Properties

Label 1815.2.a.m.1.2
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} +1.00000 q^{5} +0.193937 q^{6} -3.35026 q^{7} -0.768452 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} +1.00000 q^{5} +0.193937 q^{6} -3.35026 q^{7} -0.768452 q^{8} +1.00000 q^{9} +0.193937 q^{10} -1.96239 q^{12} -2.96239 q^{13} -0.649738 q^{14} +1.00000 q^{15} +3.77575 q^{16} +4.57452 q^{17} +0.193937 q^{18} +4.31265 q^{19} -1.96239 q^{20} -3.35026 q^{21} -6.70052 q^{23} -0.768452 q^{24} +1.00000 q^{25} -0.574515 q^{26} +1.00000 q^{27} +6.57452 q^{28} +3.61213 q^{29} +0.193937 q^{30} +9.92478 q^{31} +2.26916 q^{32} +0.887166 q^{34} -3.35026 q^{35} -1.96239 q^{36} -2.00000 q^{37} +0.836381 q^{38} -2.96239 q^{39} -0.768452 q^{40} +4.38787 q^{41} -0.649738 q^{42} +9.27504 q^{43} +1.00000 q^{45} -1.29948 q^{46} -9.92478 q^{47} +3.77575 q^{48} +4.22425 q^{49} +0.193937 q^{50} +4.57452 q^{51} +5.81336 q^{52} +4.70052 q^{53} +0.193937 q^{54} +2.57452 q^{56} +4.31265 q^{57} +0.700523 q^{58} +10.7005 q^{59} -1.96239 q^{60} +8.70052 q^{61} +1.92478 q^{62} -3.35026 q^{63} -7.11142 q^{64} -2.96239 q^{65} +5.92478 q^{67} -8.97698 q^{68} -6.70052 q^{69} -0.649738 q^{70} +9.92478 q^{71} -0.768452 q^{72} +7.73813 q^{73} -0.387873 q^{74} +1.00000 q^{75} -8.46310 q^{76} -0.574515 q^{78} -11.5369 q^{79} +3.77575 q^{80} +1.00000 q^{81} +0.850969 q^{82} -10.8872 q^{83} +6.57452 q^{84} +4.57452 q^{85} +1.79877 q^{86} +3.61213 q^{87} -2.77575 q^{89} +0.193937 q^{90} +9.92478 q^{91} +13.1490 q^{92} +9.92478 q^{93} -1.92478 q^{94} +4.31265 q^{95} +2.26916 q^{96} +0.0752228 q^{97} +0.819237 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} + q^{6} + 9 q^{8} + 3 q^{9} + O(q^{10}) \) \( 3 q + q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} + q^{6} + 9 q^{8} + 3 q^{9} + q^{10} + 5 q^{12} + 2 q^{13} - 12 q^{14} + 3 q^{15} + 13 q^{16} + 2 q^{17} + q^{18} - 8 q^{19} + 5 q^{20} + 9 q^{24} + 3 q^{25} + 10 q^{26} + 3 q^{27} + 8 q^{28} + 10 q^{29} + q^{30} + 8 q^{31} + 29 q^{32} - 30 q^{34} + 5 q^{36} - 6 q^{37} + 2 q^{39} + 9 q^{40} + 14 q^{41} - 12 q^{42} - 4 q^{43} + 3 q^{45} - 24 q^{46} - 8 q^{47} + 13 q^{48} + 11 q^{49} + q^{50} + 2 q^{51} + 30 q^{52} - 6 q^{53} + q^{54} - 4 q^{56} - 8 q^{57} - 18 q^{58} + 12 q^{59} + 5 q^{60} + 6 q^{61} - 16 q^{62} + 13 q^{64} + 2 q^{65} - 4 q^{67} - 42 q^{68} - 12 q^{70} + 8 q^{71} + 9 q^{72} + 14 q^{73} - 2 q^{74} + 3 q^{75} - 48 q^{76} + 10 q^{78} - 12 q^{79} + 13 q^{80} + 3 q^{81} + 26 q^{82} + 8 q^{84} + 2 q^{85} - 8 q^{86} + 10 q^{87} - 10 q^{89} + q^{90} + 8 q^{91} + 16 q^{92} + 8 q^{93} + 16 q^{94} - 8 q^{95} + 29 q^{96} + 22 q^{97} - 39 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.193937 0.137134 0.0685669 0.997647i \(-0.478157\pi\)
