.00000 0.267261
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.27492 0.640521
\(130\) 0 0
\(131\) 6.54983 0.572262 0.286131 0.958191i \(-0.407631\pi\)
0.286131 + 0.958191i \(0.407631\pi\)
\(132\) −1.27492 −0.110967
\(133\) 9.82475 0.851914
\(134\) −4.54983 −0.393046
\(135\) 0 0
\(136\) −2.27492 −0.195073
\(137\) 8.82475 0.753949 0.376975 0.926224i \(-0.376964\pi\)
0.376975 + 0.926224i \(0.376964\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 6.27492 0.532232 0.266116 0.963941i \(-0.414260\pi\)
0.266116 + 0.963941i \(0.414260\pi\)
\(140\) 0 0
\(141\) 8.27492 0.696874
\(142\) −8.82475 −0.740557
\(143\) −4.17525 −0.349152
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −15.5498 −1.28691
\(147\) 2.00000 0.164957
\(148\) 10.8248 0.889789
\(149\) −2.54983 −0.208891 −0.104445 0.994531i \(-0.533307\pi\)
−0.104445 + 0.994531i \(0.533307\pi\)
\(150\) 0 0
\(151\) −2.54983 −0.207503 −0.103751 0.994603i \(-0.533085\pi\)
−0.103751 + 0.994603i \(0.533085\pi\)
\(152\) 3.27492 0.265631
\(153\) −2.27492 −0.183916
\(154\) −3.82475 −0.308207
\(155\) 0 0
\(156\) 3.27492 0.262203
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) −11.8248 −0.940727
\(159\) −10.5498 −0.836656
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 1.00000 0.0785674
\(163\) −23.6495 −1.85237 −0.926186 0.377067i \(-0.876933\pi\)
−0.926186 + 0.377067i \(0.876933\pi\)
\(164\) −2.72508 −0.212793
\(165\) 0 0
\(166\) −2.45017 −0.190170
\(167\) −12.2749 −0.949862 −0.474931 0.880023i \(-0.657527\pi\)
−0.474931 + 0.880023i \(0.657527\pi\)
\(168\) 3.00000 0.231455
\(169\) −2.27492 −0.174994
\(170\) 0 0
\(171\) 3.27492 0.250439
\(172\) 7.27492 0.554707
\(173\) −15.2749 −1.16133 −0.580665 0.814142i \(-0.697207\pi\)
−0.580665 + 0.814142i \(0.697207\pi\)
\(174\) −9.54983 −0.723971
\(175\) 0 0
\(176\) −1.27492 −0.0961005
\(177\) 12.5498 0.943303
\(178\) 8.27492 0.620231
\(179\) 26.5498 1.98443 0.992214 0.124545i \(-0.0397473\pi\)
0.992214 + 0.124545i \(0.0397473\pi\)
\(180\) 0 0
\(181\) 1.72508 0.128224 0.0641122 0.997943i \(-0.479578\pi\)
0.0641122 + 0.997943i \(0.479578\pi\)
\(182\) 9.82475 0.728259
\(183\) 0.549834 0.0406449
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 2.90033 0.212093
\(188\) 8.27492 0.603510
\(189\) 3.00000 0.218218
\(190\) 0 0
\(191\) 23.2749 1.68411 0.842057 0.539389i \(-0.181345\pi\)
0.842057 + 0.539389i \(0.181345\pi\)
\(192\) 1.00000 0.0721688
\(193\) −7.72508 −0.556064 −0.278032 0.960572i \(-0.589682\pi\)
−0.278032 + 0.960572i \(0.589682\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) −4.45017 −0.317061 −0.158531 0.987354i \(-0.550676\pi\)
−0.158531 + 0.987354i \(0.550676\pi\)
\(198\) −1.27492 −0.0906044
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) −4.54983 −0.320921
\(202\) 8.27492 0.582221
\(203\) −28.6495 −2.01080
\(204\) −2.27492 −0.159276
\(205\) 0 0
\(206\) −3.00000 −0.209020
\(207\) −1.00000 −0.0695048
\(208\) 3.27492 0.227075
\(209\) −4.17525 −0.288808
\(210\) 0 0
\(211\) −20.8248 −1.43364 −0.716818 0.697261i \(-0.754402\pi\)
−0.716818 + 0.697261i \(0.754402\pi\)
\(212\) −10.5498 −0.724566
\(213\) −8.82475 −0.604662
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 18.0000 1.22192
\(218\) 2.82475 0.191316
\(219\) −15.5498 −1.05076
\(220\) 0 0
\(221\) −7.45017 −0.501152
\(222\) 10.8248 0.726510
\(223\) −14.0000 −0.937509 −0.468755 0.883328i \(-0.655297\pi\)
−0.468755 + 0.883328i \(0.655297\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) −3.72508 −0.247789
\(227\) −10.2749 −0.681970 −0.340985 0.940069i \(-0.610761\pi\)
−0.340985 + 0.940069i \(0.610761\pi\)
\(228\) 3.27492 0.216887
\(229\) −19.0997 −1.26214 −0.631071 0.775725i \(-0.717384\pi\)
−0.631071 + 0.775725i \(0.717384\pi\)
\(230\) 0 0
\(231\) −3.82475 −0.251650
\(232\) −9.54983 −0.626977
\(233\) 14.3746 0.941710 0.470855 0.882210i \(-0.343945\pi\)
