Properties

Label 3450.2.a.bn.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: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -2.47214 q^{13} +4.00000 q^{14} +1.00000 q^{16} +2.47214 q^{17} +1.00000 q^{18} +2.00000 q^{19} +4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{24} -2.47214 q^{26} +1.00000 q^{27} +4.00000 q^{28} -0.472136 q^{29} +1.00000 q^{32} +2.47214 q^{34} +1.00000 q^{36} +0.472136 q^{37} +2.00000 q^{38} -2.47214 q^{39} +10.9443 q^{41} +4.00000 q^{42} -1.00000 q^{46} +4.94427 q^{47} +1.00000 q^{48} +9.00000 q^{49} +2.47214 q^{51} -2.47214 q^{52} -8.94427 q^{53} +1.00000 q^{54} +4.00000 q^{56} +2.00000 q^{57} -0.472136 q^{58} -6.00000 q^{59} -0.472136 q^{61} +4.00000 q^{63} +1.00000 q^{64} -4.94427 q^{67} +2.47214 q^{68} -1.00000 q^{69} -7.52786 q^{71} +1.00000 q^{72} +4.94427 q^{73} +0.472136 q^{74} +2.00000 q^{76} -2.47214 q^{78} +12.4721 q^{79} +1.00000 q^{81} +10.9443 q^{82} +1.52786 q^{83} +4.00000 q^{84} -0.472136 q^{87} -16.4721 q^{89} -9.88854 q^{91} -1.00000 q^{92} +4.94427 q^{94} +1.00000 q^{96} -13.4164 q^{97} +9.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 8q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 8q^{7} + 2q^{8} + 2q^{9} + 2q^{12} + 4q^{13} + 8q^{14} + 2q^{16} - 4q^{17} + 2q^{18} + 4q^{19} + 8q^{21} - 2q^{23} + 2q^{24} + 4q^{26} + 2q^{27} + 8q^{28} + 8q^{29} + 2q^{32} - 4q^{34} + 2q^{36} - 8q^{37} + 4q^{38} + 4q^{39} + 4q^{41} + 8q^{42} - 2q^{46} - 8q^{47} + 2q^{48} + 18q^{49} - 4q^{51} + 4q^{52} + 2q^{54} + 8q^{56} + 4q^{57} + 8q^{58} - 12q^{59} + 8q^{61} + 8q^{63} + 2q^{64} + 8q^{67} - 4q^{68} - 2q^{69} - 24q^{71} + 2q^{72} - 8q^{73} - 8q^{74} + 4q^{76} + 4q^{78} + 16q^{79} + 2q^{81} + 4q^{82} + 12q^{83} + 8q^{84} + 8q^{87} - 24q^{89} + 16q^{91} - 2q^{92} - 8q^{94} + 2q^{96} + 18q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 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 \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.47214 −0.685647 −0.342824 0.939400i \(-0.611383\pi\)
−0.342824 + 0.939400i \(0.611383\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.47214 0.599581 0.299791 0.954005i \(-0.403083\pi\)
0.299791 + 0.954005i \(0.403083\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.47214 −0.484826
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) −0.472136 −0.0876734 −0.0438367 0.999039i \(-0.513958\pi\)
−0.0438367 + 0.999039i \(0.513958\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 2.47214 0.423968
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.472136 0.0776187 0.0388093 0.999247i \(-0.487644\pi\)
0.0388093 + 0.999247i \(0.487644\pi\)
\(38\) 2.00000 0.324443
\(39\) −2.47214 −0.395859
\(40\) 0 0
\(41\) 10.9443 1.70921 0.854604 0.519280i \(-0.173800\pi\)
0.854604 + 0.519280i \(0.173800\pi\)
\(42\) 4.00000 0.617213
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 4.94427 0.721196 0.360598 0.932721i \(-0.382573\pi\)
0.360598 + 0.932721i \(0.382573\pi\)
\(48\) 1.00000 0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 2.47214 0.346168
\(52\) −2.47214 −0.342824
\(53\) −8.94427 −1.22859 −0.614295 0.789076i \(-0.710560\pi\)
−0.614295 + 0.789076i \(0.710560\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) 2.00000 0.264906
\(58\) −0.472136 −0.0619945
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −0.472136 −0.0604508 −0.0302254 0.999543i \(-0.509623\pi\)
−0.0302254 + 0.999543i \(0.509623\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −4.94427 −0.604039 −0.302019 0.953302i \(-0.597661\pi\)
−0.302019 + 0.953302i \(0.597661\pi\)
\(68\) 2.47214 0.299791
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −7.52786 −0.893393 −0.446697 0.894686i \(-0.647400\pi\)
−0.446697 + 0.894686i \(0.647400\pi\)
\(72\) 1.00000 0.117851
\(73\) 4.94427 0.578683 0.289342 0.957226i \(-0.406564\pi\)
0.289342 + 0.957226i \(0.406564\pi\)
\(74\) 0.472136 0.0548847
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) −2.47214 −0.279914
\(79\) 12.4721 1.40322 0.701612 0.712559i \(-0.252464\pi\)
0.701612 + 0.712559i \(0.252464\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.9443 1.20859
\(83\) 1.52786 0.167705 0.0838524 0.996478i \(-0.473278\pi\)
0.0838524 + 0.996478i \(0.473278\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 0 0
\(87\) −0.472136 −0.0506183
\(88\) 0 0
