Properties

Label 3450.2.a.be.1.2
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.47214 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.23607 q^{11} -1.00000 q^{12} -4.47214 q^{13} -4.47214 q^{14} +1.00000 q^{16} +4.00000 q^{17} -1.00000 q^{18} +5.70820 q^{19} -4.47214 q^{21} +5.23607 q^{22} -1.00000 q^{23} +1.00000 q^{24} +4.47214 q^{26} -1.00000 q^{27} +4.47214 q^{28} -4.47214 q^{29} -2.47214 q^{31} -1.00000 q^{32} +5.23607 q^{33} -4.00000 q^{34} +1.00000 q^{36} -11.2361 q^{37} -5.70820 q^{38} +4.47214 q^{39} -2.00000 q^{41} +4.47214 q^{42} +4.76393 q^{43} -5.23607 q^{44} +1.00000 q^{46} -4.00000 q^{47} -1.00000 q^{48} +13.0000 q^{49} -4.00000 q^{51} -4.47214 q^{52} -5.23607 q^{53} +1.00000 q^{54} -4.47214 q^{56} -5.70820 q^{57} +4.47214 q^{58} -8.94427 q^{59} +0.763932 q^{61} +2.47214 q^{62} +4.47214 q^{63} +1.00000 q^{64} -5.23607 q^{66} -9.70820 q^{67} +4.00000 q^{68} +1.00000 q^{69} +8.94427 q^{71} -1.00000 q^{72} +4.47214 q^{73} +11.2361 q^{74} +5.70820 q^{76} -23.4164 q^{77} -4.47214 q^{78} +4.47214 q^{79} +1.00000 q^{81} +2.00000 q^{82} -13.2361 q^{83} -4.47214 q^{84} -4.76393 q^{86} +4.47214 q^{87} +5.23607 q^{88} -10.4721 q^{89} -20.0000 q^{91} -1.00000 q^{92} +2.47214 q^{93} +4.00000 q^{94} +1.00000 q^{96} -0.472136 q^{97} -13.0000 q^{98} -5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{8} + 2q^{9} - 6q^{11} - 2q^{12} + 2q^{16} + 8q^{17} - 2q^{18} - 2q^{19} + 6q^{22} - 2q^{23} + 2q^{24} - 2q^{27} + 4q^{31} - 2q^{32} + 6q^{33} - 8q^{34} + 2q^{36} - 18q^{37} + 2q^{38} - 4q^{41} + 14q^{43} - 6q^{44} + 2q^{46} - 8q^{47} - 2q^{48} + 26q^{49} - 8q^{51} - 6q^{53} + 2q^{54} + 2q^{57} + 6q^{61} - 4q^{62} + 2q^{64} - 6q^{66} - 6q^{67} + 8q^{68} + 2q^{69} - 2q^{72} + 18q^{74} - 2q^{76} - 20q^{77} + 2q^{81} + 4q^{82} - 22q^{83} - 14q^{86} + 6q^{88} - 12q^{89} - 40q^{91} - 2q^{92} - 4q^{93} + 8q^{94} + 2q^{96} + 8q^{97} - 26q^{98} - 6q^{99} + O(q^{100}) \)

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
\(7\) 4.47214 1.69031 0.845154 0.534522i \(-0.179509\pi\)
0.845154 + 0.534522i \(0.179509\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.23607 −1.57873 −0.789367 0.613922i \(-0.789591\pi\)
−0.789367 + 0.613922i \(0.789591\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) −4.47214 −1.19523
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.70820 1.30955 0.654776 0.755823i \(-0.272763\pi\)
0.654776 + 0.755823i \(0.272763\pi\)
\(20\) 0 0
\(21\) −4.47214 −0.975900
\(22\) 5.23607 1.11633
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 4.47214 0.877058
\(27\) −1.00000 −0.192450
\(28\) 4.47214 0.845154
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.23607 0.911482
\(34\) −4.00000 −0.685994
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −11.2361 −1.84720 −0.923599 0.383360i \(-0.874767\pi\)
−0.923599 + 0.383360i \(0.874767\pi\)
\(38\) −5.70820 −0.925993
\(39\) 4.47214 0.716115
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 4.47214 0.690066
\(43\) 4.76393 0.726493 0.363246 0.931693i \(-0.381668\pi\)
0.363246 + 0.931693i \(0.381668\pi\)
\(44\) −5.23607 −0.789367
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) −1.00000 −0.144338
\(49\) 13.0000 1.85714
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) −4.47214 −0.620174
\(53\) −5.23607 −0.719229 −0.359615 0.933101i \(-0.617092\pi\)
−0.359615 + 0.933101i \(0.617092\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −4.47214 −0.597614
\(57\) −5.70820 −0.756070
\(58\) 4.47214 0.587220
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) 0.763932 0.0978115 0.0489057 0.998803i \(-0.484427\pi\)
0.0489057 + 0.998803i \(0.484427\pi\)
\(62\) 2.47214 0.313962
\(63\) 4.47214 0.563436
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.23607 −0.644515
\(67\) −9.70820 −1.18605 −0.593023 0.805186i \(-0.702066\pi\)
−0.593023 + 0.805186i \(0.702066\pi\)
\(68\) 4.00000 0.485071
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.47214 0.523424 0.261712 0.965146i \(-0.415713\pi\)
0.261712 + 0.965146i \(0.415713\pi\)
\(74\) 11.2361 1.30617
\(75\) 0 0
\(76\) 5.70820 0.654776
\(77\) −23.4164 −2.66855
\(78\) −4.47214 −0.506370
\(79\) 4.47214 0.503155 0.251577 0.967837i \(-0.419051\pi\)
0.251577 + 0.967837i \(0.419051\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) −13.2361 −1.45285 −0.726424 0.687247i \(-0.758819\pi\)
−0.726424 + 0.687247i \(0.758819\pi\)
\(84\) −4.47214 −0.487950
\(85\) 0 0
\(86\) −4.76393 −0.513708
\(87\) 4.47214 0.479463
\(88\) 5.23607 0.558167
\(89\) −10.4721 −1.11004 −0.555022 0.831836i \(-0.687290\pi\)
−0.555022 + 0.831836i \(0.687290\pi\)
\(90\) 0 0
\(91\) −20.0000 −2.09657
\(92\) −1.00000 −0.104257