0.470855 + 0.882210i \(0.343945\pi\)
\(234\) 3.27492 0.214088
\(235\) 0 0
\(236\) 12.5498 0.816925
\(237\) −11.8248 −0.768100
\(238\) −6.82475 −0.442383
\(239\) 12.2749 0.793998 0.396999 0.917819i \(-0.370052\pi\)
0.396999 + 0.917819i \(0.370052\pi\)
\(240\) 0 0
\(241\) 10.5498 0.679575 0.339787 0.940502i \(-0.389645\pi\)
0.339787 + 0.940502i \(0.389645\pi\)
\(242\) −9.37459 −0.602621
\(243\) 1.00000 0.0641500
\(244\) 0.549834 0.0351995
\(245\) 0 0
\(246\) −2.72508 −0.173745
\(247\) 10.7251 0.682421
\(248\) 6.00000 0.381000
\(249\) −2.45017 −0.155273
\(250\) 0 0
\(251\) 26.8248 1.69316 0.846582 0.532259i \(-0.178657\pi\)
0.846582 + 0.532259i \(0.178657\pi\)
\(252\) 3.00000 0.188982
\(253\) 1.27492 0.0801534
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) 7.27492 0.452917
\(259\) 32.4743 2.01785
\(260\) 0 0
\(261\) −9.54983 −0.591120
\(262\) 6.54983 0.404650
\(263\) −4.54983 −0.280555 −0.140277 0.990112i \(-0.544799\pi\)
−0.140277 + 0.990112i \(0.544799\pi\)
\(264\) −1.27492 −0.0784657
\(265\) 0 0
\(266\) 9.82475 0.602394
\(267\) 8.27492 0.506417
\(268\) −4.54983 −0.277925
\(269\) −3.27492 −0.199675 −0.0998376 0.995004i \(-0.531832\pi\)
−0.0998376 + 0.995004i \(0.531832\pi\)
\(270\) 0 0
\(271\) 1.45017 0.0880913 0.0440456 0.999030i \(-0.485975\pi\)
0.0440456 + 0.999030i \(0.485975\pi\)
\(272\) −2.27492 −0.137937
\(273\) 9.82475 0.594621
\(274\) 8.82475 0.533123
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 5.82475 0.349975 0.174988 0.984571i \(-0.444011\pi\)
0.174988 + 0.984571i \(0.444011\pi\)
\(278\) 6.27492 0.376345
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 5.37459 0.320621 0.160310 0.987067i \(-0.448750\pi\)
0.160310 + 0.987067i \(0.448750\pi\)
\(282\) 8.27492 0.492764
\(283\) −16.0000 −0.951101 −0.475551 0.879688i \(-0.657751\pi\)
−0.475551 + 0.879688i \(0.657751\pi\)
\(284\) −8.82475 −0.523653
\(285\) 0 0
\(286\) −4.17525 −0.246888
\(287\) −8.17525 −0.482570
\(288\) 1.00000 0.0589256
\(289\) −11.8248 −0.695574
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) −15.5498 −0.909985
\(293\) −21.6495 −1.26478 −0.632389 0.774651i \(-0.717925\pi\)
−0.632389 + 0.774651i \(0.717925\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 10.8248 0.629176
\(297\) −1.27492 −0.0739782
\(298\) −2.54983 −0.147708
\(299\) −3.27492 −0.189393
\(300\) 0 0
\(301\) 21.8248 1.25796
\(302\) −2.54983 −0.146726
\(303\) 8.27492 0.475382
\(304\) 3.27492 0.187829
\(305\) 0 0
\(306\) −2.27492 −0.130048
\(307\) 6.27492 0.358128 0.179064 0.983837i \(-0.442693\pi\)
0.179064 + 0.983837i \(0.442693\pi\)
\(308\) −3.82475 −0.217935
\(309\) −3.00000 −0.170664
\(310\) 0 0
\(311\) −10.8248 −0.613815 −0.306908 0.951739i \(-0.599294\pi\)
−0.306908 + 0.951739i \(0.599294\pi\)
\(312\) 3.27492 0.185406
\(313\) −32.0000 −1.80875 −0.904373 0.426742i \(-0.859661\pi\)
−0.904373 + 0.426742i \(0.859661\pi\)
\(314\) 14.0000 0.790066
\(315\) 0 0
\(316\) −11.8248 −0.665194
\(317\) −13.5498 −0.761035 −0.380517 0.924774i \(-0.624254\pi\)
−0.380517 + 0.924774i \(0.624254\pi\)
\(318\) −10.5498 −0.591605
\(319\) 12.1752 0.681683
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) −3.00000 −0.167183
\(323\) −7.45017 −0.414538
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −23.6495 −1.30982
\(327\) 2.82475 0.156209
\(328\) −2.72508 −0.150468
\(329\) 24.8248 1.36863
\(330\) 0 0
\(331\) 16.8248 0.924772 0.462386 0.886679i \(-0.346993\pi\)
0.462386 + 0.886679i \(0.346993\pi\)
\(332\) −2.45017 −0.134470
\(333\) 10.8248 0.593193
\(334\) −12.2749 −0.671654
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 9.45017 0.514783 0.257392 0.966307i \(-0.417137\pi\)
0.257392 + 0.966307i \(0.417137\pi\)
\(338\) −2.27492 −0.123739
\(339\) −3.72508 −0.202319
\(340\) 0 0
\(341\) −7.64950 −0.414244
\(342\) 3.27492 0.177087
\(343\) −15.0000 −0.809924
\(344\) 7.27492 0.392237