0.0685669 + 0.997647i \(0.478157\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.96239 −0.981194
\(5\) 1.00000 0.447214
\(6\) 0.193937 0.0791743
\(7\) −3.35026 −1.26628 −0.633140 0.774037i \(-0.718234\pi\)
−0.633140 + 0.774037i \(0.718234\pi\)
\(8\) −0.768452 −0.271689
\(9\) 1.00000 0.333333
\(10\) 0.193937 0.0613281
\(11\) 0 0
\(12\) −1.96239 −0.566493
\(13\) −2.96239 −0.821619 −0.410809 0.911721i \(-0.634754\pi\)
−0.410809 + 0.911721i \(0.634754\pi\)
\(14\) −0.649738 −0.173650
\(15\) 1.00000 0.258199
\(16\) 3.77575 0.943937
\(17\) 4.57452 1.10948 0.554741 0.832023i \(-0.312817\pi\)
0.554741 + 0.832023i \(0.312817\pi\)
\(18\) 0.193937 0.0457113
\(19\) 4.31265 0.989390 0.494695 0.869067i \(-0.335280\pi\)
0.494695 + 0.869067i \(0.335280\pi\)
\(20\) −1.96239 −0.438803
\(21\) −3.35026 −0.731087
\(22\) 0 0
\(23\) −6.70052 −1.39716 −0.698578 0.715534i \(-0.746183\pi\)
−0.698578 + 0.715534i \(0.746183\pi\)
\(24\) −0.768452 −0.156860
\(25\) 1.00000 0.200000
\(26\) −0.574515 −0.112672
\(27\) 1.00000 0.192450
\(28\) 6.57452 1.24247
\(29\) 3.61213 0.670755 0.335378 0.942084i \(-0.391136\pi\)
0.335378 + 0.942084i \(0.391136\pi\)
\(30\) 0.193937 0.0354078
\(31\) 9.92478 1.78254 0.891271 0.453470i \(-0.149814\pi\)
0.891271 + 0.453470i \(0.149814\pi\)
\(32\) 2.26916 0.401134
\(33\) 0 0
\(34\) 0.887166 0.152148
\(35\) −3.35026 −0.566298
\(36\) −1.96239 −0.327065
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0.836381 0.135679
\(39\) −2.96239 −0.474362
\(40\) −0.768452 −0.121503
\(41\) 4.38787 0.685271 0.342635 0.939468i \(-0.388680\pi\)
0.342635 + 0.939468i \(0.388680\pi\)
\(42\) −0.649738 −0.100257
\(43\) 9.27504 1.41443 0.707215 0.706998i \(-0.249951\pi\)
0.707215 + 0.706998i \(0.249951\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −1.29948 −0.191597
\(47\) −9.92478 −1.44768 −0.723839 0.689969i \(-0.757624\pi\)
−0.723839 + 0.689969i \(0.757624\pi\)
\(48\) 3.77575 0.544982
\(49\) 4.22425 0.603465
\(50\) 0.193937 0.0274268
\(51\) 4.57452 0.640560
\(52\) 5.81336 0.806168
\(53\) 4.70052 0.645667 0.322833 0.946456i \(-0.395365\pi\)
0.322833 + 0.946456i \(0.395365\pi\)
\(54\) 0.193937 0.0263914
\(55\) 0 0
\(56\) 2.57452 0.344034
\(57\) 4.31265 0.571224
\(58\) 0.700523 0.0919832
\(59\) 10.7005 1.39309 0.696545 0.717513i \(-0.254720\pi\)
0.696545 + 0.717513i \(0.254720\pi\)
\(60\) −1.96239 −0.253343
\(61\) 8.70052 1.11399 0.556994 0.830517i \(-0.311955\pi\)
0.556994 + 0.830517i \(0.311955\pi\)
\(62\) 1.92478 0.244447
\(63\) −3.35026 −0.422093
\(64\) −7.11142 −0.888927
\(65\) −2.96239 −0.367439
\(66\) 0 0
\(67\) 5.92478 0.723827 0.361913 0.932212i \(-0.382124\pi\)
0.361913 + 0.932212i \(0.382124\pi\)
\(68\) −8.97698 −1.08862
\(69\) −6.70052 −0.806648