\(93\) 2.47214 0.256349
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −0.472136 −0.0479381 −0.0239691 0.999713i \(-0.507630\pi\)
−0.0239691 + 0.999713i \(0.507630\pi\)
\(98\) −13.0000 −1.31320
\(99\) −5.23607 −0.526245
\(100\) 0 0
\(101\) 4.47214 0.444994 0.222497 0.974933i \(-0.428579\pi\)
0.222497 + 0.974933i \(0.428579\pi\)
\(102\) 4.00000 0.396059
\(103\) 6.00000 0.591198 0.295599 0.955312i \(-0.404481\pi\)
0.295599 + 0.955312i \(0.404481\pi\)
\(104\) 4.47214 0.438529
\(105\) 0 0
\(106\) 5.23607 0.508572
\(107\) 12.6525 1.22316 0.611581 0.791182i \(-0.290534\pi\)
0.611581 + 0.791182i \(0.290534\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.76393 0.456302 0.228151 0.973626i \(-0.426732\pi\)
0.228151 + 0.973626i \(0.426732\pi\)
\(110\) 0 0
\(111\) 11.2361 1.06648
\(112\) 4.47214 0.422577
\(113\) 5.52786 0.520018 0.260009 0.965606i \(-0.416275\pi\)
0.260009 + 0.965606i \(0.416275\pi\)
\(114\) 5.70820 0.534622
\(115\) 0 0
\(116\) −4.47214 −0.415227
\(117\) −4.47214 −0.413449
\(118\) 8.94427 0.823387
\(119\) 17.8885 1.63984
\(120\) 0 0
\(121\) 16.4164 1.49240
\(122\) −0.763932 −0.0691632
\(123\) 2.00000 0.180334
\(124\) −2.47214 −0.222004
\(125\) 0 0
\(126\) −4.47214 −0.398410
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −4.76393 −0.419441
\(130\) 0 0
\(131\) −9.52786 −0.832453 −0.416227 0.909261i \(-0.636648\pi\)
−0.416227 + 0.909261i \(0.636648\pi\)
\(132\) 5.23607 0.455741
\(133\) 25.5279 2.21355
\(134\) 9.70820 0.838661
\(135\) 0 0
\(136\) −4.00000 −0.342997
\(137\) −3.05573 −0.261068 −0.130534 0.991444i \(-0.541669\pi\)
−0.130534 + 0.991444i \(0.541669\pi\)
\(138\) −1.00000 −0.0851257
\(139\) −16.9443 −1.43719 −0.718597 0.695427i \(-0.755215\pi\)
−0.718597 + 0.695427i \(0.755215\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) −8.94427 −0.750587
\(143\) 23.4164 1.95818
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.47214 −0.370117
\(147\) −13.0000 −1.07222
\(148\) −11.2361 −0.923599
\(149\) −11.7082 −0.959173 −0.479587 0.877494i \(-0.659213\pi\)
−0.479587 + 0.877494i \(0.659213\pi\)
\(150\) 0 0
\(151\) −14.4721 −1.17773 −0.588863 0.808233i \(-0.700424\pi\)
−0.588863 + 0.808233i \(0.700424\pi\)
\(152\) −5.70820 −0.462996
\(153\) 4.00000 0.323381
\(154\) 23.4164 1.88695
\(155\) 0 0
\(156\) 4.47214 0.358057
\(157\) 6.65248 0.530925 0.265463 0.964121i \(-0.414475\pi\)
0.265463 + 0.964121i \(0.414475\pi\)
\(158\) −4.47214 −0.355784
\(159\) 5.23607 0.415247
\(160\) 0 0
\(161\) −4.47214 −0.352454
\(162\) −1.00000 −0.0785674
\(163\) 2.47214 0.193633 0.0968163 0.995302i \(-0.469134\pi\)
0.0968163 + 0.995302i \(0.469134\pi\)
\(164\) −2.00000 −0.156174
\(165\) 0 0
\(166\) 13.2361 1.02732
\(167\) −16.9443 −1.31119 −0.655594 0.755114i \(-0.727582\pi\)
−0.655594 + 0.755114i \(0.727582\pi\)
\(168\) 4.47214 0.345033
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 5.70820 0.436517
\(172\) 4.76393 0.363246
\(173\) 17.4164 1.32414 0.662072 0.749440i \(-0.269677\pi\)
0.662072 + 0.749440i \(0.269677\pi\)
\(174\) −4.47214 −0.339032
\(175\) 0 0
\(176\) −5.23607 −0.394683
\(177\) 8.94427 0.672293
\(178\) 10.4721 0.784920
\(179\) 19.4164 1.45125 0.725625 0.688090i \(-0.241551\pi\)
0.725625 + 0.688090i \(0.241551\pi\)
\(180\) 0 0
\(181\) −11.2361 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(182\) 20.0000 1.48250
\(183\) −0.763932 −0.0564715
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −2.47214 −0.181266
\(187\) −20.9443 −1.53160
\(188\) −4.00000 −0.291730
\(189\) −4.47214 −0.325300
\(190\) 0 0
\(191\) 6.47214 0.468307 0.234154 0.972200i \(-0.424768\pi\)
0.234154 + 0.972200i \(0.424768\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −23.8885 −1.71954 −0.859768 0.510686i \(-0.829392\pi\)
−0.859768 + 0.510686i \(0.829392\pi\)
\(194\) 0.472136 0.0338974
\(195\) 0 0
\(196\) 13.0000 0.928571
\(197\) −2.94427 −0.209771 −0.104885 0.994484i \(-0.533448\pi\)
−0.104885 + 0.994484i \(0.533448\pi\)
\(198\) 5.23607 0.372111
\(199\) 17.4164 1.23462 0.617308 0.786721i \(-0.288223\pi\)
0.617308 + 0.786721i \(0.288223\pi\)
\(200\) 0 0
\(201\) 9.70820 0.684764
\(202\) −4.47214 −0.314658
\(203\) −20.0000 −1.40372
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) −1.00000 −0.0695048
\(208\) −4.47214 −0.310087
\(209\) −29.8885 −2.06743
\(210\) 0 0
\(211\) −23.4164 −1.61205 −0.806026 0.591880i \(-0.798386\pi\)
−0.806026 + 0.591880i \(0.798386\pi\)
\(212\) −5.23607 −0.359615
\(213\) −8.94427 −0.612851
\(214\) −12.6525 −0.864905
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −11.0557 −0.750512
\(218\) −4.76393 −0.322654
\(219\) −4.47214 −0.302199
\(220\) 0 0
\(221\) −17.8885 −1.20331