\(345\) 0 0
\(346\) −15.2749 −0.821185
\(347\) −19.4502 −1.04414 −0.522070 0.852903i \(-0.674840\pi\)
−0.522070 + 0.852903i \(0.674840\pi\)
\(348\) −9.54983 −0.511925
\(349\) −27.2749 −1.45999 −0.729996 0.683451i \(-0.760478\pi\)
−0.729996 + 0.683451i \(0.760478\pi\)
\(350\) 0 0
\(351\) 3.27492 0.174802
\(352\) −1.27492 −0.0679533
\(353\) 24.3746 1.29733 0.648664 0.761075i \(-0.275328\pi\)
0.648664 + 0.761075i \(0.275328\pi\)
\(354\) 12.5498 0.667016
\(355\) 0 0
\(356\) 8.27492 0.438570
\(357\) −6.82475 −0.361204
\(358\) 26.5498 1.40320
\(359\) −28.9244 −1.52657 −0.763286 0.646060i \(-0.776415\pi\)
−0.763286 + 0.646060i \(0.776415\pi\)
\(360\) 0 0
\(361\) −8.27492 −0.435522
\(362\) 1.72508 0.0906683
\(363\) −9.37459 −0.492038
\(364\) 9.82475 0.514957
\(365\) 0 0
\(366\) 0.549834 0.0287403
\(367\) 9.27492 0.484147 0.242073 0.970258i \(-0.422173\pi\)
0.242073 + 0.970258i \(0.422173\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.72508 −0.141862
\(370\) 0 0
\(371\) −31.6495 −1.64316
\(372\) 6.00000 0.311086
\(373\) −23.3746 −1.21029 −0.605145 0.796115i \(-0.706885\pi\)
−0.605145 + 0.796115i \(0.706885\pi\)
\(374\) 2.90033 0.149973
\(375\) 0 0
\(376\) 8.27492 0.426746
\(377\) −31.2749 −1.61074
\(378\) 3.00000 0.154303
\(379\) 12.0000 0.616399 0.308199 0.951322i \(-0.400274\pi\)
0.308199 + 0.951322i \(0.400274\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 23.2749 1.19085
\(383\) 0.175248 0.00895477 0.00447739 0.999990i \(-0.498575\pi\)
0.00447739 + 0.999990i \(0.498575\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −7.72508 −0.393196
\(387\) 7.27492 0.369805
\(388\) 14.0000 0.710742
\(389\) 23.0997 1.17120 0.585600 0.810600i \(-0.300859\pi\)
0.585600 + 0.810600i \(0.300859\pi\)
\(390\) 0 0
\(391\) 2.27492 0.115048
\(392\) 2.00000 0.101015
\(393\) 6.54983 0.330395
\(394\) −4.45017 −0.224196
\(395\) 0 0
\(396\) −1.27492 −0.0640670
\(397\) −21.4502 −1.07655 −0.538276 0.842768i \(-0.680924\pi\)
−0.538276 + 0.842768i \(0.680924\pi\)
\(398\) 11.0000 0.551380
\(399\) 9.82475 0.491853
\(400\) 0 0
\(401\) −9.64950 −0.481873 −0.240937 0.970541i \(-0.577455\pi\)
−0.240937 + 0.970541i \(0.577455\pi\)
\(402\) −4.54983 −0.226925
\(403\) 19.6495 0.978811
\(404\) 8.27492 0.411693
\(405\) 0 0
\(406\) −28.6495 −1.42185
\(407\) −13.8007 −0.684073
\(408\) −2.27492 −0.112625
\(409\) −7.00000 −0.346128 −0.173064 0.984911i \(-0.555367\pi\)
−0.173064 + 0.984911i \(0.555367\pi\)
\(410\) 0 0
\(411\) 8.82475 0.435293
\(412\) −3.00000 −0.147799
\(413\) 37.6495 1.85261
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 3.27492 0.160566
\(417\) 6.27492 0.307284
\(418\) −4.17525 −0.204218
\(419\) 3.00000 0.146560 0.0732798 0.997311i \(-0.476653\pi\)
0.0732798 + 0.997311i \(0.476653\pi\)
\(420\) 0 0
\(421\) 31.0997 1.51571 0.757853 0.652426i \(-0.226249\pi\)
0.757853 + 0.652426i \(0.226249\pi\)
\(422\) −20.8248 −1.01373
\(423\) 8.27492 0.402340
\(424\) −10.5498 −0.512345
\(425\) 0 0
\(426\) −8.82475 −0.427561
\(427\) 1.64950 0.0798251
\(428\) −12.0000 −0.580042
\(429\) −4.17525 −0.201583
\(430\) 0 0
\(431\) −12.5498 −0.604504 −0.302252 0.953228i \(-0.597738\pi\)
−0.302252 + 0.953228i \(0.597738\pi\)
\(432\) 1.00000 0.0481125
\(433\) 17.6495 0.848181 0.424091 0.905620i \(-0.360594\pi\)
0.424091 + 0.905620i \(0.360594\pi\)
\(434\) 18.0000 0.864028
\(435\) 0 0
\(436\) 2.82475 0.135281
\(437\) −3.27492 −0.156661
\(438\) −15.5498 −0.743000
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 2.00000 0.0952381
\(442\) −7.45017 −0.354368
\(443\) 15.6495 0.743530 0.371765 0.928327i \(-0.378753\pi\)
0.371765 + 0.928327i \(0.378753\pi\)
\(444\) 10.8248 0.513720
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) −2.54983 −0.120603
\(448\) 3.00000 0.141737
\(449\) 15.0997 0.712597 0.356299 0.934372i \(-0.384039\pi\)
0.356299 + 0.934372i \(0.384039\pi\)
\(450\) 0 0