\(89\) −16.4721 −1.74604 −0.873021 0.487682i \(-0.837843\pi\)
−0.873021 + 0.487682i \(0.837843\pi\)
\(90\) 0 0
\(91\) −9.88854 −1.03660
\(92\) −1.00000 −0.104257
\(93\) 0 0
\(94\) 4.94427 0.509963
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −13.4164 −1.36223 −0.681115 0.732177i \(-0.738505\pi\)
−0.681115 + 0.732177i \(0.738505\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) 0 0
\(101\) 0.472136 0.0469793 0.0234896 0.999724i \(-0.492522\pi\)
0.0234896 + 0.999724i \(0.492522\pi\)
\(102\) 2.47214 0.244778
\(103\) 16.9443 1.66957 0.834784 0.550577i \(-0.185592\pi\)
0.834784 + 0.550577i \(0.185592\pi\)
\(104\) −2.47214 −0.242413
\(105\) 0 0
\(106\) −8.94427 −0.868744
\(107\) −2.47214 −0.238990 −0.119495 0.992835i \(-0.538128\pi\)
−0.119495 + 0.992835i \(0.538128\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0.472136 0.0452224 0.0226112 0.999744i \(-0.492802\pi\)
0.0226112 + 0.999744i \(0.492802\pi\)
\(110\) 0 0
\(111\) 0.472136 0.0448132
\(112\) 4.00000 0.377964
\(113\) 19.4164 1.82654 0.913271 0.407353i \(-0.133548\pi\)
0.913271 + 0.407353i \(0.133548\pi\)
\(114\) 2.00000 0.187317
\(115\) 0 0
\(116\) −0.472136 −0.0438367
\(117\) −2.47214 −0.228549
\(118\) −6.00000 −0.552345
\(119\) 9.88854 0.906481
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −0.472136 −0.0427452
\(123\) 10.9443 0.986812
\(124\) 0 0
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 11.4164 1.01304 0.506521 0.862228i \(-0.330931\pi\)
0.506521 + 0.862228i \(0.330931\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −6.94427 −0.606724 −0.303362 0.952875i \(-0.598109\pi\)
−0.303362 + 0.952875i \(0.598109\pi\)
\(132\) 0 0
\(133\) 8.00000 0.693688
\(134\) −4.94427 −0.427120
\(135\) 0 0
\(136\) 2.47214 0.211984
\(137\) −19.4164 −1.65886 −0.829428 0.558614i \(-0.811333\pi\)
−0.829428 + 0.558614i \(0.811333\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 4.94427 0.416383
\(142\) −7.52786 −0.631724
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 4.94427 0.409191
\(147\) 9.00000 0.742307
\(148\) 0.472136 0.0388093
\(149\) −13.4164 −1.09911 −0.549557 0.835456i \(-0.685204\pi\)
−0.549557 + 0.835456i \(0.685204\pi\)
\(150\) 0 0
\(151\) 16.9443 1.37891 0.689453 0.724331i \(-0.257851\pi\)
0.689453 + 0.724331i \(0.257851\pi\)
\(152\) 2.00000 0.162221
\(153\) 2.47214 0.199860
\(154\) 0 0
\(155\) 0 0
\(156\) −2.47214 −0.197929
\(157\) −11.5279 −0.920024 −0.460012 0.887913i \(-0.652155\pi\)
−0.460012 + 0.887913i \(0.652155\pi\)
\(158\) 12.4721 0.992230
\(159\) −8.94427 −0.709327
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 1.00000 0.0785674
\(163\) −4.94427 −0.387265 −0.193633 0.981074i \(-0.562027\pi\)
−0.193633 + 0.981074i \(0.562027\pi\)
\(164\) 10.9443 0.854604
\(165\) 0 0
\(166\) 1.52786 0.118585
\(167\) 3.05573 0.236459 0.118230 0.992986i \(-0.462278\pi\)
0.118230 + 0.992986i \(0.462278\pi\)
\(168\) 4.00000 0.308607
\(169\) −6.88854 −0.529888
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) −14.9443 −1.13619 −0.568096 0.822962i \(-0.692320\pi\)
−0.568096 + 0.822962i \(0.692320\pi\)
\(174\) −0.472136 −0.0357925
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) −16.4721 −1.23464
\(179\) −18.9443 −1.41596 −0.707981 0.706232i \(-0.750394\pi\)
−0.707981 + 0.706232i \(0.750394\pi\)
\(180\) 0 0
\(181\) 8.47214 0.629729 0.314864 0.949137i \(-0.398041\pi\)
0.314864 + 0.949137i \(0.398041\pi\)
\(182\) −9.88854 −0.732988
\(183\) −0.472136 −0.0349013
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 4.94427 0.360598
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −25.8885 −1.87323 −0.936615 0.350361i \(-0.886059\pi\)
−0.936615 + 0.350361i \(0.886059\pi\)
\(192\) 1.00000 0.0721688
\(193\) 20.0000 1.43963 0.719816 0.694165i \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −13.4164 −0.963242
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) 18.9443 1.34972 0.674862 0.737944i \(-0.264203\pi\)
0.674862 + 0.737944i \(0.264203\pi\)
\(198\) 0 0
\(199\) 17.4164 1.23462 0.617308 0.786721i \(-0.288223\pi\)
0.617308 + 0.786721i \(0.288223\pi\)
\(200\) 0 0
\(201\) −4.94427 −0.348742
\(202\) 0.472136 0.0332194
\(203\) −1.88854 −0.132550
\(204\) 2.47214 0.173084
\(205\) 0 0
\(206\) 16.9443 1.18056
\(207\) −1.00000 −0.0695048
\(208\) −2.47214 −0.171412
\(209\) 0 0
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −8.94427 −0.614295