\(222\) −11.2361 −0.754116
\(223\) −19.4164 −1.30022 −0.650109 0.759841i \(-0.725277\pi\)
−0.650109 + 0.759841i \(0.725277\pi\)
\(224\) −4.47214 −0.298807
\(225\) 0 0
\(226\) −5.52786 −0.367708
\(227\) 9.23607 0.613019 0.306510 0.951868i \(-0.400839\pi\)
0.306510 + 0.951868i \(0.400839\pi\)
\(228\) −5.70820 −0.378035
\(229\) 17.7082 1.17019 0.585096 0.810964i \(-0.301057\pi\)
0.585096 + 0.810964i \(0.301057\pi\)
\(230\) 0 0
\(231\) 23.4164 1.54069
\(232\) 4.47214 0.293610
\(233\) −19.8885 −1.30294 −0.651471 0.758674i \(-0.725848\pi\)
−0.651471 + 0.758674i \(0.725848\pi\)
\(234\) 4.47214 0.292353
\(235\) 0 0
\(236\) −8.94427 −0.582223
\(237\) −4.47214 −0.290496
\(238\) −17.8885 −1.15954
\(239\) −4.94427 −0.319818 −0.159909 0.987132i \(-0.551120\pi\)
−0.159909 + 0.987132i \(0.551120\pi\)
\(240\) 0 0
\(241\) −12.4721 −0.803401 −0.401700 0.915771i \(-0.631581\pi\)
−0.401700 + 0.915771i \(0.631581\pi\)
\(242\) −16.4164 −1.05529
\(243\) −1.00000 −0.0641500
\(244\) 0.763932 0.0489057
\(245\) 0 0
\(246\) −2.00000 −0.127515
\(247\) −25.5279 −1.62430
\(248\) 2.47214 0.156981
\(249\) 13.2361 0.838802
\(250\) 0 0
\(251\) −19.7082 −1.24397 −0.621985 0.783029i \(-0.713674\pi\)
−0.621985 + 0.783029i \(0.713674\pi\)
\(252\) 4.47214 0.281718
\(253\) 5.23607 0.329189
\(254\) −4.00000 −0.250982
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 11.8885 0.741587 0.370793 0.928715i \(-0.379086\pi\)
0.370793 + 0.928715i \(0.379086\pi\)
\(258\) 4.76393 0.296589
\(259\) −50.2492 −3.12233
\(260\) 0 0
\(261\) −4.47214 −0.276818
\(262\) 9.52786 0.588633
\(263\) −24.9443 −1.53813 −0.769065 0.639171i \(-0.779278\pi\)
−0.769065 + 0.639171i \(0.779278\pi\)
\(264\) −5.23607 −0.322258
\(265\) 0 0
\(266\) −25.5279 −1.56521
\(267\) 10.4721 0.640884
\(268\) −9.70820 −0.593023
\(269\) −13.0557 −0.796022 −0.398011 0.917381i \(-0.630299\pi\)
−0.398011 + 0.917381i \(0.630299\pi\)
\(270\) 0 0
\(271\) −16.9443 −1.02929 −0.514646 0.857403i \(-0.672076\pi\)
−0.514646 + 0.857403i \(0.672076\pi\)
\(272\) 4.00000 0.242536
\(273\) 20.0000 1.21046
\(274\) 3.05573 0.184603
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 20.4721 1.23005 0.615026 0.788507i \(-0.289146\pi\)
0.615026 + 0.788507i \(0.289146\pi\)
\(278\) 16.9443 1.01625
\(279\) −2.47214 −0.148003
\(280\) 0 0
\(281\) −13.5279 −0.807005 −0.403502 0.914979i \(-0.632207\pi\)
−0.403502 + 0.914979i \(0.632207\pi\)
\(282\) −4.00000 −0.238197
\(283\) 3.81966 0.227055 0.113528 0.993535i \(-0.463785\pi\)
0.113528 + 0.993535i \(0.463785\pi\)
\(284\) 8.94427 0.530745
\(285\) 0 0
\(286\) −23.4164 −1.38464
\(287\) −8.94427 −0.527964
\(288\) −1.00000 −0.0589256
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0.472136 0.0276771
\(292\) 4.47214 0.261712
\(293\) −0.291796 −0.0170469 −0.00852345 0.999964i \(-0.502713\pi\)
−0.00852345 + 0.999964i \(0.502713\pi\)
\(294\) 13.0000 0.758175
\(295\) 0 0
\(296\) 11.2361 0.653083
\(297\) 5.23607 0.303827
\(298\) 11.7082 0.678238
\(299\) 4.47214 0.258630
\(300\) 0 0
\(301\) 21.3050 1.22800
\(302\) 14.4721 0.832778
\(303\) −4.47214 −0.256917
\(304\) 5.70820 0.327388
\(305\) 0 0
\(306\) −4.00000 −0.228665
\(307\) −15.4164 −0.879861 −0.439930 0.898032i \(-0.644997\pi\)
−0.439930 + 0.898032i \(0.644997\pi\)
\(308\) −23.4164 −1.33427
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) −20.9443 −1.18764 −0.593820 0.804598i \(-0.702381\pi\)
−0.593820 + 0.804598i \(0.702381\pi\)
\(312\) −4.47214 −0.253185
\(313\) −15.5279 −0.877687 −0.438843 0.898564i \(-0.644612\pi\)
−0.438843 + 0.898564i \(0.644612\pi\)
\(314\) −6.65248 −0.375421
\(315\) 0 0
\(316\) 4.47214 0.251577
\(317\) −19.5279 −1.09679 −0.548397 0.836218i \(-0.684762\pi\)
−0.548397 + 0.836218i \(0.684762\pi\)
\(318\) −5.23607 −0.293624
\(319\) 23.4164 1.31107
\(320\) 0 0
\(321\) −12.6525 −0.706192
\(322\) 4.47214 0.249222
\(323\) 22.8328 1.27045
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −2.47214 −0.136919
\(327\) −4.76393 −0.263446
\(328\) 2.00000 0.110432
\(329\) −17.8885 −0.986227
\(330\) 0 0
\(331\) −10.4721 −0.575601 −0.287800 0.957690i \(-0.592924\pi\)
−0.287800 + 0.957690i \(0.592924\pi\)
\(332\) −13.2361 −0.726424
\(333\) −11.2361 −0.615733
\(334\) 16.9443 0.927149
\(335\) 0 0
\(336\) −4.47214 −0.243975
\(337\) 19.8885 1.08340 0.541699 0.840573i \(-0.317781\pi\)
0.541699 + 0.840573i \(0.317781\pi\)
\(338\) −7.00000 −0.380750
\(339\) −5.52786 −0.300232
\(340\) 0 0
\(341\) 12.9443 0.700972
\(342\) −5.70820 −0.308664
\(343\) 26.8328 1.44884
\(344\) −4.76393 −0.256854
\(345\) 0 0
\(346\) −17.4164 −0.936312
\(347\) −30.4721 −1.63583 −0.817915 0.575339i \(-0.804870\pi\)
−0.817915 + 0.575339i \(0.804870\pi\)
\(348\) 4.47214 0.239732