\(451\) 3.47425 0.163596
\(452\) −3.72508 −0.175213
\(453\) −2.54983 −0.119802
\(454\) −10.2749 −0.482226
\(455\) 0 0
\(456\) 3.27492 0.153362
\(457\) −39.6495 −1.85473 −0.927363 0.374163i \(-0.877930\pi\)
−0.927363 + 0.374163i \(0.877930\pi\)
\(458\) −19.0997 −0.892469
\(459\) −2.27492 −0.106184
\(460\) 0 0
\(461\) 0.0996689 0.00464204 0.00232102 0.999997i \(-0.499261\pi\)
0.00232102 + 0.999997i \(0.499261\pi\)
\(462\) −3.82475 −0.177944
\(463\) −29.0997 −1.35238 −0.676188 0.736729i \(-0.736369\pi\)
−0.676188 + 0.736729i \(0.736369\pi\)
\(464\) −9.54983 −0.443340
\(465\) 0 0
\(466\) 14.3746 0.665890
\(467\) −21.0000 −0.971764 −0.485882 0.874024i \(-0.661502\pi\)
−0.485882 + 0.874024i \(0.661502\pi\)
\(468\) 3.27492 0.151383
\(469\) −13.6495 −0.630276
\(470\) 0 0
\(471\) 14.0000 0.645086
\(472\) 12.5498 0.577653
\(473\) −9.27492 −0.426461
\(474\) −11.8248 −0.543129
\(475\) 0 0
\(476\) −6.82475 −0.312812
\(477\) −10.5498 −0.483044
\(478\) 12.2749 0.561442
\(479\) 7.27492 0.332399 0.166200 0.986092i \(-0.446850\pi\)
0.166200 + 0.986092i \(0.446850\pi\)
\(480\) 0 0
\(481\) 35.4502 1.61639
\(482\) 10.5498 0.480532
\(483\) −3.00000 −0.136505
\(484\) −9.37459 −0.426118
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −6.54983 −0.296801 −0.148401 0.988927i \(-0.547413\pi\)
−0.148401 + 0.988927i \(0.547413\pi\)
\(488\) 0.549834 0.0248898
\(489\) −23.6495 −1.06947
\(490\) 0 0
\(491\) −8.54983 −0.385849 −0.192924 0.981214i \(-0.561797\pi\)
−0.192924 + 0.981214i \(0.561797\pi\)
\(492\) −2.72508 −0.122856
\(493\) 21.7251 0.978449
\(494\) 10.7251 0.482544
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −26.4743 −1.18753
\(498\) −2.45017 −0.109795
\(499\) 11.7251 0.524887 0.262443 0.964947i \(-0.415472\pi\)
0.262443 + 0.964947i \(0.415472\pi\)
\(500\) 0 0
\(501\) −12.2749 −0.548403
\(502\) 26.8248 1.19725
\(503\) −18.3746 −0.819282 −0.409641 0.912247i \(-0.634346\pi\)
−0.409641 + 0.912247i \(0.634346\pi\)
\(504\) 3.00000 0.133631
\(505\) 0 0
\(506\) 1.27492 0.0566770
\(507\) −2.27492 −0.101033
\(508\) 8.00000 0.354943
\(509\) −30.4743 −1.35075 −0.675374 0.737476i \(-0.736018\pi\)
−0.675374 + 0.737476i \(0.736018\pi\)
\(510\) 0 0
\(511\) −46.6495 −2.06365
\(512\) 1.00000 0.0441942
\(513\) 3.27492 0.144591
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 7.27492 0.320260
\(517\) −10.5498 −0.463981
\(518\) 32.4743 1.42684
\(519\) −15.2749 −0.670494
\(520\) 0 0
\(521\) 3.37459 0.147843 0.0739217 0.997264i \(-0.476449\pi\)
0.0739217 + 0.997264i \(0.476449\pi\)
\(522\) −9.54983 −0.417985
\(523\) 26.3746 1.15328 0.576640 0.816998i \(-0.304364\pi\)
0.576640 + 0.816998i \(0.304364\pi\)
\(524\) 6.54983 0.286131
\(525\) 0 0
\(526\) −4.54983 −0.198382
\(527\) −13.6495 −0.594582
\(528\) −1.27492 −0.0554837
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 12.5498 0.544616
\(532\) 9.82475 0.425957
\(533\) −8.92442 −0.386560
\(534\) 8.27492 0.358091
\(535\) 0 0
\(536\) −4.54983 −0.196523
\(537\) 26.5498 1.14571
\(538\) −3.27492 −0.141192
\(539\) −2.54983 −0.109829
\(540\) 0 0
\(541\) −31.8248 −1.36825 −0.684126 0.729363i \(-0.739816\pi\)
−0.684126 + 0.729363i \(0.739816\pi\)
\(542\) 1.45017 0.0622899
\(543\) 1.72508 0.0740304
\(544\) −2.27492 −0.0975363
\(545\) 0 0
\(546\) 9.82475 0.420461
\(547\) −44.8248 −1.91657 −0.958284 0.285818i \(-0.907735\pi\)
−0.958284 + 0.285818i \(0.907735\pi\)
\(548\) 8.82475 0.376975
\(549\) 0.549834 0.0234664
\(550\) 0 0
\(551\) −31.2749 −1.33236
\(552\) −1.00000 −0.0425628
\(553\) −35.4743 −1.50852
\(554\) 5.82475 0.247470
\(555\) 0 0
\(556\) 6.27492 0.266116
\(557\) −20.5498 −0.870724 −0.435362 0.900255i \(-0.643380\pi\)
−0.435362 + 0.900255i \(0.643380\pi\)
\(558\) 6.00000 0.254000
\(559\) 23.8248 1.00768
\(560\) 0 0
\(561\) 2.90033 0.122452
\(562\) 5.37459 0.226713
\(563\) 6.72508 0.283428 0.141714 0.989908i \(-0.454739\pi\)