\(213\) −7.52786 −0.515801
\(214\) −2.47214 −0.168992
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 0.472136 0.0319771
\(219\) 4.94427 0.334103
\(220\) 0 0
\(221\) −6.11146 −0.411101
\(222\) 0.472136 0.0316877
\(223\) 11.4164 0.764499 0.382250 0.924059i \(-0.375149\pi\)
0.382250 + 0.924059i \(0.375149\pi\)
\(224\) 4.00000 0.267261
\(225\) 0 0
\(226\) 19.4164 1.29156
\(227\) 18.4721 1.22604 0.613019 0.790068i \(-0.289955\pi\)
0.613019 + 0.790068i \(0.289955\pi\)
\(228\) 2.00000 0.132453
\(229\) 25.4164 1.67956 0.839782 0.542924i \(-0.182683\pi\)
0.839782 + 0.542924i \(0.182683\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.472136 −0.0309972
\(233\) 14.9443 0.979032 0.489516 0.871994i \(-0.337174\pi\)
0.489516 + 0.871994i \(0.337174\pi\)
\(234\) −2.47214 −0.161609
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) 12.4721 0.810152
\(238\) 9.88854 0.640979
\(239\) −12.4721 −0.806755 −0.403378 0.915034i \(-0.632164\pi\)
−0.403378 + 0.915034i \(0.632164\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) −11.0000 −0.707107
\(243\) 1.00000 0.0641500
\(244\) −0.472136 −0.0302254
\(245\) 0 0
\(246\) 10.9443 0.697781
\(247\) −4.94427 −0.314596
\(248\) 0 0
\(249\) 1.52786 0.0968244
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 4.00000 0.251976
\(253\) 0 0
\(254\) 11.4164 0.716329
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 1.88854 0.117348
\(260\) 0 0
\(261\) −0.472136 −0.0292245
\(262\) −6.94427 −0.429019
\(263\) 20.9443 1.29148 0.645740 0.763558i \(-0.276549\pi\)
0.645740 + 0.763558i \(0.276549\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 8.00000 0.490511
\(267\) −16.4721 −1.00808
\(268\) −4.94427 −0.302019
\(269\) 0.472136 0.0287866 0.0143933 0.999896i \(-0.495418\pi\)
0.0143933 + 0.999896i \(0.495418\pi\)
\(270\) 0 0
\(271\) 0.944272 0.0573604 0.0286802 0.999589i \(-0.490870\pi\)
0.0286802 + 0.999589i \(0.490870\pi\)
\(272\) 2.47214 0.149895
\(273\) −9.88854 −0.598482
\(274\) −19.4164 −1.17299
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −27.4164 −1.64729 −0.823646 0.567104i \(-0.808064\pi\)
−0.823646 + 0.567104i \(0.808064\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.52786 0.210455 0.105227 0.994448i \(-0.466443\pi\)
0.105227 + 0.994448i \(0.466443\pi\)
\(282\) 4.94427 0.294427
\(283\) 24.0000 1.42665 0.713326 0.700832i \(-0.247188\pi\)
0.713326 + 0.700832i \(0.247188\pi\)
\(284\) −7.52786 −0.446697
\(285\) 0 0
\(286\) 0 0
\(287\) 43.7771 2.58408
\(288\) 1.00000 0.0589256
\(289\) −10.8885 −0.640503
\(290\) 0 0
\(291\) −13.4164 −0.786484
\(292\) 4.94427 0.289342
\(293\) −0.944272 −0.0551650 −0.0275825 0.999620i \(-0.508781\pi\)
−0.0275825 + 0.999620i \(0.508781\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) 0.472136 0.0274423
\(297\) 0 0
\(298\) −13.4164 −0.777192
\(299\) 2.47214 0.142967
\(300\) 0 0
\(301\) 0 0
\(302\) 16.9443 0.975033
\(303\) 0.472136 0.0271235
\(304\) 2.00000 0.114708
\(305\) 0 0
\(306\) 2.47214 0.141323
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 16.9443 0.963926
\(310\) 0 0
\(311\) −24.4721 −1.38769 −0.693844 0.720126i \(-0.744084\pi\)
−0.693844 + 0.720126i \(0.744084\pi\)
\(312\) −2.47214 −0.139957
\(313\) −25.4164 −1.43662 −0.718310 0.695723i \(-0.755084\pi\)
−0.718310 + 0.695723i \(0.755084\pi\)
\(314\) −11.5279 −0.650555
\(315\) 0 0
\(316\) 12.4721 0.701612
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −8.94427 −0.501570
\(319\) 0 0
\(320\) 0 0
\(321\) −2.47214 −0.137981
\(322\) −4.00000 −0.222911
\(323\) 4.94427 0.275107
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −4.94427 −0.273838
\(327\) 0.472136 0.0261092
\(328\) 10.9443 0.604296
\(329\) 19.7771 1.09035
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 1.52786 0.0838524
\(333\) 0.472136 0.0258729
\(334\) 3.05573 0.167202
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 25.4164 1.38452 0.692260 0.721648i \(-0.256615\pi\)
0.692260 + 0.721648i \(0.256615\pi\)
\(338\) −6.88854 −0.374687
\(339\) 19.4164 1.05455
\(340\) 0 0
\(341\) 0 0
\(342\) 2.00000 0.108148
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) −14.9443 −0.803409
\(347\) 8.94427 0.480154 0.240077 0.970754i \(-0.422827\pi\)
0.240077 + 0.970754i \(0.422827\pi\)
\(348\) −0.472136 −0.0253091
\(349\) −27.8885 −1.49284 −0.746420 0.665475i \(-0.768229\pi\)