\(70\) −0.649738 −0.0776586
\(71\) 9.92478 1.17785 0.588927 0.808186i \(-0.299550\pi\)
0.588927 + 0.808186i \(0.299550\pi\)
\(72\) −0.768452 −0.0905629
\(73\) 7.73813 0.905680 0.452840 0.891592i \(-0.350411\pi\)
0.452840 + 0.891592i \(0.350411\pi\)
\(74\) −0.387873 −0.0450893
\(75\) 1.00000 0.115470
\(76\) −8.46310 −0.970784
\(77\) 0 0
\(78\) −0.574515 −0.0650511
\(79\) −11.5369 −1.29800 −0.649002 0.760787i \(-0.724813\pi\)
−0.649002 + 0.760787i \(0.724813\pi\)
\(80\) 3.77575 0.422141
\(81\) 1.00000 0.111111
\(82\) 0.850969 0.0939738
\(83\) −10.8872 −1.19502 −0.597511 0.801861i \(-0.703844\pi\)
−0.597511 + 0.801861i \(0.703844\pi\)
\(84\) 6.57452 0.717338
\(85\) 4.57452 0.496176
\(86\) 1.79877 0.193966
\(87\) 3.61213 0.387261
\(88\) 0 0
\(89\) −2.77575 −0.294229 −0.147114 0.989120i \(-0.546999\pi\)
−0.147114 + 0.989120i \(0.546999\pi\)
\(90\) 0.193937 0.0204427
\(91\) 9.92478 1.04040
\(92\) 13.1490 1.37088
\(93\) 9.92478 1.02915
\(94\) −1.92478 −0.198526
\(95\) 4.31265 0.442469
\(96\) 2.26916 0.231595
\(97\) 0.0752228 0.00763772 0.00381886 0.999993i \(-0.498784\pi\)
0.00381886 + 0.999993i \(0.498784\pi\)
\(98\) 0.819237 0.0827555
\(99\) 0 0
\(100\) −1.96239 −0.196239
\(101\) 15.0884 1.50135 0.750676 0.660671i \(-0.229728\pi\)
0.750676 + 0.660671i \(0.229728\pi\)
\(102\) 0.887166 0.0878425
\(103\) −3.22425 −0.317695 −0.158848 0.987303i \(-0.550778\pi\)
−0.158848 + 0.987303i \(0.550778\pi\)
\(104\) 2.27645 0.223225
\(105\) −3.35026 −0.326952
\(106\) 0.911603 0.0885427
\(107\) 0.962389 0.0930376 0.0465188 0.998917i \(-0.485187\pi\)
0.0465188 + 0.998917i \(0.485187\pi\)
\(108\) −1.96239 −0.188831
\(109\) −11.4010 −1.09202 −0.546011 0.837778i \(-0.683854\pi\)
−0.546011 + 0.837778i \(0.683854\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) −12.6497 −1.19529
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0.836381 0.0783342
\(115\) −6.70052 −0.624827
\(116\) −7.08840 −0.658141
\(117\) −2.96239 −0.273873
\(118\) 2.07522 0.191040
\(119\) −15.3258 −1.40492
\(120\) −0.768452 −0.0701498
\(121\) 0 0
\(122\) 1.68735 0.152765
\(123\) 4.38787 0.395641
\(124\) −19.4763 −1.74902
\(125\) 1.00000 0.0894427
\(126\) −0.649738 −0.0578833
\(127\) 14.5745 1.29328 0.646640 0.762796i \(-0.276174\pi\)
0.646640 + 0.762796i \(0.276174\pi\)
\(128\) −5.91748 −0.523037
\(129\) 9.27504 0.816622
\(130\) −0.574515 −0.0503883
\(131\) 5.92478 0.517650 0.258825 0.965924i \(-0.416665\pi\)
0.258825 + 0.965924i \(0.416665\pi\)
\(132\) 0 0
\(133\) −14.4485 −1.25284
\(134\) 1.14903 0.0992612
\(135\) 1.00000 0.0860663
\(136\) −3.51530 −0.301434
\(137\) 13.8496 1.18325 0.591624 0.806214i \(-0.298487\pi\)
0.591624 + 0.806214i \(0.298487\pi\)
\(138\) −1.29948 −0.110619