\(349\) 3.88854 0.208149 0.104074 0.994570i \(-0.466812\pi\)
0.104074 + 0.994570i \(0.466812\pi\)
\(350\) 0 0
\(351\) 4.47214 0.238705
\(352\) 5.23607 0.279083
\(353\) 3.88854 0.206966 0.103483 0.994631i \(-0.467001\pi\)
0.103483 + 0.994631i \(0.467001\pi\)
\(354\) −8.94427 −0.475383
\(355\) 0 0
\(356\) −10.4721 −0.555022
\(357\) −17.8885 −0.946762
\(358\) −19.4164 −1.02619
\(359\) 29.3050 1.54666 0.773328 0.634006i \(-0.218591\pi\)
0.773328 + 0.634006i \(0.218591\pi\)
\(360\) 0 0
\(361\) 13.5836 0.714926
\(362\) 11.2361 0.590555
\(363\) −16.4164 −0.861638
\(364\) −20.0000 −1.04828
\(365\) 0 0
\(366\) 0.763932 0.0399314
\(367\) −9.41641 −0.491532 −0.245766 0.969329i \(-0.579040\pi\)
−0.245766 + 0.969329i \(0.579040\pi\)
\(368\) −1.00000 −0.0521286
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −23.4164 −1.21572
\(372\) 2.47214 0.128174
\(373\) 35.5967 1.84313 0.921565 0.388224i \(-0.126911\pi\)
0.921565 + 0.388224i \(0.126911\pi\)
\(374\) 20.9443 1.08300
\(375\) 0 0
\(376\) 4.00000 0.206284
\(377\) 20.0000 1.03005
\(378\) 4.47214 0.230022
\(379\) 13.7082 0.704143 0.352072 0.935973i \(-0.385477\pi\)
0.352072 + 0.935973i \(0.385477\pi\)
\(380\) 0 0
\(381\) −4.00000 −0.204926
\(382\) −6.47214 −0.331143
\(383\) −7.05573 −0.360531 −0.180265 0.983618i \(-0.557696\pi\)
−0.180265 + 0.983618i \(0.557696\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 23.8885 1.21589
\(387\) 4.76393 0.242164
\(388\) −0.472136 −0.0239691
\(389\) 23.7082 1.20205 0.601027 0.799229i \(-0.294758\pi\)
0.601027 + 0.799229i \(0.294758\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) −13.0000 −0.656599
\(393\) 9.52786 0.480617
\(394\) 2.94427 0.148330
\(395\) 0 0
\(396\) −5.23607 −0.263122
\(397\) −26.9443 −1.35229 −0.676147 0.736767i \(-0.736352\pi\)
−0.676147 + 0.736767i \(0.736352\pi\)
\(398\) −17.4164 −0.873006
\(399\) −25.5279 −1.27799
\(400\) 0 0
\(401\) 8.94427 0.446656 0.223328 0.974743i \(-0.428308\pi\)
0.223328 + 0.974743i \(0.428308\pi\)
\(402\) −9.70820 −0.484201
\(403\) 11.0557 0.550725
\(404\) 4.47214 0.222497
\(405\) 0 0
\(406\) 20.0000 0.992583
\(407\) 58.8328 2.91623
\(408\) 4.00000 0.198030
\(409\) −35.8885 −1.77457 −0.887287 0.461217i \(-0.847413\pi\)
−0.887287 + 0.461217i \(0.847413\pi\)
\(410\) 0 0
\(411\) 3.05573 0.150728
\(412\) 6.00000 0.295599
\(413\) −40.0000 −1.96827
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) 4.47214 0.219265
\(417\) 16.9443 0.829765
\(418\) 29.8885 1.46190
\(419\) 28.0689 1.37125 0.685627 0.727953i \(-0.259528\pi\)
0.685627 + 0.727953i \(0.259528\pi\)
\(420\) 0 0
\(421\) −0.763932 −0.0372318 −0.0186159 0.999827i \(-0.505926\pi\)
−0.0186159 + 0.999827i \(0.505926\pi\)
\(422\) 23.4164 1.13989
\(423\) −4.00000 −0.194487
\(424\) 5.23607 0.254286
\(425\) 0 0
\(426\) 8.94427 0.433351
\(427\) 3.41641 0.165332
\(428\) 12.6525 0.611581
\(429\) −23.4164 −1.13055
\(430\) 0 0
\(431\) −23.4164 −1.12793 −0.563964 0.825799i \(-0.690724\pi\)
−0.563964 + 0.825799i \(0.690724\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 21.4164 1.02921 0.514603 0.857428i \(-0.327939\pi\)
0.514603 + 0.857428i \(0.327939\pi\)
\(434\) 11.0557 0.530692
\(435\) 0 0
\(436\) 4.76393 0.228151
\(437\) −5.70820 −0.273060
\(438\) 4.47214 0.213687
\(439\) 19.0557 0.909480 0.454740 0.890624i \(-0.349732\pi\)
0.454740 + 0.890624i \(0.349732\pi\)
\(440\) 0 0
\(441\) 13.0000 0.619048
\(442\) 17.8885 0.850871
\(443\) 15.0557 0.715319 0.357660 0.933852i \(-0.383575\pi\)
0.357660 + 0.933852i \(0.383575\pi\)
\(444\) 11.2361 0.533240
\(445\) 0 0
\(446\) 19.4164 0.919394
\(447\) 11.7082 0.553779
\(448\) 4.47214 0.211289
\(449\) 15.8885 0.749827 0.374913 0.927060i \(-0.377672\pi\)
0.374913 + 0.927060i \(0.377672\pi\)
\(450\) 0 0
\(451\) 10.4721 0.493114
\(452\) 5.52786 0.260009
\(453\) 14.4721 0.679960
\(454\) −9.23607 −0.433470
\(455\) 0 0
\(456\) 5.70820 0.267311
\(457\) −27.5279 −1.28770 −0.643850 0.765152i \(-0.722664\pi\)
−0.643850 + 0.765152i \(0.722664\pi\)
\(458\) −17.7082 −0.827450
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 22.0000 1.02464 0.512321 0.858794i \(-0.328786\pi\)
0.512321 + 0.858794i \(0.328786\pi\)
\(462\) −23.4164 −1.08943
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −4.47214 −0.207614
\(465\) 0 0
\(466\) 19.8885 0.921319
\(467\) 17.8197 0.824596 0.412298 0.911049i \(-0.364726\pi\)
0.412298 + 0.911049i \(0.364726\pi\)
\(468\) −4.47214 −0.206725
\(469\) −43.4164 −2.00478
\(470\) 0 0
\(471\) −6.65248 −0.306530
\(472\) 8.94427 0.411693
\(473\) −24.9443 −1.14694
\(474\) 4.47214 0.205412
\(475\) 0 0
\(476\) 17.8885 0.819920
\(477\) −5.23607 −0.239743
\(478\) 4.94427 0.226146