−0.746420 + 0.665475i \(0.768229\pi\)
\(350\) 0 0
\(351\) −2.47214 −0.131953
\(352\) 0 0
\(353\) 2.94427 0.156708 0.0783539 0.996926i \(-0.475034\pi\)
0.0783539 + 0.996926i \(0.475034\pi\)
\(354\) −6.00000 −0.318896
\(355\) 0 0
\(356\) −16.4721 −0.873021
\(357\) 9.88854 0.523357
\(358\) −18.9443 −1.00124
\(359\) −32.9443 −1.73873 −0.869366 0.494169i \(-0.835473\pi\)
−0.869366 + 0.494169i \(0.835473\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 8.47214 0.445286
\(363\) −11.0000 −0.577350
\(364\) −9.88854 −0.518301
\(365\) 0 0
\(366\) −0.472136 −0.0246789
\(367\) −12.0000 −0.626395 −0.313197 0.949688i \(-0.601400\pi\)
−0.313197 + 0.949688i \(0.601400\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 10.9443 0.569736
\(370\) 0 0
\(371\) −35.7771 −1.85745
\(372\) 0 0
\(373\) −21.4164 −1.10890 −0.554450 0.832217i \(-0.687071\pi\)
−0.554450 + 0.832217i \(0.687071\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 4.94427 0.254981
\(377\) 1.16718 0.0601130
\(378\) 4.00000 0.205738
\(379\) 16.8328 0.864644 0.432322 0.901719i \(-0.357694\pi\)
0.432322 + 0.901719i \(0.357694\pi\)
\(380\) 0 0
\(381\) 11.4164 0.584880
\(382\) −25.8885 −1.32457
\(383\) −13.8885 −0.709671 −0.354836 0.934929i \(-0.615463\pi\)
−0.354836 + 0.934929i \(0.615463\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 20.0000 1.01797
\(387\) 0 0
\(388\) −13.4164 −0.681115
\(389\) 4.47214 0.226746 0.113373 0.993552i \(-0.463834\pi\)
0.113373 + 0.993552i \(0.463834\pi\)
\(390\) 0 0
\(391\) −2.47214 −0.125021
\(392\) 9.00000 0.454569
\(393\) −6.94427 −0.350292
\(394\) 18.9443 0.954399
\(395\) 0 0
\(396\) 0 0
\(397\) 29.3050 1.47077 0.735387 0.677648i \(-0.237001\pi\)
0.735387 + 0.677648i \(0.237001\pi\)
\(398\) 17.4164 0.873006
\(399\) 8.00000 0.400501
\(400\) 0 0
\(401\) 25.4164 1.26923 0.634617 0.772826i \(-0.281158\pi\)
0.634617 + 0.772826i \(0.281158\pi\)
\(402\) −4.94427 −0.246598
\(403\) 0 0
\(404\) 0.472136 0.0234896
\(405\) 0 0
\(406\) −1.88854 −0.0937269
\(407\) 0 0
\(408\) 2.47214 0.122389
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) −19.4164 −0.957741
\(412\) 16.9443 0.834784
\(413\) −24.0000 −1.18096
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −2.47214 −0.121206
\(417\) 0 0
\(418\) 0 0
\(419\) −37.8885 −1.85098 −0.925488 0.378776i \(-0.876345\pi\)
−0.925488 + 0.378776i \(0.876345\pi\)
\(420\) 0 0
\(421\) −37.4164 −1.82356 −0.911782 0.410674i \(-0.865293\pi\)
−0.911782 + 0.410674i \(0.865293\pi\)
\(422\) −12.0000 −0.584151
\(423\) 4.94427 0.240399
\(424\) −8.94427 −0.434372
\(425\) 0 0
\(426\) −7.52786 −0.364726
\(427\) −1.88854 −0.0913930
\(428\) −2.47214 −0.119495
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000 0.0481125
\(433\) −21.4164 −1.02921 −0.514603 0.857428i \(-0.672061\pi\)
−0.514603 + 0.857428i \(0.672061\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.472136 0.0226112
\(437\) −2.00000 −0.0956730
\(438\) 4.94427 0.236246
\(439\) −26.8328 −1.28066 −0.640330 0.768100i \(-0.721202\pi\)
−0.640330 + 0.768100i \(0.721202\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) −6.11146 −0.290692
\(443\) −8.94427 −0.424955 −0.212478 0.977166i \(-0.568153\pi\)
−0.212478 + 0.977166i \(0.568153\pi\)
\(444\) 0.472136 0.0224066
\(445\) 0 0
\(446\) 11.4164 0.540583
\(447\) −13.4164 −0.634574
\(448\) 4.00000 0.188982
\(449\) −34.9443 −1.64912 −0.824561 0.565773i \(-0.808578\pi\)
−0.824561 + 0.565773i \(0.808578\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 19.4164 0.913271
\(453\) 16.9443 0.796111
\(454\) 18.4721 0.866940
\(455\) 0 0
\(456\) 2.00000 0.0936586
\(457\) 25.4164 1.18893 0.594465 0.804122i \(-0.297364\pi\)
0.594465 + 0.804122i \(0.297364\pi\)
\(458\) 25.4164 1.18763
\(459\) 2.47214 0.115389
\(460\) 0 0
\(461\) −14.3607 −0.668844 −0.334422 0.942424i \(-0.608541\pi\)
−0.334422 + 0.942424i \(0.608541\pi\)
\(462\) 0 0
\(463\) −15.4164 −0.716461 −0.358231 0.933633i \(-0.616620\pi\)
−0.358231 + 0.933633i \(0.616620\pi\)
\(464\) −0.472136 −0.0219184
\(465\) 0 0
\(466\) 14.9443 0.692280
\(467\) 26.4721 1.22498 0.612492 0.790477i \(-0.290167\pi\)
0.612492 + 0.790477i \(0.290167\pi\)
\(468\) −2.47214 −0.114275
\(469\) −19.7771 −0.913221
\(470\) 0 0
\(471\) −11.5279 −0.531176
\(472\) −6.00000 −0.276172
\(473\) 0 0
\(474\) 12.4721 0.572864
\(475\) 0 0
\(476\) 9.88854 0.453241