\(139\) −13.6121 −1.15457 −0.577283 0.816544i \(-0.695887\pi\)
−0.577283 + 0.816544i \(0.695887\pi\)
\(140\) 6.57452 0.555648
\(141\) −9.92478 −0.835817
\(142\) 1.92478 0.161524
\(143\) 0 0
\(144\) 3.77575 0.314646
\(145\) 3.61213 0.299971
\(146\) 1.50071 0.124199
\(147\) 4.22425 0.348411
\(148\) 3.92478 0.322615
\(149\) −1.53690 −0.125908 −0.0629540 0.998016i \(-0.520052\pi\)
−0.0629540 + 0.998016i \(0.520052\pi\)
\(150\) 0.193937 0.0158349
\(151\) 6.76116 0.550215 0.275108 0.961413i \(-0.411287\pi\)
0.275108 + 0.961413i \(0.411287\pi\)
\(152\) −3.31406 −0.268806
\(153\) 4.57452 0.369828
\(154\) 0 0
\(155\) 9.92478 0.797177
\(156\) 5.81336 0.465441
\(157\) −5.47627 −0.437054 −0.218527 0.975831i \(-0.570125\pi\)
−0.218527 + 0.975831i \(0.570125\pi\)
\(158\) −2.23743 −0.178000
\(159\) 4.70052 0.372776
\(160\) 2.26916 0.179393
\(161\) 22.4485 1.76919
\(162\) 0.193937 0.0152371
\(163\) 12.6253 0.988890 0.494445 0.869209i \(-0.335371\pi\)
0.494445 + 0.869209i \(0.335371\pi\)
\(164\) −8.61071 −0.672384
\(165\) 0 0
\(166\) −2.11142 −0.163878
\(167\) −18.3634 −1.42101 −0.710503 0.703695i \(-0.751532\pi\)
−0.710503 + 0.703695i \(0.751532\pi\)
\(168\) 2.57452 0.198628
\(169\) −4.22425 −0.324943
\(170\) 0.887166 0.0680425
\(171\) 4.31265 0.329797
\(172\) −18.2012 −1.38783
\(173\) 8.57452 0.651908 0.325954 0.945386i \(-0.394314\pi\)
0.325954 + 0.945386i \(0.394314\pi\)
\(174\) 0.700523 0.0531065
\(175\) −3.35026 −0.253256
\(176\) 0 0
\(177\) 10.7005 0.804301
\(178\) −0.538319 −0.0403487
\(179\) 14.1768 1.05962 0.529812 0.848115i \(-0.322263\pi\)
0.529812 + 0.848115i \(0.322263\pi\)
\(180\) −1.96239 −0.146268
\(181\) −5.22425 −0.388316 −0.194158 0.980970i \(-0.562197\pi\)
−0.194158 + 0.980970i \(0.562197\pi\)
\(182\) 1.92478 0.142674
\(183\) 8.70052 0.643161
\(184\) 5.14903 0.379592
\(185\) −2.00000 −0.147043
\(186\) 1.92478 0.141132
\(187\) 0 0
\(188\) 19.4763 1.42045
\(189\) −3.35026 −0.243696
\(190\) 0.836381 0.0606774
\(191\) −16.6253 −1.20296 −0.601482 0.798886i \(-0.705423\pi\)
−0.601482 + 0.798886i \(0.705423\pi\)
\(192\) −7.11142 −0.513222
\(193\) 16.3634 1.17787 0.588933 0.808182i \(-0.299548\pi\)
0.588933 + 0.808182i \(0.299548\pi\)
\(194\) 0.0145884 0.00104739
\(195\) −2.96239 −0.212141
\(196\) −8.28963 −0.592116
\(197\) 20.4241 1.45515 0.727577 0.686026i \(-0.240646\pi\)
0.727577 + 0.686026i \(0.240646\pi\)
\(198\) 0 0
\(199\) −8.62530 −0.611431 −0.305716 0.952123i \(-0.598896\pi\)
−0.305716 + 0.952123i \(0.598896\pi\)
\(200\) −0.768452 −0.0543378
\(201\) 5.92478 0.417902
\(202\) 2.92619 0.205886
\(203\) −12.1016 −0.849364
\(204\) −8.97698 −0.628514
\(205\) 4.38787 0.306462
\(206\) −0.625301 −0.0435668
\(207\) −6.70052 −0.465719
\(208\) −11.1852 −0.775556
\(209\) 0 0
\(210\) −0.649738 −0.0448362