0.141714 + 0.989908i \(0.454739\pi\)
\(564\) 8.27492 0.348437
\(565\) 0 0
\(566\) −16.0000 −0.672530
\(567\) 3.00000 0.125988
\(568\) −8.82475 −0.370278
\(569\) −7.45017 −0.312327 −0.156164 0.987731i \(-0.549913\pi\)
−0.156164 + 0.987731i \(0.549913\pi\)
\(570\) 0 0
\(571\) −32.5498 −1.36217 −0.681084 0.732205i \(-0.738491\pi\)
−0.681084 + 0.732205i \(0.738491\pi\)
\(572\) −4.17525 −0.174576
\(573\) 23.2749 0.972324
\(574\) −8.17525 −0.341228
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −1.82475 −0.0759654 −0.0379827 0.999278i \(-0.512093\pi\)
−0.0379827 + 0.999278i \(0.512093\pi\)
\(578\) −11.8248 −0.491845
\(579\) −7.72508 −0.321043
\(580\) 0 0
\(581\) −7.35050 −0.304950
\(582\) 14.0000 0.580319
\(583\) 13.4502 0.557049
\(584\) −15.5498 −0.643457
\(585\) 0 0
\(586\) −21.6495 −0.894333
\(587\) −41.0997 −1.69636 −0.848182 0.529704i \(-0.822303\pi\)
−0.848182 + 0.529704i \(0.822303\pi\)
\(588\) 2.00000 0.0824786
\(589\) 19.6495 0.809644
\(590\) 0 0
\(591\) −4.45017 −0.183055
\(592\) 10.8248 0.444895
\(593\) 8.17525 0.335717 0.167859 0.985811i \(-0.446315\pi\)
0.167859 + 0.985811i \(0.446315\pi\)
\(594\) −1.27492 −0.0523105
\(595\) 0 0
\(596\) −2.54983 −0.104445
\(597\) 11.0000 0.450200
\(598\) −3.27492 −0.133921
\(599\) 36.7492 1.50153 0.750765 0.660569i \(-0.229685\pi\)
0.750765 + 0.660569i \(0.229685\pi\)
\(600\) 0 0
\(601\) 19.7251 0.804603 0.402301 0.915507i \(-0.368210\pi\)
0.402301 + 0.915507i \(0.368210\pi\)
\(602\) 21.8248 0.889510
\(603\) −4.54983 −0.185284
\(604\) −2.54983 −0.103751
\(605\) 0 0
\(606\) 8.27492 0.336146
\(607\) 35.0997 1.42465 0.712326 0.701849i \(-0.247642\pi\)
0.712326 + 0.701849i \(0.247642\pi\)
\(608\) 3.27492 0.132815
\(609\) −28.6495 −1.16094
\(610\) 0 0
\(611\) 27.0997 1.09634
\(612\) −2.27492 −0.0919581
\(613\) 26.2749 1.06123 0.530617 0.847612i \(-0.321960\pi\)
0.530617 + 0.847612i \(0.321960\pi\)
\(614\) 6.27492 0.253235
\(615\) 0 0
\(616\) −3.82475 −0.154104
\(617\) 37.6495 1.51571 0.757856 0.652422i \(-0.226247\pi\)
0.757856 + 0.652422i \(0.226247\pi\)
\(618\) −3.00000 −0.120678
\(619\) −28.0000 −1.12542 −0.562708 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −10.8248 −0.434033
\(623\) 24.8248 0.994583
\(624\) 3.27492 0.131102
\(625\) 0 0
\(626\) −32.0000 −1.27898
\(627\) −4.17525 −0.166743
\(628\) 14.0000 0.558661
\(629\) −24.6254 −0.981880
\(630\) 0 0
\(631\) −1.90033 −0.0756510 −0.0378255 0.999284i \(-0.512043\pi\)
−0.0378255 + 0.999284i \(0.512043\pi\)
\(632\) −11.8248 −0.470363
\(633\) −20.8248 −0.827710
\(634\) −13.5498 −0.538133
\(635\) 0 0
\(636\) −10.5498 −0.418328
\(637\) 6.54983 0.259514
\(638\) 12.1752 0.482023
\(639\) −8.82475 −0.349102
\(640\) 0 0
\(641\) 8.62541 0.340683 0.170342 0.985385i \(-0.445513\pi\)
0.170342 + 0.985385i \(0.445513\pi\)
\(642\) −12.0000 −0.473602
\(643\) 7.27492 0.286895 0.143447 0.989658i \(-0.454181\pi\)
0.143447 + 0.989658i \(0.454181\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) −7.45017 −0.293123
\(647\) −15.1752 −0.596601 −0.298300 0.954472i \(-0.596420\pi\)
−0.298300 + 0.954472i \(0.596420\pi\)
\(648\) 1.00000 0.0392837
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 18.0000 0.705476
\(652\) −23.6495 −0.926186
\(653\) 38.0997 1.49096 0.745478 0.666531i \(-0.232221\pi\)
0.745478 + 0.666531i \(0.232221\pi\)
\(654\) 2.82475 0.110457
\(655\) 0 0
\(656\) −2.72508 −0.106397
\(657\) −15.5498 −0.606657
\(658\) 24.8248 0.967770
\(659\) −26.0997 −1.01670 −0.508349 0.861151i \(-0.669744\pi\)
−0.508349 + 0.861151i \(0.669744\pi\)
\(660\) 0 0
\(661\) 40.8248 1.58790 0.793949 0.607984i \(-0.208021\pi\)
0.793949 + 0.607984i \(0.208021\pi\)
\(662\) 16.8248 0.653913
\(663\) −7.45017 −0.289340
\(664\) −2.45017 −0.0950849
\(665\) 0 0
\(666\) 10.8248 0.419451
\(667\) 9.54983 0.369771
\(668\) −12.2749 −0.474931
\(669\) −14.0000 −0.541271
\(670\) 0 0