\(477\) −8.94427 −0.409530
\(478\) −12.4721 −0.570462
\(479\) −16.9443 −0.774204 −0.387102 0.922037i \(-0.626524\pi\)
−0.387102 + 0.922037i \(0.626524\pi\)
\(480\) 0 0
\(481\) −1.16718 −0.0532190
\(482\) −2.00000 −0.0910975
\(483\) −4.00000 −0.182006
\(484\) −11.0000 −0.500000
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −31.4164 −1.42361 −0.711807 0.702375i \(-0.752123\pi\)
−0.711807 + 0.702375i \(0.752123\pi\)
\(488\) −0.472136 −0.0213726
\(489\) −4.94427 −0.223588
\(490\) 0 0
\(491\) 27.8885 1.25859 0.629296 0.777166i \(-0.283343\pi\)
0.629296 + 0.777166i \(0.283343\pi\)
\(492\) 10.9443 0.493406
\(493\) −1.16718 −0.0525673
\(494\) −4.94427 −0.222453
\(495\) 0 0
\(496\) 0 0
\(497\) −30.1115 −1.35068
\(498\) 1.52786 0.0684652
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 3.05573 0.136520
\(502\) −12.0000 −0.535586
\(503\) 13.8885 0.619260 0.309630 0.950857i \(-0.399795\pi\)
0.309630 + 0.950857i \(0.399795\pi\)
\(504\) 4.00000 0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) −6.88854 −0.305931
\(508\) 11.4164 0.506521
\(509\) 1.41641 0.0627812 0.0313906 0.999507i \(-0.490006\pi\)
0.0313906 + 0.999507i \(0.490006\pi\)
\(510\) 0 0
\(511\) 19.7771 0.874887
\(512\) 1.00000 0.0441942
\(513\) 2.00000 0.0883022
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 1.88854 0.0829779
\(519\) −14.9443 −0.655981
\(520\) 0 0
\(521\) −19.5279 −0.855531 −0.427766 0.903890i \(-0.640699\pi\)
−0.427766 + 0.903890i \(0.640699\pi\)
\(522\) −0.472136 −0.0206648
\(523\) 4.94427 0.216198 0.108099 0.994140i \(-0.465524\pi\)
0.108099 + 0.994140i \(0.465524\pi\)
\(524\) −6.94427 −0.303362
\(525\) 0 0
\(526\) 20.9443 0.913214
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −6.00000 −0.260378
\(532\) 8.00000 0.346844
\(533\) −27.0557 −1.17191
\(534\) −16.4721 −0.712819
\(535\) 0 0
\(536\) −4.94427 −0.213560
\(537\) −18.9443 −0.817506
\(538\) 0.472136 0.0203552
\(539\) 0 0
\(540\) 0 0
\(541\) 40.8328 1.75554 0.877770 0.479082i \(-0.159030\pi\)
0.877770 + 0.479082i \(0.159030\pi\)
\(542\) 0.944272 0.0405600
\(543\) 8.47214 0.363574
\(544\) 2.47214 0.105992
\(545\) 0 0
\(546\) −9.88854 −0.423191
\(547\) −22.8328 −0.976261 −0.488130 0.872771i \(-0.662321\pi\)
−0.488130 + 0.872771i \(0.662321\pi\)
\(548\) −19.4164 −0.829428
\(549\) −0.472136 −0.0201503
\(550\) 0 0
\(551\) −0.944272 −0.0402273
\(552\) −1.00000 −0.0425628
\(553\) 49.8885 2.12148
\(554\) −27.4164 −1.16481
\(555\) 0 0
\(556\) 0 0
\(557\) −29.8885 −1.26642 −0.633209 0.773981i \(-0.718263\pi\)
−0.633209 + 0.773981i \(0.718263\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 3.52786 0.148814
\(563\) −37.3050 −1.57222 −0.786108 0.618089i \(-0.787907\pi\)
−0.786108 + 0.618089i \(0.787907\pi\)
\(564\) 4.94427 0.208191
\(565\) 0 0
\(566\) 24.0000 1.00880
\(567\) 4.00000 0.167984
\(568\) −7.52786 −0.315862
\(569\) 39.3050 1.64775 0.823875 0.566772i \(-0.191808\pi\)
0.823875 + 0.566772i \(0.191808\pi\)
\(570\) 0 0
\(571\) 32.8328 1.37401 0.687005 0.726652i \(-0.258925\pi\)
0.687005 + 0.726652i \(0.258925\pi\)
\(572\) 0 0
\(573\) −25.8885 −1.08151
\(574\) 43.7771 1.82722
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 16.9443 0.705399 0.352700 0.935737i \(-0.385264\pi\)
0.352700 + 0.935737i \(0.385264\pi\)
\(578\) −10.8885 −0.452904
\(579\) 20.0000 0.831172
\(580\) 0 0
\(581\) 6.11146 0.253546
\(582\) −13.4164 −0.556128
\(583\) 0 0
\(584\) 4.94427 0.204595
\(585\) 0 0
\(586\) −0.944272 −0.0390075
\(587\) −29.8885 −1.23363 −0.616816 0.787107i \(-0.711578\pi\)
−0.616816 + 0.787107i \(0.711578\pi\)
\(588\) 9.00000 0.371154
\(589\) 0 0
\(590\) 0 0
\(591\) 18.9443 0.779263
\(592\) 0.472136 0.0194047
\(593\) 23.8885 0.980985 0.490492 0.871445i \(-0.336817\pi\)
0.490492 + 0.871445i \(0.336817\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13.4164 −0.549557
\(597\) 17.4164 0.712806
\(598\) 2.47214 0.101093
\(599\) 26.3607 1.07707 0.538534 0.842604i \(-0.318978\pi\)
0.538534 + 0.842604i \(0.318978\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) −4.94427 −0.201346
\(604\) 16.9443 0.689453
\(605\) 0 0
\(606\) 0.472136 0.0191792
\(607\) 7.41641 0.301023 0.150511 0.988608i \(-0.451908\pi\)
0.150511 + 0.988608i \(0.451908\pi\)
\(608\) 2.00000 0.0811107
\(609\) −1.88854 −0.0765277
\(610\) 0 0
\(611\) −12.2229 −0.494486
\(612\) 2.47214 0.0999302