\(211\) −9.08840 −0.625671 −0.312836 0.949807i \(-0.601279\pi\)
−0.312836 + 0.949807i \(0.601279\pi\)
\(212\) −9.22425 −0.633524
\(213\) 9.92478 0.680035
\(214\) 0.186642 0.0127586
\(215\) 9.27504 0.632552
\(216\) −0.768452 −0.0522865
\(217\) −33.2506 −2.25720
\(218\) −2.21108 −0.149753
\(219\) 7.73813 0.522895
\(220\) 0 0
\(221\) −13.5515 −0.911572
\(222\) −0.387873 −0.0260323
\(223\) −6.70052 −0.448700 −0.224350 0.974509i \(-0.572026\pi\)
−0.224350 + 0.974509i \(0.572026\pi\)
\(224\) −7.60228 −0.507949
\(225\) 1.00000 0.0666667
\(226\) −1.16362 −0.0774028
\(227\) −16.9624 −1.12583 −0.562917 0.826514i \(-0.690321\pi\)
−0.562917 + 0.826514i \(0.690321\pi\)
\(228\) −8.46310 −0.560482
\(229\) 25.8496 1.70819 0.854093 0.520120i \(-0.174113\pi\)
0.854093 + 0.520120i \(0.174113\pi\)
\(230\) −1.29948 −0.0856849
\(231\) 0 0
\(232\) −2.77575 −0.182237
\(233\) 19.2750 1.26275 0.631375 0.775478i \(-0.282491\pi\)
0.631375 + 0.775478i \(0.282491\pi\)
\(234\) −0.574515 −0.0375573
\(235\) −9.92478 −0.647421
\(236\) −20.9986 −1.36689
\(237\) −11.5369 −0.749402
\(238\) −2.97224 −0.192662
\(239\) −26.5501 −1.71738 −0.858691 0.512494i \(-0.828722\pi\)
−0.858691 + 0.512494i \(0.828722\pi\)
\(240\) 3.77575 0.243723
\(241\) −28.5501 −1.83907 −0.919536 0.393006i \(-0.871435\pi\)
−0.919536 + 0.393006i \(0.871435\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −17.0738 −1.09304
\(245\) 4.22425 0.269878
\(246\) 0.850969 0.0542558
\(247\) −12.7757 −0.812901
\(248\) −7.62672 −0.484297
\(249\) −10.8872 −0.689946
\(250\) 0.193937 0.0122656
\(251\) 29.9248 1.88884 0.944418 0.328748i \(-0.106627\pi\)
0.944418 + 0.328748i \(0.106627\pi\)
\(252\) 6.57452 0.414156
\(253\) 0 0
\(254\) 2.82653 0.177352
\(255\) 4.57452 0.286467
\(256\) 13.0752 0.817201
\(257\) 8.70052 0.542724 0.271362 0.962477i \(-0.412526\pi\)
0.271362 + 0.962477i \(0.412526\pi\)
\(258\) 1.79877 0.111986
\(259\) 6.70052 0.416350
\(260\) 5.81336 0.360529
\(261\) 3.61213 0.223585
\(262\) 1.14903 0.0709874
\(263\) −12.2882 −0.757724 −0.378862 0.925453i \(-0.623684\pi\)
−0.378862 + 0.925453i \(0.623684\pi\)
\(264\) 0 0
\(265\) 4.70052 0.288751
\(266\) −2.80209 −0.171807
\(267\) −2.77575 −0.169873
\(268\) −11.6267 −0.710215
\(269\) −5.84955 −0.356654 −0.178327 0.983971i \(-0.557068\pi\)
−0.178327 + 0.983971i \(0.557068\pi\)
\(270\) 0.193937 0.0118026
\(271\) 5.08840 0.309098 0.154549 0.987985i \(-0.450608\pi\)
0.154549 + 0.987985i \(0.450608\pi\)
\(272\) 17.2722 1.04728
\(273\) 9.92478 0.600675
\(274\) 2.68594 0.162263
\(275\) 0 0
\(276\) 13.1490 0.791479
\(277\) −1.41090 −0.0847725 −0.0423863 0.999101i \(-0.513496\pi\)
−0.0423863 + 0.999101i \(0.513496\pi\)
\(278\) −2.63989 −0.158330
\(279\) 9.92478 0.594181