\(671\) −0.700993 −0.0270615
\(672\) 3.00000 0.115728
\(673\) 6.45017 0.248636 0.124318 0.992242i \(-0.460326\pi\)
0.124318 + 0.992242i \(0.460326\pi\)
\(674\) 9.45017 0.364007
\(675\) 0 0
\(676\) −2.27492 −0.0874968
\(677\) 17.0997 0.657194 0.328597 0.944470i \(-0.393424\pi\)
0.328597 + 0.944470i \(0.393424\pi\)
\(678\) −3.72508 −0.143061
\(679\) 42.0000 1.61181
\(680\) 0 0
\(681\) −10.2749 −0.393736
\(682\) −7.64950 −0.292915
\(683\) −22.5498 −0.862845 −0.431423 0.902150i \(-0.641988\pi\)
−0.431423 + 0.902150i \(0.641988\pi\)
\(684\) 3.27492 0.125220
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) −19.0997 −0.728698
\(688\) 7.27492 0.277354
\(689\) −34.5498 −1.31624
\(690\) 0 0
\(691\) 17.7251 0.674294 0.337147 0.941452i \(-0.390538\pi\)
0.337147 + 0.941452i \(0.390538\pi\)
\(692\) −15.2749 −0.580665
\(693\) −3.82475 −0.145290
\(694\) −19.4502 −0.738318
\(695\) 0 0
\(696\) −9.54983 −0.361986
\(697\) 6.19934 0.234817
\(698\) −27.2749 −1.03237
\(699\) 14.3746 0.543697
\(700\) 0 0
\(701\) 17.6495 0.666613 0.333306 0.942819i \(-0.391836\pi\)
0.333306 + 0.942819i \(0.391836\pi\)
\(702\) 3.27492 0.123604
\(703\) 35.4502 1.33703
\(704\) −1.27492 −0.0480503
\(705\) 0 0
\(706\) 24.3746 0.917350
\(707\) 24.8248 0.933631
\(708\) 12.5498 0.471652
\(709\) −27.1752 −1.02059 −0.510294 0.860000i \(-0.670463\pi\)
−0.510294 + 0.860000i \(0.670463\pi\)
\(710\) 0 0
\(711\) −11.8248 −0.443463
\(712\) 8.27492 0.310116
\(713\) −6.00000 −0.224702
\(714\) −6.82475 −0.255410
\(715\) 0 0
\(716\) 26.5498 0.992214
\(717\) 12.2749 0.458415
\(718\) −28.9244 −1.07945
\(719\) 15.3746 0.573375 0.286688 0.958024i \(-0.407446\pi\)
0.286688 + 0.958024i \(0.407446\pi\)
\(720\) 0 0
\(721\) −9.00000 −0.335178
\(722\) −8.27492 −0.307961
\(723\) 10.5498 0.392353
\(724\) 1.72508 0.0641122
\(725\) 0 0
\(726\) −9.37459 −0.347924
\(727\) 34.1993 1.26838 0.634192 0.773176i \(-0.281333\pi\)
0.634192 + 0.773176i \(0.281333\pi\)
\(728\) 9.82475 0.364130
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.5498 −0.612118
\(732\) 0.549834 0.0203225
\(733\) 33.9244 1.25303 0.626514 0.779411i \(-0.284481\pi\)
0.626514 + 0.779411i \(0.284481\pi\)
\(734\) 9.27492 0.342343
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 5.80066 0.213670
\(738\) −2.72508 −0.100312
\(739\) −33.9244 −1.24793 −0.623965 0.781452i \(-0.714479\pi\)
−0.623965 + 0.781452i \(0.714479\pi\)
\(740\) 0 0
\(741\) 10.7251 0.393996
\(742\) −31.6495 −1.16189
\(743\) 21.8248 0.800672 0.400336 0.916368i \(-0.368893\pi\)
0.400336 + 0.916368i \(0.368893\pi\)
\(744\) 6.00000 0.219971
\(745\) 0 0
\(746\) −23.3746 −0.855804
\(747\) −2.45017 −0.0896469
\(748\) 2.90033 0.106047
\(749\) −36.0000 −1.31541
\(750\) 0 0
\(751\) −8.64950 −0.315625 −0.157812 0.987469i \(-0.550444\pi\)
−0.157812 + 0.987469i \(0.550444\pi\)
\(752\) 8.27492 0.301755
\(753\) 26.8248 0.977548
\(754\) −31.2749 −1.13897
\(755\) 0 0
\(756\) 3.00000 0.109109
\(757\) 21.7251 0.789612 0.394806 0.918765i \(-0.370812\pi\)
0.394806 + 0.918765i \(0.370812\pi\)
\(758\) 12.0000 0.435860
\(759\) 1.27492 0.0462766
\(760\) 0 0
\(761\) −13.8248 −0.501147 −0.250573 0.968098i \(-0.580619\pi\)
−0.250573 + 0.968098i \(0.580619\pi\)
\(762\) 8.00000 0.289809
\(763\) 8.47425 0.306789
\(764\) 23.2749 0.842057
\(765\) 0 0
\(766\) 0.175248 0.00633198
\(767\) 41.0997 1.48402
\(768\) 1.00000 0.0360844
\(769\) 8.54983 0.308315 0.154157 0.988046i \(-0.450734\pi\)
0.154157 + 0.988046i \(0.450734\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) −7.72508 −0.278032
\(773\) −46.7492 −1.68145 −0.840725 0.541462i \(-0.817871\pi\)
−0.840725 + 0.541462i \(0.817871\pi\)
\(774\) 7.27492 0.261491
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 32.4743 1.16501
\(778\) 23.0997 0.828163
\(779\) −8.92442 −0.319751
\(780\) 0 0
\(781\) 11.2508 0.402586
\(782\) 2.27492 0.0813509