\(613\) 9.41641 0.380325 0.190163 0.981753i \(-0.439098\pi\)
0.190163 + 0.981753i \(0.439098\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.4721 0.582626 0.291313 0.956628i \(-0.405908\pi\)
0.291313 + 0.956628i \(0.405908\pi\)
\(618\) 16.9443 0.681599
\(619\) 40.8328 1.64121 0.820605 0.571496i \(-0.193637\pi\)
0.820605 + 0.571496i \(0.193637\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −24.4721 −0.981243
\(623\) −65.8885 −2.63977
\(624\) −2.47214 −0.0989646
\(625\) 0 0
\(626\) −25.4164 −1.01584
\(627\) 0 0
\(628\) −11.5279 −0.460012
\(629\) 1.16718 0.0465387
\(630\) 0 0
\(631\) −2.36068 −0.0939772 −0.0469886 0.998895i \(-0.514962\pi\)
−0.0469886 + 0.998895i \(0.514962\pi\)
\(632\) 12.4721 0.496115
\(633\) −12.0000 −0.476957
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −8.94427 −0.354663
\(637\) −22.2492 −0.881546
\(638\) 0 0
\(639\) −7.52786 −0.297798
\(640\) 0 0
\(641\) −11.5279 −0.455323 −0.227662 0.973740i \(-0.573108\pi\)
−0.227662 + 0.973740i \(0.573108\pi\)
\(642\) −2.47214 −0.0975674
\(643\) 12.9443 0.510472 0.255236 0.966879i \(-0.417847\pi\)
0.255236 + 0.966879i \(0.417847\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) 4.94427 0.194530
\(647\) −41.8885 −1.64681 −0.823404 0.567455i \(-0.807928\pi\)
−0.823404 + 0.567455i \(0.807928\pi\)
\(648\) 1.00000 0.0392837
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −4.94427 −0.193633
\(653\) 20.8328 0.815251 0.407626 0.913149i \(-0.366357\pi\)
0.407626 + 0.913149i \(0.366357\pi\)
\(654\) 0.472136 0.0184620
\(655\) 0 0
\(656\) 10.9443 0.427302
\(657\) 4.94427 0.192894
\(658\) 19.7771 0.770991
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) −36.2492 −1.40993 −0.704966 0.709241i \(-0.749038\pi\)
−0.704966 + 0.709241i \(0.749038\pi\)
\(662\) 0 0
\(663\) −6.11146 −0.237349
\(664\) 1.52786 0.0592926
\(665\) 0 0
\(666\) 0.472136 0.0182949
\(667\) 0.472136 0.0182812
\(668\) 3.05573 0.118230
\(669\) 11.4164 0.441384
\(670\) 0 0
\(671\) 0 0
\(672\) 4.00000 0.154303
\(673\) −38.8328 −1.49690 −0.748448 0.663194i \(-0.769200\pi\)
−0.748448 + 0.663194i \(0.769200\pi\)
\(674\) 25.4164 0.979003
\(675\) 0 0
\(676\) −6.88854 −0.264944
\(677\) −25.8885 −0.994978 −0.497489 0.867470i \(-0.665744\pi\)
−0.497489 + 0.867470i \(0.665744\pi\)
\(678\) 19.4164 0.745683
\(679\) −53.6656 −2.05950
\(680\) 0 0
\(681\) 18.4721 0.707854
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 2.00000 0.0764719
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 25.4164 0.969696
\(688\) 0 0
\(689\) 22.1115 0.842379
\(690\) 0 0
\(691\) −39.7771 −1.51319 −0.756596 0.653883i \(-0.773139\pi\)
−0.756596 + 0.653883i \(0.773139\pi\)
\(692\) −14.9443 −0.568096
\(693\) 0 0
\(694\) 8.94427 0.339520
\(695\) 0 0
\(696\) −0.472136 −0.0178963
\(697\) 27.0557 1.02481
\(698\) −27.8885 −1.05560
\(699\) 14.9443 0.565244
\(700\) 0 0
\(701\) 37.4164 1.41320 0.706599 0.707614i \(-0.250228\pi\)
0.706599 + 0.707614i \(0.250228\pi\)
\(702\) −2.47214 −0.0933048
\(703\) 0.944272 0.0356139
\(704\) 0 0
\(705\) 0 0
\(706\) 2.94427 0.110809
\(707\) 1.88854 0.0710260
\(708\) −6.00000 −0.225494
\(709\) 20.4721 0.768847 0.384424 0.923157i \(-0.374400\pi\)
0.384424 + 0.923157i \(0.374400\pi\)
\(710\) 0 0
\(711\) 12.4721 0.467742
\(712\) −16.4721 −0.617319
\(713\) 0 0
\(714\) 9.88854 0.370069
\(715\) 0 0
\(716\) −18.9443 −0.707981
\(717\) −12.4721 −0.465780
\(718\) −32.9443 −1.22947
\(719\) −35.5279 −1.32497 −0.662483 0.749077i \(-0.730497\pi\)
−0.662483 + 0.749077i \(0.730497\pi\)
\(720\) 0 0
\(721\) 67.7771 2.52415
\(722\) −15.0000 −0.558242
\(723\) −2.00000 −0.0743808
\(724\) 8.47214 0.314864
\(725\) 0 0
\(726\) −11.0000 −0.408248
\(727\) 0.944272 0.0350211 0.0175106 0.999847i \(-0.494426\pi\)
0.0175106 + 0.999847i \(0.494426\pi\)
\(728\) −9.88854 −0.366494
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) −0.472136 −0.0174506
\(733\) −0.472136 −0.0174387 −0.00871937 0.999962i \(-0.502775\pi\)
−0.00871937 + 0.999962i \(0.502775\pi\)
\(734\) −12.0000 −0.442928
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 0 0
\(738\) 10.9443 0.402864
\(739\) 1.88854 0.0694712 0.0347356 0.999397i \(-0.488941\pi\)
0.0347356 + 0.999397i \(0.488941\pi\)
\(740\) 0 0
\(741\) −4.94427 −0.181632
\(742\) −35.7771 −1.31342
\(743\) 2.11146 0.0774618 0.0387309 0.999250i \(-0.487668\pi\)