\(280\) 2.57452 0.153857
\(281\) 4.38787 0.261759 0.130879 0.991398i \(-0.458220\pi\)
0.130879 + 0.991398i \(0.458220\pi\)
\(282\) −1.92478 −0.114619
\(283\) −26.5745 −1.57969 −0.789845 0.613306i \(-0.789839\pi\)
−0.789845 + 0.613306i \(0.789839\pi\)
\(284\) −19.4763 −1.15570
\(285\) 4.31265 0.255459
\(286\) 0 0
\(287\) −14.7005 −0.867744
\(288\) 2.26916 0.133711
\(289\) 3.92619 0.230952
\(290\) 0.700523 0.0411362
\(291\) 0.0752228 0.00440964
\(292\) −15.1852 −0.888648
\(293\) 3.42548 0.200119 0.100059 0.994981i \(-0.468097\pi\)
0.100059 + 0.994981i \(0.468097\pi\)
\(294\) 0.819237 0.0477789
\(295\) 10.7005 0.623009
\(296\) 1.53690 0.0893307
\(297\) 0 0
\(298\) −0.298062 −0.0172663
\(299\) 19.8496 1.14793
\(300\) −1.96239 −0.113299
\(301\) −31.0738 −1.79106
\(302\) 1.31124 0.0754531
\(303\) 15.0884 0.866806
\(304\) 16.2835 0.933921
\(305\) 8.70052 0.498191
\(306\) 0.887166 0.0507159
\(307\) 16.6497 0.950251 0.475125 0.879918i \(-0.342403\pi\)
0.475125 + 0.879918i \(0.342403\pi\)
\(308\) 0 0
\(309\) −3.22425 −0.183421
\(310\) 1.92478 0.109320
\(311\) 32.9986 1.87118 0.935589 0.353091i \(-0.114869\pi\)
0.935589 + 0.353091i \(0.114869\pi\)
\(312\) 2.27645 0.128879
\(313\) 15.4010 0.870519 0.435259 0.900305i \(-0.356657\pi\)
0.435259 + 0.900305i \(0.356657\pi\)
\(314\) −1.06205 −0.0599349
\(315\) −3.35026 −0.188766
\(316\) 22.6399 1.27359
\(317\) 2.15045 0.120781 0.0603905 0.998175i \(-0.480765\pi\)
0.0603905 + 0.998175i \(0.480765\pi\)
\(318\) 0.911603 0.0511202
\(319\) 0 0
\(320\) −7.11142 −0.397540
\(321\) 0.962389 0.0537153
\(322\) 4.35359 0.242616
\(323\) 19.7283 1.09771
\(324\) −1.96239 −0.109022
\(325\) −2.96239 −0.164324
\(326\) 2.44851 0.135610
\(327\) −11.4010 −0.630479
\(328\) −3.37187 −0.186180
\(329\) 33.2506 1.83316
\(330\) 0 0
\(331\) −14.5501 −0.799745 −0.399872 0.916571i \(-0.630946\pi\)
−0.399872 + 0.916571i \(0.630946\pi\)
\(332\) 21.3649 1.17255
\(333\) −2.00000 −0.109599
\(334\) −3.56134 −0.194868
\(335\) 5.92478 0.323705
\(336\) −12.6497 −0.690100
\(337\) −16.2619 −0.885840 −0.442920 0.896561i \(-0.646057\pi\)
−0.442920 + 0.896561i \(0.646057\pi\)
\(338\) −0.819237 −0.0445606
\(339\) −6.00000 −0.325875
\(340\) −8.97698 −0.486845
\(341\) 0 0
\(342\) 0.836381 0.0452263
\(343\) 9.29948 0.502125
\(344\) −7.12742 −0.384285
\(345\) −6.70052 −0.360744
\(346\) 1.66291 0.0893987
\(347\) 0.962389 0.0516637 0.0258319 0.999666i \(-0.491777\pi\)
0.0258319 + 0.999666i \(0.491777\pi\)
\(348\) −7.08840 −0.379978
\(349\) −20.7005 −1.10807 −0.554037 0.832492i \(-0.686913\pi\)
−0.554037 + 0.832492i \(0.686913\pi\)
\(350\) −0.649738 −0.0347300
\(351\) −2.96239 −0.158121
\(352\) 0 0
\(353\) 20.5501 1.09377 0.546885 0.837208i \(-0.315813\pi\)
0.546885 + 0.837208i \(0.315813\pi\)