\(783\) −9.54983 −0.341283
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 6.54983 0.233625
\(787\) −23.8248 −0.849261 −0.424630 0.905367i \(-0.639596\pi\)
−0.424630 + 0.905367i \(0.639596\pi\)
\(788\) −4.45017 −0.158531
\(789\) −4.54983 −0.161978
\(790\) 0 0
\(791\) −11.1752 −0.397346
\(792\) −1.27492 −0.0453022
\(793\) 1.80066 0.0639434
\(794\) −21.4502 −0.761238
\(795\) 0 0
\(796\) 11.0000 0.389885
\(797\) 12.5498 0.444538 0.222269 0.974985i \(-0.428654\pi\)
0.222269 + 0.974985i \(0.428654\pi\)
\(798\) 9.82475 0.347792
\(799\) −18.8248 −0.665972
\(800\) 0 0
\(801\) 8.27492 0.292380
\(802\) −9.64950 −0.340736
\(803\) 19.8248 0.699600
\(804\) −4.54983 −0.160460
\(805\) 0 0
\(806\) 19.6495 0.692124
\(807\) −3.27492 −0.115283
\(808\) 8.27492 0.291111
\(809\) 13.4743 0.473730 0.236865 0.971543i \(-0.423880\pi\)
0.236865 + 0.971543i \(0.423880\pi\)
\(810\) 0 0
\(811\) −41.0997 −1.44320 −0.721602 0.692308i \(-0.756594\pi\)
−0.721602 + 0.692308i \(0.756594\pi\)
\(812\) −28.6495 −1.00540
\(813\) 1.45017 0.0508595
\(814\) −13.8007 −0.483713
\(815\) 0 0
\(816\) −2.27492 −0.0796380
\(817\) 23.8248 0.833523
\(818\) −7.00000 −0.244749
\(819\) 9.82475 0.343305
\(820\) 0 0
\(821\) −52.0241 −1.81565 −0.907827 0.419346i \(-0.862260\pi\)
−0.907827 + 0.419346i \(0.862260\pi\)
\(822\) 8.82475 0.307799
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) −3.00000 −0.104510
\(825\) 0 0
\(826\) 37.6495 1.30999
\(827\) 48.0997 1.67259 0.836295 0.548280i \(-0.184717\pi\)
0.836295 + 0.548280i \(0.184717\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −4.37459 −0.151936 −0.0759678 0.997110i \(-0.524205\pi\)
−0.0759678 + 0.997110i \(0.524205\pi\)
\(830\) 0 0
\(831\) 5.82475 0.202058
\(832\) 3.27492 0.113537
\(833\) −4.54983 −0.157642
\(834\) 6.27492 0.217283
\(835\) 0 0
\(836\) −4.17525 −0.144404
\(837\) 6.00000 0.207390
\(838\) 3.00000 0.103633
\(839\) 3.82475 0.132045 0.0660225 0.997818i \(-0.478969\pi\)
0.0660225 + 0.997818i \(0.478969\pi\)
\(840\) 0 0
\(841\) 62.1993 2.14480
\(842\) 31.0997 1.07177
\(843\) 5.37459 0.185111
\(844\) −20.8248 −0.716818
\(845\) 0 0
\(846\) 8.27492 0.284498
\(847\) −28.1238 −0.966344
\(848\) −10.5498 −0.362283
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −10.8248 −0.371068
\(852\) −8.82475 −0.302331
\(853\) 18.7251 0.641135 0.320567 0.947226i \(-0.396126\pi\)
0.320567 + 0.947226i \(0.396126\pi\)
\(854\) 1.64950 0.0564448
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −3.09967 −0.105883 −0.0529413 0.998598i \(-0.516860\pi\)
−0.0529413 + 0.998598i \(0.516860\pi\)
\(858\) −4.17525 −0.142541
\(859\) −4.00000 −0.136478 −0.0682391 0.997669i \(-0.521738\pi\)
−0.0682391 + 0.997669i \(0.521738\pi\)
\(860\) 0 0
\(861\) −8.17525 −0.278612
\(862\) −12.5498 −0.427449
\(863\) −53.9244 −1.83561 −0.917804 0.397033i \(-0.870040\pi\)
−0.917804 + 0.397033i \(0.870040\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 17.6495 0.599755
\(867\) −11.8248 −0.401590
\(868\) 18.0000 0.610960
\(869\) 15.0756 0.511404
\(870\) 0 0
\(871\) −14.9003 −0.504878
\(872\) 2.82475 0.0956582
\(873\) 14.0000 0.473828
\(874\) −3.27492 −0.110776
\(875\) 0 0
\(876\) −15.5498 −0.525380
\(877\) 0.900331 0.0304020 0.0152010 0.999884i \(-0.495161\pi\)
0.0152010 + 0.999884i \(0.495161\pi\)
\(878\) −20.0000 −0.674967
\(879\) −21.6495 −0.730220
\(880\) 0 0
\(881\) −30.7492 −1.03597 −0.517983 0.855391i \(-0.673317\pi\)
−0.517983 + 0.855391i \(0.673317\pi\)
\(882\) 2.00000 0.0673435
\(883\) −30.2749 −1.01883 −0.509416 0.860520i \(-0.670139\pi\)
−0.509416 + 0.860520i \(0.670139\pi\)
\(884\) −7.45017 −0.250576
\(885\) 0 0
\(886\) 15.6495 0.525755
\(887\) −47.3746 −1.59068 −0.795341 0.606162i \(-0.792708\pi\)
−0.795341 + 0.606162i \(0.792708\pi\)
\(888\) 10.8248 0.363255
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) −1.27492 −0.0427113
\(892\) −14.0000 −0.468755