0.0387309 + 0.999250i \(0.487668\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −21.4164 −0.784110
\(747\) 1.52786 0.0559016
\(748\) 0 0
\(749\) −9.88854 −0.361320
\(750\) 0 0
\(751\) 36.4721 1.33089 0.665444 0.746448i \(-0.268242\pi\)
0.665444 + 0.746448i \(0.268242\pi\)
\(752\) 4.94427 0.180299
\(753\) −12.0000 −0.437304
\(754\) 1.16718 0.0425063
\(755\) 0 0
\(756\) 4.00000 0.145479
\(757\) 45.4164 1.65069 0.825344 0.564631i \(-0.190981\pi\)
0.825344 + 0.564631i \(0.190981\pi\)
\(758\) 16.8328 0.611395
\(759\) 0 0
\(760\) 0 0
\(761\) −17.7771 −0.644419 −0.322209 0.946668i \(-0.604426\pi\)
−0.322209 + 0.946668i \(0.604426\pi\)
\(762\) 11.4164 0.413573
\(763\) 1.88854 0.0683699
\(764\) −25.8885 −0.936615
\(765\) 0 0
\(766\) −13.8885 −0.501813
\(767\) 14.8328 0.535582
\(768\) 1.00000 0.0360844
\(769\) 34.9443 1.26012 0.630061 0.776545i \(-0.283030\pi\)
0.630061 + 0.776545i \(0.283030\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 20.0000 0.719816
\(773\) −3.05573 −0.109907 −0.0549535 0.998489i \(-0.517501\pi\)
−0.0549535 + 0.998489i \(0.517501\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −13.4164 −0.481621
\(777\) 1.88854 0.0677511
\(778\) 4.47214 0.160334
\(779\) 21.8885 0.784238
\(780\) 0 0
\(781\) 0 0
\(782\) −2.47214 −0.0884034
\(783\) −0.472136 −0.0168728
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) −6.94427 −0.247694
\(787\) −1.16718 −0.0416056 −0.0208028 0.999784i \(-0.506622\pi\)
−0.0208028 + 0.999784i \(0.506622\pi\)
\(788\) 18.9443 0.674862
\(789\) 20.9443 0.745636
\(790\) 0 0
\(791\) 77.6656 2.76147
\(792\) 0 0
\(793\) 1.16718 0.0414479
\(794\) 29.3050 1.03999
\(795\) 0 0
\(796\) 17.4164 0.617308
\(797\) −37.8885 −1.34208 −0.671041 0.741421i \(-0.734152\pi\)
−0.671041 + 0.741421i \(0.734152\pi\)
\(798\) 8.00000 0.283197
\(799\) 12.2229 0.432416
\(800\) 0 0
\(801\) −16.4721 −0.582014
\(802\) 25.4164 0.897485
\(803\) 0 0
\(804\) −4.94427 −0.174371
\(805\) 0 0
\(806\) 0 0
\(807\) 0.472136 0.0166200
\(808\) 0.472136 0.0166097
\(809\) −3.88854 −0.136714 −0.0683570 0.997661i \(-0.521776\pi\)
−0.0683570 + 0.997661i \(0.521776\pi\)
\(810\) 0 0
\(811\) −49.8885 −1.75182 −0.875912 0.482471i \(-0.839739\pi\)
−0.875912 + 0.482471i \(0.839739\pi\)
\(812\) −1.88854 −0.0662749
\(813\) 0.944272 0.0331171
\(814\) 0 0
\(815\) 0 0
\(816\) 2.47214 0.0865421
\(817\) 0 0
\(818\) −18.0000 −0.629355
\(819\) −9.88854 −0.345534
\(820\) 0 0
\(821\) −16.4721 −0.574882 −0.287441 0.957798i \(-0.592804\pi\)
−0.287441 + 0.957798i \(0.592804\pi\)
\(822\) −19.4164 −0.677225
\(823\) 20.3607 0.709729 0.354864 0.934918i \(-0.384527\pi\)
0.354864 + 0.934918i \(0.384527\pi\)
\(824\) 16.9443 0.590282
\(825\) 0 0
\(826\) −24.0000 −0.835067
\(827\) −42.4721 −1.47690 −0.738450 0.674308i \(-0.764442\pi\)
−0.738450 + 0.674308i \(0.764442\pi\)
\(828\) −1.00000 −0.0347524
\(829\) 30.9443 1.07474 0.537369 0.843347i \(-0.319418\pi\)
0.537369 + 0.843347i \(0.319418\pi\)
\(830\) 0 0
\(831\) −27.4164 −0.951065
\(832\) −2.47214 −0.0857059
\(833\) 22.2492 0.770890
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −37.8885 −1.30884
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −28.7771 −0.992313
\(842\) −37.4164 −1.28945
\(843\) 3.52786 0.121506
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 4.94427 0.169988
\(847\) −44.0000 −1.51186
\(848\) −8.94427 −0.307148
\(849\) 24.0000 0.823678
\(850\) 0 0
\(851\) −0.472136 −0.0161846
\(852\) −7.52786 −0.257900
\(853\) 9.52786 0.326228 0.163114 0.986607i \(-0.447846\pi\)
0.163114 + 0.986607i \(0.447846\pi\)
\(854\) −1.88854 −0.0646246
\(855\) 0 0
\(856\) −2.47214 −0.0844959
\(857\) −23.8885 −0.816017 −0.408009 0.912978i \(-0.633777\pi\)
−0.408009 + 0.912978i \(0.633777\pi\)
\(858\) 0 0
\(859\) −39.7771 −1.35718 −0.678588 0.734519i \(-0.737408\pi\)
−0.678588 + 0.734519i \(0.737408\pi\)
\(860\) 0 0
\(861\) 43.7771 1.49192
\(862\) 0 0
\(863\) 3.05573 0.104018 0.0520091 0.998647i \(-0.483438\pi\)
0.0520091 + 0.998647i \(0.483438\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −21.4164 −0.727759
\(867\) −10.8885 −0.369794
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 12.2229 0.414158
\(872\) 0.472136 0.0159885
\(873\) −13.4164 −0.454077
\(874\) −2.00000 −0.0676510
\(875\) 0 0
\(876\) 4.94427 0.167051
\(877\) 20.3607 0.687531 0.343766 0.939055i \(-0.388297\pi\)
0.343766 + 0.939055i \(0.388297\pi\)