\(354\) 2.07522 0.110297
\(355\) 9.92478 0.526752
\(356\) 5.44709 0.288695
\(357\) −15.3258 −0.811129
\(358\) 2.74940 0.145310
\(359\) −17.9248 −0.946034 −0.473017 0.881053i \(-0.656835\pi\)
−0.473017 + 0.881053i \(0.656835\pi\)
\(360\) −0.768452 −0.0405010
\(361\) −0.401047 −0.0211077
\(362\) −1.01317 −0.0532512
\(363\) 0 0
\(364\) −19.4763 −1.02083
\(365\) 7.73813 0.405032
\(366\) 1.68735 0.0881992
\(367\) −29.6531 −1.54788 −0.773939 0.633261i \(-0.781716\pi\)
−0.773939 + 0.633261i \(0.781716\pi\)
\(368\) −25.2995 −1.31883
\(369\) 4.38787 0.228424
\(370\) −0.387873 −0.0201646
\(371\) −15.7480 −0.817595
\(372\) −19.4763 −1.00980
\(373\) 9.13918 0.473209 0.236604 0.971606i \(-0.423965\pi\)
0.236604 + 0.971606i \(0.423965\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 7.62672 0.393318
\(377\) −10.7005 −0.551105
\(378\) −0.649738 −0.0334189
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) −8.46310 −0.434148
\(381\) 14.5745 0.746675
\(382\) −3.22425 −0.164967
\(383\) −34.9234 −1.78450 −0.892250 0.451541i \(-0.850874\pi\)
−0.892250 + 0.451541i \(0.850874\pi\)
\(384\) −5.91748 −0.301975
\(385\) 0 0
\(386\) 3.17347 0.161525
\(387\) 9.27504 0.471477
\(388\) −0.147616 −0.00749408
\(389\) 2.77575 0.140736 0.0703680 0.997521i \(-0.477583\pi\)
0.0703680 + 0.997521i \(0.477583\pi\)
\(390\) −0.574515 −0.0290917
\(391\) −30.6516 −1.55012
\(392\) −3.24614 −0.163955
\(393\) 5.92478 0.298865
\(394\) 3.96097 0.199551
\(395\) −11.5369 −0.580485
\(396\) 0 0
\(397\) −19.9248 −0.999996 −0.499998 0.866027i \(-0.666666\pi\)
−0.499998 + 0.866027i \(0.666666\pi\)
\(398\) −1.67276 −0.0838479
\(399\) −14.4485 −0.723330
\(400\) 3.77575 0.188787
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 1.14903 0.0573085
\(403\) −29.4010 −1.46457
\(404\) −29.6093 −1.47312
\(405\) 1.00000 0.0496904
\(406\) −2.34694 −0.116477
\(407\) 0 0
\(408\) −3.51530 −0.174033
\(409\) 13.0738 0.646458 0.323229 0.946321i \(-0.395232\pi\)
0.323229 + 0.946321i \(0.395232\pi\)
\(410\) 0.850969 0.0420264
\(411\) 13.8496 0.683148
\(412\) 6.32724 0.311721
\(413\) −35.8496 −1.76404
\(414\) −1.29948 −0.0638658
\(415\) −10.8872 −0.534430
\(416\) −6.72213 −0.329580
\(417\) −13.6121 −0.666589
\(418\) 0 0
\(419\) 7.22425 0.352928 0.176464 0.984307i \(-0.443534\pi\)
0.176464 + 0.984307i \(0.443534\pi\)
\(420\) 6.57452 0.320804
\(421\) 30.6253 1.49259 0.746293 0.665618i \(-0.231832\pi\)
0.746293 + 0.665618i \(0.231832\pi\)
\(422\) −1.76257 −0.0858007
\(423\) −9.92478 −0.482559
\(424\) −3.61213 −0.175420
\(425\) 4.57452 0.221897
\(426\) 1.92478 0.0932558
\(427\) −29.1490 −1.41062
\(428\) −1.88858 −0.0912880
\(429\) 0 0
\(430\) 1.79877 0.0867444
\(431\) 33.8759 1.63174 0.815872 0.578232i \(-0.196257\pi\)