\(893\) 27.0997 0.906856
\(894\) −2.54983 −0.0852792
\(895\) 0 0
\(896\) 3.00000 0.100223
\(897\) −3.27492 −0.109346
\(898\) 15.0997 0.503882
\(899\) −57.2990 −1.91103
\(900\) 0 0
\(901\) 24.0000 0.799556
\(902\) 3.47425 0.115680
\(903\) 21.8248 0.726282
\(904\) −3.72508 −0.123894
\(905\) 0 0
\(906\) −2.54983 −0.0847126
\(907\) −0.374586 −0.0124379 −0.00621896 0.999981i \(-0.501980\pi\)
−0.00621896 + 0.999981i \(0.501980\pi\)
\(908\) −10.2749 −0.340985
\(909\) 8.27492 0.274462
\(910\) 0 0
\(911\) −43.8248 −1.45198 −0.725989 0.687706i \(-0.758618\pi\)
−0.725989 + 0.687706i \(0.758618\pi\)
\(912\) 3.27492 0.108443
\(913\) 3.12376 0.103381
\(914\) −39.6495 −1.31149
\(915\) 0 0
\(916\) −19.0997 −0.631071
\(917\) 19.6495 0.648884
\(918\) −2.27492 −0.0750835
\(919\) 48.4743 1.59902 0.799509 0.600654i \(-0.205093\pi\)
0.799509 + 0.600654i \(0.205093\pi\)
\(920\) 0 0
\(921\) 6.27492 0.206766
\(922\) 0.0996689 0.00328242
\(923\) −28.9003 −0.951266
\(924\) −3.82475 −0.125825
\(925\) 0 0
\(926\) −29.0997 −0.956274
\(927\) −3.00000 −0.0985329
\(928\) −9.54983 −0.313489
\(929\) −8.37459 −0.274761 −0.137381 0.990518i \(-0.543868\pi\)
−0.137381 + 0.990518i \(0.543868\pi\)
\(930\) 0 0
\(931\) 6.54983 0.214662
\(932\) 14.3746 0.470855
\(933\) −10.8248 −0.354386
\(934\) −21.0000 −0.687141
\(935\) 0 0
\(936\) 3.27492 0.107044
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) −13.6495 −0.445672
\(939\) −32.0000 −1.04428
\(940\) 0 0
\(941\) −16.1993 −0.528083 −0.264042 0.964511i \(-0.585056\pi\)
−0.264042 + 0.964511i \(0.585056\pi\)
\(942\) 14.0000 0.456145
\(943\) 2.72508 0.0887409
\(944\) 12.5498 0.408462
\(945\) 0 0
\(946\) −9.27492 −0.301554
\(947\) −51.6495 −1.67838 −0.839192 0.543836i \(-0.816971\pi\)
−0.839192 + 0.543836i \(0.816971\pi\)
\(948\) −11.8248 −0.384050
\(949\) −50.9244 −1.65308
\(950\) 0 0
\(951\) −13.5498 −0.439383
\(952\) −6.82475 −0.221191
\(953\) 35.9244 1.16371 0.581853 0.813294i \(-0.302328\pi\)
0.581853 + 0.813294i \(0.302328\pi\)
\(954\) −10.5498 −0.341564
\(955\) 0 0
\(956\) 12.2749 0.396999
\(957\) 12.1752 0.393570
\(958\) 7.27492 0.235042
\(959\) 26.4743 0.854898
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 35.4502 1.14296
\(963\) −12.0000 −0.386695
\(964\) 10.5498 0.339787
\(965\) 0 0
\(966\) −3.00000 −0.0965234
\(967\) 25.6495 0.824832 0.412416 0.910996i \(-0.364685\pi\)
0.412416 + 0.910996i \(0.364685\pi\)
\(968\) −9.37459 −0.301311
\(969\) −7.45017 −0.239334
\(970\) 0 0
\(971\) 4.09967 0.131565 0.0657823 0.997834i \(-0.479046\pi\)
0.0657823 + 0.997834i \(0.479046\pi\)
\(972\) 1.00000 0.0320750
\(973\) 18.8248 0.603494
\(974\) −6.54983 −0.209870
\(975\) 0 0
\(976\) 0.549834 0.0175998
\(977\) 33.9244 1.08534 0.542669 0.839947i \(-0.317414\pi\)
0.542669 + 0.839947i \(0.317414\pi\)
\(978\) −23.6495 −0.756228
\(979\) −10.5498 −0.337174
\(980\) 0 0
\(981\) 2.82475 0.0901874
\(982\) −8.54983 −0.272836
\(983\) 36.9244 1.17771 0.588853 0.808240i \(-0.299580\pi\)
0.588853 + 0.808240i \(0.299580\pi\)
\(984\) −2.72508 −0.0868725
\(985\) 0 0
\(986\) 21.7251 0.691868
\(987\) 24.8248 0.790181
\(988\) 10.7251 0.341210
\(989\) −7.27492 −0.231329
\(990\) 0 0
\(991\) 10.5498 0.335127 0.167563 0.985861i \(-0.446410\pi\)
0.167563 + 0.985861i \(0.446410\pi\)
\(992\) 6.00000 0.190500
\(993\) 16.8248 0.533917
\(994\) −26.4743 −0.839712
\(995\) 0 0
\(996\) −2.45017 −0.0776365
\(997\) 8.72508 0.276326 0.138163 0.990409i \(-0.455880\pi\)
0.138163 + 0.990409i \(0.455880\pi\)
\(998\) 11.7251 0.371151
\(999\) 10.8248 0.342480
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.bm.1.1 yes 2
5.2 odd 4 3450.2.d.z.2899.3 4
5.3 odd 4 3450.2.d.z.2899.1 4
5.4 even 2 3450.2.a.bd.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bd.1.1 2 5.4 even 2
3450.2.a.bm.1.1 yes 2 1.1 even 1 trivial
3450.2.d.z.2899.1 4 5.3 odd 4
3450.2.d.z.2899.3 4 5.2 odd 4