\(479\) −28.9443 −1.32250 −0.661249 0.750167i \(-0.729973\pi\)
−0.661249 + 0.750167i \(0.729973\pi\)
\(480\) 0 0
\(481\) 50.2492 2.29117
\(482\) 12.4721 0.568090
\(483\) 4.47214 0.203489
\(484\) 16.4164 0.746200
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) −9.88854 −0.448093 −0.224046 0.974578i \(-0.571927\pi\)
−0.224046 + 0.974578i \(0.571927\pi\)
\(488\) −0.763932 −0.0345816
\(489\) −2.47214 −0.111794
\(490\) 0 0
\(491\) 29.3050 1.32251 0.661257 0.750159i \(-0.270023\pi\)
0.661257 + 0.750159i \(0.270023\pi\)
\(492\) 2.00000 0.0901670
\(493\) −17.8885 −0.805659
\(494\) 25.5279 1.14855
\(495\) 0 0
\(496\) −2.47214 −0.111002
\(497\) 40.0000 1.79425
\(498\) −13.2361 −0.593122
\(499\) 0.583592 0.0261252 0.0130626 0.999915i \(-0.495842\pi\)
0.0130626 + 0.999915i \(0.495842\pi\)
\(500\) 0 0
\(501\) 16.9443 0.757014
\(502\) 19.7082 0.879620
\(503\) −10.4721 −0.466929 −0.233465 0.972365i \(-0.575006\pi\)
−0.233465 + 0.972365i \(0.575006\pi\)
\(504\) −4.47214 −0.199205
\(505\) 0 0
\(506\) −5.23607 −0.232772
\(507\) −7.00000 −0.310881
\(508\) 4.00000 0.177471
\(509\) −33.4164 −1.48116 −0.740578 0.671970i \(-0.765448\pi\)
−0.740578 + 0.671970i \(0.765448\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) −1.00000 −0.0441942
\(513\) −5.70820 −0.252023
\(514\) −11.8885 −0.524381
\(515\) 0 0
\(516\) −4.76393 −0.209720
\(517\) 20.9443 0.921128
\(518\) 50.2492 2.20782
\(519\) −17.4164 −0.764495
\(520\) 0 0
\(521\) 16.0000 0.700973 0.350486 0.936568i \(-0.386016\pi\)
0.350486 + 0.936568i \(0.386016\pi\)
\(522\) 4.47214 0.195740
\(523\) −25.7082 −1.12414 −0.562071 0.827089i \(-0.689995\pi\)
−0.562071 + 0.827089i \(0.689995\pi\)
\(524\) −9.52786 −0.416227
\(525\) 0 0
\(526\) 24.9443 1.08762
\(527\) −9.88854 −0.430752
\(528\) 5.23607 0.227871
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.94427 −0.388148
\(532\) 25.5279 1.10677
\(533\) 8.94427 0.387419
\(534\) −10.4721 −0.453174
\(535\) 0 0
\(536\) 9.70820 0.419331
\(537\) −19.4164 −0.837880
\(538\) 13.0557 0.562872
\(539\) −68.0689 −2.93193
\(540\) 0 0
\(541\) 8.11146 0.348739 0.174369 0.984680i \(-0.444211\pi\)
0.174369 + 0.984680i \(0.444211\pi\)
\(542\) 16.9443 0.727819
\(543\) 11.2361 0.482186
\(544\) −4.00000 −0.171499
\(545\) 0 0
\(546\) −20.0000 −0.855921
\(547\) −41.3050 −1.76607 −0.883036 0.469305i \(-0.844504\pi\)
−0.883036 + 0.469305i \(0.844504\pi\)
\(548\) −3.05573 −0.130534
\(549\) 0.763932 0.0326038
\(550\) 0 0
\(551\) −25.5279 −1.08752
\(552\) −1.00000 −0.0425628
\(553\) 20.0000 0.850487
\(554\) −20.4721 −0.869778
\(555\) 0 0
\(556\) −16.9443 −0.718597
\(557\) 15.1246 0.640850 0.320425 0.947274i \(-0.396174\pi\)
0.320425 + 0.947274i \(0.396174\pi\)
\(558\) 2.47214 0.104654
\(559\) −21.3050 −0.901103
\(560\) 0 0
\(561\) 20.9443 0.884268
\(562\) 13.5279 0.570639
\(563\) −24.6525 −1.03898 −0.519489 0.854477i \(-0.673878\pi\)
−0.519489 + 0.854477i \(0.673878\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −3.81966 −0.160552
\(567\) 4.47214 0.187812
\(568\) −8.94427 −0.375293
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) −14.2918 −0.598093 −0.299047 0.954239i \(-0.596669\pi\)
−0.299047 + 0.954239i \(0.596669\pi\)
\(572\) 23.4164 0.979089
\(573\) −6.47214 −0.270377
\(574\) 8.94427 0.373327
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 34.3607 1.43045 0.715227 0.698892i \(-0.246323\pi\)
0.715227 + 0.698892i \(0.246323\pi\)
\(578\) 1.00000 0.0415945
\(579\) 23.8885 0.992774
\(580\) 0 0
\(581\) −59.1935 −2.45576
\(582\) −0.472136 −0.0195707
\(583\) 27.4164 1.13547
\(584\) −4.47214 −0.185058
\(585\) 0 0
\(586\) 0.291796 0.0120540
\(587\) −6.47214 −0.267134 −0.133567 0.991040i \(-0.542643\pi\)
−0.133567 + 0.991040i \(0.542643\pi\)
\(588\) −13.0000 −0.536111
\(589\) −14.1115 −0.581452
\(590\) 0 0
\(591\) 2.94427 0.121111
\(592\) −11.2361 −0.461800
\(593\) −33.7771 −1.38706 −0.693529 0.720428i \(-0.743945\pi\)
−0.693529 + 0.720428i \(0.743945\pi\)
\(594\) −5.23607 −0.214838
\(595\) 0 0
\(596\) −11.7082 −0.479587
\(597\) −17.4164 −0.712806
\(598\) −4.47214 −0.182879
\(599\) 33.8885 1.38465 0.692324 0.721587i \(-0.256587\pi\)
0.692324 + 0.721587i \(0.256587\pi\)
\(600\) 0 0
\(601\) −10.3607 −0.422621 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(602\) −21.3050 −0.868325
\(603\) −9.70820 −0.395349
\(604\) −14.4721 −0.588863
\(605\) 0 0
\(606\) 4.47214 0.181668
\(607\) 17.5279 0.711434 0.355717 0.934594i \(-0.384237\pi\)
0.355717 + 0.934594i \(0.384237\pi\)
\(608\) −5.70820 −0.231498
\(609\) 20.0000 0.810441
\(610\) 0 0
\(611\) 17.8885 0.723693
\(612\) 4.00000 0.161690
\(613\) 25.1246 1.01477 0.507387 0.861718i \(-0.330612\pi\)
0.507387 + 0.861718i \(0.330612\pi\)