\(878\) −26.8328 −0.905564
\(879\) −0.944272 −0.0318495
\(880\) 0 0
\(881\) −4.47214 −0.150670 −0.0753350 0.997158i \(-0.524003\pi\)
−0.0753350 + 0.997158i \(0.524003\pi\)
\(882\) 9.00000 0.303046
\(883\) −19.7771 −0.665552 −0.332776 0.943006i \(-0.607985\pi\)
−0.332776 + 0.943006i \(0.607985\pi\)
\(884\) −6.11146 −0.205551
\(885\) 0 0
\(886\) −8.94427 −0.300489
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0.472136 0.0158438
\(889\) 45.6656 1.53158
\(890\) 0 0
\(891\) 0 0
\(892\) 11.4164 0.382250
\(893\) 9.88854 0.330908
\(894\) −13.4164 −0.448712
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 2.47214 0.0825422
\(898\) −34.9443 −1.16611
\(899\) 0 0
\(900\) 0 0
\(901\) −22.1115 −0.736639
\(902\) 0 0
\(903\) 0 0
\(904\) 19.4164 0.645780
\(905\) 0 0
\(906\) 16.9443 0.562936
\(907\) 14.8328 0.492516 0.246258 0.969204i \(-0.420799\pi\)
0.246258 + 0.969204i \(0.420799\pi\)
\(908\) 18.4721 0.613019
\(909\) 0.472136 0.0156598
\(910\) 0 0
\(911\) 24.9443 0.826441 0.413220 0.910631i \(-0.364404\pi\)
0.413220 + 0.910631i \(0.364404\pi\)
\(912\) 2.00000 0.0662266
\(913\) 0 0
\(914\) 25.4164 0.840700
\(915\) 0 0
\(916\) 25.4164 0.839782
\(917\) −27.7771 −0.917280
\(918\) 2.47214 0.0815926
\(919\) −9.41641 −0.310619 −0.155309 0.987866i \(-0.549637\pi\)
−0.155309 + 0.987866i \(0.549637\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −14.3607 −0.472944
\(923\) 18.6099 0.612552
\(924\) 0 0
\(925\) 0 0
\(926\) −15.4164 −0.506615
\(927\) 16.9443 0.556523
\(928\) −0.472136 −0.0154986
\(929\) −10.9443 −0.359070 −0.179535 0.983752i \(-0.557459\pi\)
−0.179535 + 0.983752i \(0.557459\pi\)
\(930\) 0 0
\(931\) 18.0000 0.589926
\(932\) 14.9443 0.489516
\(933\) −24.4721 −0.801182
\(934\) 26.4721 0.866195
\(935\) 0 0
\(936\) −2.47214 −0.0808043
\(937\) 6.58359 0.215077 0.107538 0.994201i \(-0.465703\pi\)
0.107538 + 0.994201i \(0.465703\pi\)
\(938\) −19.7771 −0.645745
\(939\) −25.4164 −0.829433
\(940\) 0 0
\(941\) 36.4721 1.18896 0.594479 0.804111i \(-0.297358\pi\)
0.594479 + 0.804111i \(0.297358\pi\)
\(942\) −11.5279 −0.375598
\(943\) −10.9443 −0.356395
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 12.4721 0.405076
\(949\) −12.2229 −0.396773
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 9.88854 0.320490
\(953\) −12.3607 −0.400402 −0.200201 0.979755i \(-0.564159\pi\)
−0.200201 + 0.979755i \(0.564159\pi\)
\(954\) −8.94427 −0.289581
\(955\) 0 0
\(956\) −12.4721 −0.403378
\(957\) 0 0
\(958\) −16.9443 −0.547445
\(959\) −77.6656 −2.50795
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −1.16718 −0.0376315
\(963\) −2.47214 −0.0796635
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) −4.00000 −0.128698
\(967\) 55.4164 1.78207 0.891036 0.453933i \(-0.149979\pi\)
0.891036 + 0.453933i \(0.149979\pi\)
\(968\) −11.0000 −0.353553
\(969\) 4.94427 0.158833
\(970\) 0 0
\(971\) −18.1115 −0.581224 −0.290612 0.956841i \(-0.593859\pi\)
−0.290612 + 0.956841i \(0.593859\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −31.4164 −1.00665
\(975\) 0 0
\(976\) −0.472136 −0.0151127
\(977\) 1.52786 0.0488807 0.0244404 0.999701i \(-0.492220\pi\)
0.0244404 + 0.999701i \(0.492220\pi\)
\(978\) −4.94427 −0.158100
\(979\) 0 0
\(980\) 0 0
\(981\) 0.472136 0.0150741
\(982\) 27.8885 0.889959
\(983\) −24.0000 −0.765481 −0.382741 0.923856i \(-0.625020\pi\)
−0.382741 + 0.923856i \(0.625020\pi\)
\(984\) 10.9443 0.348891
\(985\) 0 0
\(986\) −1.16718 −0.0371707
\(987\) 19.7771 0.629512
\(988\) −4.94427 −0.157298
\(989\) 0 0
\(990\) 0 0
\(991\) 53.6656 1.70474 0.852372 0.522935i \(-0.175163\pi\)
0.852372 + 0.522935i \(0.175163\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −30.1115 −0.955077
\(995\) 0 0
\(996\) 1.52786 0.0484122
\(997\) −7.41641 −0.234880 −0.117440 0.993080i \(-0.537469\pi\)
−0.117440 + 0.993080i \(0.537469\pi\)
\(998\) −24.0000 −0.759707
\(999\) 0.472136 0.0149377
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.bn.1.1 2
5.2 odd 4 690.2.d.a.139.4 yes 4
5.3 odd 4 690.2.d.a.139.2 4
5.4 even 2 3450.2.a.bc.1.2 2
15.2 even 4 2070.2.d.b.829.1 4
15.8 even 4 2070.2.d.b.829.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.a.139.2 4 5.3 odd 4
690.2.d.a.139.4 yes 4 5.2 odd 4
2070.2.d.b.829.1 4 15.2 even 4
2070.2.d.b.829.3 4 15.8 even 4
3450.2.a.bc.1.2 2 5.4 even 2
3450.2.a.bn.1.1 2 1.1 even 1 trivial