\(614\) 15.4164 0.622156
\(615\) 0 0
\(616\) 23.4164 0.943474
\(617\) −20.3607 −0.819690 −0.409845 0.912155i \(-0.634417\pi\)
−0.409845 + 0.912155i \(0.634417\pi\)
\(618\) 6.00000 0.241355
\(619\) 18.2918 0.735209 0.367605 0.929982i \(-0.380178\pi\)
0.367605 + 0.929982i \(0.380178\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 20.9443 0.839789
\(623\) −46.8328 −1.87632
\(624\) 4.47214 0.179029
\(625\) 0 0
\(626\) 15.5279 0.620618
\(627\) 29.8885 1.19363
\(628\) 6.65248 0.265463
\(629\) −44.9443 −1.79205
\(630\) 0 0
\(631\) −6.00000 −0.238856 −0.119428 0.992843i \(-0.538106\pi\)
−0.119428 + 0.992843i \(0.538106\pi\)
\(632\) −4.47214 −0.177892
\(633\) 23.4164 0.930719
\(634\) 19.5279 0.775551
\(635\) 0 0
\(636\) 5.23607 0.207624
\(637\) −58.1378 −2.30350
\(638\) −23.4164 −0.927064
\(639\) 8.94427 0.353830
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 12.6525 0.499353
\(643\) 32.5410 1.28329 0.641646 0.767001i \(-0.278252\pi\)
0.641646 + 0.767001i \(0.278252\pi\)
\(644\) −4.47214 −0.176227
\(645\) 0 0
\(646\) −22.8328 −0.898345
\(647\) −12.9443 −0.508892 −0.254446 0.967087i \(-0.581893\pi\)
−0.254446 + 0.967087i \(0.581893\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 46.8328 1.83835
\(650\) 0 0
\(651\) 11.0557 0.433308
\(652\) 2.47214 0.0968163
\(653\) −9.41641 −0.368493 −0.184246 0.982880i \(-0.558984\pi\)
−0.184246 + 0.982880i \(0.558984\pi\)
\(654\) 4.76393 0.186284
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 4.47214 0.174475
\(658\) 17.8885 0.697368
\(659\) −32.6525 −1.27196 −0.635980 0.771706i \(-0.719404\pi\)
−0.635980 + 0.771706i \(0.719404\pi\)
\(660\) 0 0
\(661\) 3.81966 0.148568 0.0742838 0.997237i \(-0.476333\pi\)
0.0742838 + 0.997237i \(0.476333\pi\)
\(662\) 10.4721 0.407011
\(663\) 17.8885 0.694733
\(664\) 13.2361 0.513659
\(665\) 0 0
\(666\) 11.2361 0.435389
\(667\) 4.47214 0.173162
\(668\) −16.9443 −0.655594
\(669\) 19.4164 0.750682
\(670\) 0 0
\(671\) −4.00000 −0.154418
\(672\) 4.47214 0.172516
\(673\) −4.47214 −0.172388 −0.0861941 0.996278i \(-0.527471\pi\)
−0.0861941 + 0.996278i \(0.527471\pi\)
\(674\) −19.8885 −0.766078
\(675\) 0 0
\(676\) 7.00000 0.269231
\(677\) −19.1246 −0.735019 −0.367509 0.930020i \(-0.619789\pi\)
−0.367509 + 0.930020i \(0.619789\pi\)
\(678\) 5.52786 0.212296
\(679\) −2.11146 −0.0810303
\(680\) 0 0
\(681\) −9.23607 −0.353927
\(682\) −12.9443 −0.495662
\(683\) −14.4721 −0.553761 −0.276880 0.960904i \(-0.589301\pi\)
−0.276880 + 0.960904i \(0.589301\pi\)
\(684\) 5.70820 0.218259
\(685\) 0 0
\(686\) −26.8328 −1.02448
\(687\) −17.7082 −0.675610
\(688\) 4.76393 0.181623
\(689\) 23.4164 0.892094
\(690\) 0 0
\(691\) 16.5836 0.630870 0.315435 0.948947i \(-0.397850\pi\)
0.315435 + 0.948947i \(0.397850\pi\)
\(692\) 17.4164 0.662072
\(693\) −23.4164 −0.889516
\(694\) 30.4721 1.15671
\(695\) 0 0
\(696\) −4.47214 −0.169516
\(697\) −8.00000 −0.303022
\(698\) −3.88854 −0.147184
\(699\) 19.8885 0.752254
\(700\) 0 0
\(701\) −3.12461 −0.118015 −0.0590075 0.998258i \(-0.518794\pi\)
−0.0590075 + 0.998258i \(0.518794\pi\)
\(702\) −4.47214 −0.168790
\(703\) −64.1378 −2.41900
\(704\) −5.23607 −0.197342
\(705\) 0 0
\(706\) −3.88854 −0.146347
\(707\) 20.0000 0.752177
\(708\) 8.94427 0.336146
\(709\) 35.0132 1.31495 0.657473 0.753478i \(-0.271625\pi\)
0.657473 + 0.753478i \(0.271625\pi\)
\(710\) 0 0
\(711\) 4.47214 0.167718
\(712\) 10.4721 0.392460
\(713\) 2.47214 0.0925822
\(714\) 17.8885 0.669462
\(715\) 0 0
\(716\) 19.4164 0.725625
\(717\) 4.94427 0.184647
\(718\) −29.3050 −1.09365
\(719\) −3.05573 −0.113959 −0.0569797 0.998375i \(-0.518147\pi\)
−0.0569797 + 0.998375i \(0.518147\pi\)
\(720\) 0 0
\(721\) 26.8328 0.999306
\(722\) −13.5836 −0.505529
\(723\) 12.4721 0.463844
\(724\) −11.2361 −0.417585
\(725\) 0 0
\(726\) 16.4164 0.609270
\(727\) 19.3050 0.715981 0.357991 0.933725i \(-0.383462\pi\)
0.357991 + 0.933725i \(0.383462\pi\)
\(728\) 20.0000 0.741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 19.0557 0.704802
\(732\) −0.763932 −0.0282357
\(733\) 21.7082 0.801811 0.400905 0.916119i \(-0.368696\pi\)
0.400905 + 0.916119i \(0.368696\pi\)
\(734\) 9.41641 0.347566
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 50.8328 1.87245
\(738\) 2.00000 0.0736210
\(739\) −32.9443 −1.21187 −0.605937 0.795512i \(-0.707202\pi\)
−0.605937 + 0.795512i \(0.707202\pi\)
\(740\) 0 0
\(741\) 25.5279 0.937790
\(742\) 23.4164 0.859643
\(743\) 4.00000 0.146746 0.0733729 0.997305i \(-0.476624\pi\)
0.0733729 + 0.997305i \(0.476624\pi\)
\(744\) −2.47214 −0.0906329
\(745\) 0 0
\(746\) −35.5967 −1.30329
\(747\) −13.2361 −0.484282
\(748\) −20.9443 −0.765798
\(749\) 56.5836 2.06752
\(750\) 0 0
\(751\) −18.0000 −0.656829 −0.328415 0.944534i \(-0.606514\pi\)
−0.328415 + 0.944534i \(0.606514\pi\)
\(752\) −4.00000 −0.145865
\(753\) 19.7082 0.718207
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) −4.47214 −0.162650
\(757\) 1.34752 0.0489766 0.0244883 0.999700i \(-0.492204\pi\)
0.0244883 + 0.999700i \(0.492204\pi\)
\(758\) −13.7082 −0.497904
\(759\) −5.23607 −0.190057
\(760\) 0 0
\(761\) −5.05573 −0.183270 −0.0916350 0.995793i \(-0.529209\pi\)
−0.0916350 + 0.995793i \(0.529209\pi\)
\(762\) 4.00000 0.144905
\(763\) 21.3050 0.771291
\(764\) 6.47214 0.234154
\(765\) 0 0
\(766\) 7.05573 0.254934
\(767\) 40.0000 1.44432
\(768\) −1.00000 −0.0360844
\(769\) 12.4721 0.449757 0.224878 0.974387i \(-0.427802\pi\)
0.224878 + 0.974387i \(0.427802\pi\)
\(770\) 0 0
\(771\) −11.8885 −0.428155
\(772\) −23.8885 −0.859768
\(773\) 38.1803 1.37325 0.686626 0.727011i \(-0.259091\pi\)
0.686626 + 0.727011i \(0.259091\pi\)
\(774\) −4.76393 −0.171236
\(775\) 0 0
\(776\) 0.472136 0.0169487
\(777\) 50.2492 1.80268
\(778\) −23.7082 −0.849980
\(779\) −11.4164 −0.409035
\(780\) 0 0
\(781\) −46.8328 −1.67581
\(782\) 4.00000 0.143040
\(783\) 4.47214 0.159821
\(784\) 13.0000 0.464286
\(785\) 0 0
\(786\) −9.52786 −0.339848
\(787\) −35.2361 −1.25603 −0.628015 0.778201i \(-0.716132\pi\)
−0.628015 + 0.778201i \(0.716132\pi\)
\(788\) −2.94427 −0.104885
\(789\) 24.9443 0.888040
\(790\) 0 0
\(791\) 24.7214 0.878990
\(792\) 5.23607 0.186056
\(793\) −3.41641 −0.121320
\(794\) 26.9443 0.956216
\(795\) 0 0
\(796\) 17.4164 0.617308
\(797\) −41.5967 −1.47343 −0.736716 0.676202i \(-0.763625\pi\)
−0.736716 + 0.676202i \(0.763625\pi\)
\(798\) 25.5279 0.903677
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −10.4721 −0.370015
\(802\) −8.94427 −0.315833
\(803\) −23.4164 −0.826347
\(804\) 9.70820 0.342382
\(805\) 0 0
\(806\) −11.0557 −0.389421
\(807\) 13.0557 0.459583
\(808\) −4.47214 −0.157329
\(809\) 42.9443 1.50984 0.754920 0.655817i \(-0.227676\pi\)
0.754920 + 0.655817i \(0.227676\pi\)
\(810\) 0 0
\(811\) 41.3050 1.45041 0.725207 0.688531i \(-0.241744\pi\)
0.725207 + 0.688531i \(0.241744\pi\)
\(812\) −20.0000 −0.701862
\(813\) 16.9443 0.594262
\(814\) −58.8328 −2.06209
\(815\) 0 0
\(816\) −4.00000 −0.140028
\(817\) 27.1935 0.951380
\(818\) 35.8885 1.25481
\(819\) −20.0000 −0.698857
\(820\) 0 0
\(821\) 39.5279 1.37953 0.689766 0.724032i \(-0.257713\pi\)
0.689766 + 0.724032i \(0.257713\pi\)
\(822\) −3.05573 −0.106581
\(823\) 52.3607 1.82518 0.912589 0.408877i \(-0.134080\pi\)
0.912589 + 0.408877i \(0.134080\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) 40.0000 1.39178
\(827\) 21.5967 0.750993 0.375496 0.926824i \(-0.377472\pi\)
0.375496 + 0.926824i \(0.377472\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −6.00000 −0.208389 −0.104194 0.994557i \(-0.533226\pi\)
−0.104194 + 0.994557i \(0.533226\pi\)
\(830\) 0 0
\(831\) −20.4721 −0.710171
\(832\) −4.47214 −0.155043
\(833\) 52.0000 1.80169
\(834\) −16.9443 −0.586732
\(835\) 0 0
\(836\) −29.8885 −1.03372
\(837\) 2.47214 0.0854495
\(838\) −28.0689 −0.969623
\(839\) −45.8885 −1.58425 −0.792124 0.610360i \(-0.791025\pi\)
−0.792124 + 0.610360i \(0.791025\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0.763932 0.0263268
\(843\) 13.5279 0.465924
\(844\) −23.4164 −0.806026
\(845\) 0 0
\(846\) 4.00000 0.137523
\(847\) 73.4164 2.52262
\(848\) −5.23607 −0.179807
\(849\) −3.81966 −0.131090
\(850\) 0 0
\(851\) 11.2361 0.385167
\(852\) −8.94427 −0.306426
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) −3.41641 −0.116907
\(855\) 0 0
\(856\) −12.6525 −0.432453
\(857\) 22.0000 0.751506 0.375753 0.926720i \(-0.377384\pi\)
0.375753 + 0.926720i \(0.377384\pi\)
\(858\) 23.4164 0.799423
\(859\) −20.0000 −0.682391 −0.341196 0.939992i \(-0.610832\pi\)
−0.341196 + 0.939992i \(0.610832\pi\)
\(860\) 0 0
\(861\) 8.94427 0.304820
\(862\) 23.4164 0.797566
\(863\) −14.8328 −0.504915 −0.252457 0.967608i \(-0.581239\pi\)
−0.252457 + 0.967608i \(0.581239\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −21.4164 −0.727759
\(867\) 1.00000 0.0339618
\(868\) −11.0557 −0.375256
\(869\) −23.4164 −0.794347
\(870\) 0 0
\(871\) 43.4164 1.47111
\(872\) −4.76393 −0.161327
\(873\) −0.472136 −0.0159794
\(874\) 5.70820 0.193083
\(875\) 0 0
\(876\) −4.47214 −0.151099
\(877\) 48.2492 1.62926 0.814630 0.579981i \(-0.196940\pi\)
0.814630 + 0.579981i \(0.196940\pi\)
\(878\) −19.0557 −0.643100
\(879\) 0.291796 0.00984204
\(880\) 0 0
\(881\) 39.4164 1.32797 0.663986 0.747745i \(-0.268863\pi\)
0.663986 + 0.747745i \(0.268863\pi\)
\(882\) −13.0000 −0.437733
\(883\) −13.5279 −0.455249 −0.227624 0.973749i \(-0.573096\pi\)
−0.227624 + 0.973749i \(0.573096\pi\)
\(884\) −17.8885 −0.601657
\(885\) 0 0
\(886\) −15.0557 −0.505807
\(887\) 16.9443 0.568933 0.284466 0.958686i \(-0.408184\pi\)
0.284466 + 0.958686i \(0.408184\pi\)
\(888\) −11.2361 −0.377058
\(889\) 17.8885 0.599963
\(890\) 0 0
\(891\) −5.23607 −0.175415
\(892\) −19.4164 −0.650109
\(893\) −22.8328 −0.764071
\(894\) −11.7082 −0.391581
\(895\) 0 0
\(896\) −4.47214 −0.149404
\(897\) −4.47214 −0.149320
\(898\) −15.8885 −0.530208
\(899\) 11.0557 0.368729
\(900\) 0 0
\(901\) −20.9443 −0.697755
\(902\) −10.4721 −0.348684
\(903\) −21.3050 −0.708984
\(904\) −5.52786 −0.183854
\(905\) 0 0
\(906\) −14.4721 −0.480805
\(907\) −4.18034 −0.138806 −0.0694030 0.997589i \(-0.522109\pi\)
−0.0694030 + 0.997589i \(0.522109\pi\)
\(908\) 9.23607 0.306510
\(909\) 4.47214 0.148331
\(910\) 0 0
\(911\) 16.5836 0.549439 0.274719 0.961524i \(-0.411415\pi\)
0.274719 + 0.961524i \(0.411415\pi\)
\(912\) −5.70820 −0.189018
\(913\) 69.3050 2.29366
\(914\) 27.5279 0.910541
\(915\) 0 0
\(916\) 17.7082 0.585096
\(917\) −42.6099 −1.40710
\(918\) 4.00000 0.132020
\(919\) −16.4721 −0.543366 −0.271683 0.962387i \(-0.587580\pi\)
−0.271683 + 0.962387i \(0.587580\pi\)
\(920\) 0 0
\(921\) 15.4164 0.507988
\(922\) −22.0000 −0.724531
\(923\) −40.0000 −1.31662
\(924\) 23.4164 0.770343
\(925\) 0 0
\(926\) 24.0000 0.788689
\(927\) 6.00000 0.197066
\(928\) 4.47214 0.146805
\(929\) −24.8328 −0.814738 −0.407369 0.913264i \(-0.633554\pi\)
−0.407369 + 0.913264i \(0.633554\pi\)
\(930\) 0 0
\(931\) 74.2067 2.43202
\(932\) −19.8885 −0.651471
\(933\) 20.9443 0.685685
\(934\) −17.8197 −0.583077
\(935\) 0 0
\(936\) 4.47214 0.146176
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 43.4164 1.41760
\(939\) 15.5279 0.506733
\(940\) 0 0
\(941\) 38.1803 1.24464 0.622322 0.782762i \(-0.286190\pi\)
0.622322 + 0.782762i \(0.286190\pi\)
\(942\) 6.65248 0.216749
\(943\) 2.00000 0.0651290
\(944\) −8.94427 −0.291111
\(945\) 0 0
\(946\) 24.9443 0.811008
\(947\) 47.1935 1.53358 0.766791 0.641897i \(-0.221852\pi\)
0.766791 + 0.641897i \(0.221852\pi\)
\(948\) −4.47214 −0.145248
\(949\) −20.0000 −0.649227
\(950\) 0 0
\(951\) 19.5279 0.633234
\(952\) −17.8885 −0.579771
\(953\) 47.7771 1.54765 0.773826 0.633398i \(-0.218341\pi\)
0.773826 + 0.633398i \(0.218341\pi\)
\(954\) 5.23607 0.169524
\(955\) 0 0
\(956\) −4.94427 −0.159909
\(957\) −23.4164 −0.756945
\(958\) 28.9443 0.935147
\(959\) −13.6656 −0.441286
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) −50.2492 −1.62010
\(963\) 12.6525 0.407720
\(964\) −12.4721 −0.401700
\(965\) 0 0
\(966\) −4.47214 −0.143889
\(967\) −15.4164 −0.495758 −0.247879 0.968791i \(-0.579734\pi\)
−0.247879 + 0.968791i \(0.579734\pi\)
\(968\) −16.4164 −0.527643
\(969\) −22.8328 −0.733496
\(970\) 0 0
\(971\) −19.1246 −0.613738 −0.306869 0.951752i \(-0.599281\pi\)
−0.306869 + 0.951752i \(0.599281\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −75.7771 −2.42930
\(974\) 9.88854 0.316849
\(975\) 0 0
\(976\) 0.763932 0.0244529
\(977\) −1.16718 −0.0373415 −0.0186708 0.999826i \(-0.505943\pi\)
−0.0186708 + 0.999826i \(0.505943\pi\)
\(978\) 2.47214 0.0790502
\(979\) 54.8328 1.75246
\(980\) 0 0
\(981\) 4.76393 0.152101
\(982\) −29.3050 −0.935159
\(983\) −0.583592 −0.0186137 −0.00930685 0.999957i \(-0.502963\pi\)
−0.00930685 + 0.999957i \(0.502963\pi\)
\(984\) −2.00000 −0.0637577
\(985\) 0 0
\(986\) 17.8885 0.569687
\(987\) 17.8885 0.569399
\(988\) −25.5279 −0.812150
\(989\) −4.76393 −0.151484
\(990\) 0 0
\(991\) −10.4721 −0.332658 −0.166329 0.986070i \(-0.553191\pi\)
−0.166329 + 0.986070i \(0.553191\pi\)
\(992\) 2.47214 0.0784904
\(993\) 10.4721 0.332323
\(994\) −40.0000 −1.26872
\(995\) 0 0
\(996\) 13.2361 0.419401
\(997\) −5.05573 −0.160117 −0.0800583 0.996790i \(-0.525511\pi\)
−0.0800583 + 0.996790i \(0.525511\pi\)
\(998\) −0.583592 −0.0184733
\(999\) 11.2361 0.355493
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.be.1.2 2
5.2 odd 4 3450.2.d.x.2899.2 4
5.3 odd 4 3450.2.d.x.2899.3 4
5.4 even 2 138.2.a.d.1.2 2
15.14 odd 2 414.2.a.f.1.1 2
20.19 odd 2 1104.2.a.j.1.2 2
35.34 odd 2 6762.2.a.cb.1.1 2
40.19 odd 2 4416.2.a.bl.1.1 2
40.29 even 2 4416.2.a.bh.1.1 2
60.59 even 2 3312.2.a.bc.1.1 2
115.114 odd 2 3174.2.a.s.1.1 2
345.344 even 2 9522.2.a.q.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.a.d.1.2 2 5.4 even 2
414.2.a.f.1.1 2 15.14 odd 2
1104.2.a.j.1.2 2 20.19 odd 2
3174.2.a.s.1.1 2 115.114 odd 2
3312.2.a.bc.1.1 2 60.59 even 2
3450.2.a.be.1.2 2 1.1 even 1 trivial
3450.2.d.x.2899.2 4 5.2 odd 4
3450.2.d.x.2899.3 4 5.3 odd 4
4416.2.a.bh.1.1 2 40.29 even 2
4416.2.a.bl.1.1 2 40.19 odd 2
6762.2.a.cb.1.1 2 35.34 odd 2
9522.2.a.q.1.2 2 345.344 even 2