0.815872 + 0.578232i \(0.196257\pi\)
\(432\) 3.77575 0.181661
\(433\) −9.47627 −0.455400 −0.227700 0.973731i \(-0.573121\pi\)
−0.227700 + 0.973731i \(0.573121\pi\)
\(434\) −6.44851 −0.309538
\(435\) 3.61213 0.173188
\(436\) 22.3733 1.07149
\(437\) −28.8970 −1.38233
\(438\) 1.50071 0.0717066
\(439\) 29.4617 1.40613 0.703065 0.711126i \(-0.251814\pi\)
0.703065 + 0.711126i \(0.251814\pi\)
\(440\) 0 0
\(441\) 4.22425 0.201155
\(442\) −2.62813 −0.125007
\(443\) −19.0738 −0.906224 −0.453112 0.891454i \(-0.649686\pi\)
−0.453112 + 0.891454i \(0.649686\pi\)
\(444\) 3.92478 0.186262
\(445\) −2.77575 −0.131583
\(446\) −1.29948 −0.0615320
\(447\) −1.53690 −0.0726931
\(448\) 23.8251 1.12563
\(449\) 35.8759 1.69309 0.846544 0.532318i \(-0.178679\pi\)
0.846544 + 0.532318i \(0.178679\pi\)
\(450\) 0.193937 0.00914226
\(451\) 0 0
\(452\) 11.7743 0.553818
\(453\) 6.76116 0.317667
\(454\) −3.28963 −0.154390
\(455\) 9.92478 0.465281
\(456\) −3.31406 −0.155195
\(457\) −5.28963 −0.247438 −0.123719 0.992317i \(-0.539482\pi\)
−0.123719 + 0.992317i \(0.539482\pi\)
\(458\) 5.01317 0.234250
\(459\) 4.57452 0.213520
\(460\) 13.1490 0.613077
\(461\) −36.3390 −1.69248 −0.846238 0.532805i \(-0.821138\pi\)
−0.846238 + 0.532805i \(0.821138\pi\)
\(462\) 0 0
\(463\) 10.5501 0.490304 0.245152 0.969485i \(-0.421162\pi\)
0.245152 + 0.969485i \(0.421162\pi\)
\(464\) 13.6385 0.633150
\(465\) 9.92478 0.460251
\(466\) 3.73813 0.173166
\(467\) 18.7005 0.865357 0.432679 0.901548i \(-0.357569\pi\)
0.432679 + 0.901548i \(0.357569\pi\)
\(468\) 5.81336 0.268723
\(469\) −19.8496 −0.916567
\(470\) −1.92478 −0.0887834
\(471\) −5.47627 −0.252333
\(472\) −8.22284 −0.378487
\(473\) 0 0
\(474\) −2.23743 −0.102768
\(475\) 4.31265 0.197878
\(476\) 30.0752 1.37850
\(477\) 4.70052 0.215222
\(478\) −5.14903 −0.235511
\(479\) 9.29948 0.424904 0.212452 0.977172i \(-0.431855\pi\)
0.212452 + 0.977172i \(0.431855\pi\)
\(480\) 2.26916 0.103572
\(481\) 5.92478 0.270147
\(482\) −5.53690 −0.252199
\(483\) 22.4485 1.02144
\(484\) 0 0
\(485\) 0.0752228 0.00341569
\(486\) 0.193937 0.00879714
\(487\) −35.4763 −1.60758 −0.803792 0.594911i \(-0.797187\pi\)
−0.803792 + 0.594911i \(0.797187\pi\)
\(488\) −6.68594 −0.302658
\(489\) 12.6253 0.570936
\(490\) 0.819237 0.0370094
\(491\) −24.7757 −1.11811 −0.559057 0.829129i \(-0.688837\pi\)
−0.559057 + 0.829129i \(0.688837\pi\)
\(492\) −8.61071 −0.388201
\(493\) 16.5237 0.744191
\(494\) −2.47768 −0.111476
\(495\) 0 0
\(496\) 37.4734 1.68261
\(497\) −33.2506 −1.49149
\(498\) −2.11142 −0.0946150
\(499\) 14.1768 0.634640 0.317320 0.948318i \(-0.397217\pi\)
0.317320 + 0.948318i \(0.397217\pi\)
\(500\) −1.96239 −0.0877607
\(501\) −18.3634 −0.820418
\(502\) 5.80351 0.259023
\(503\) 8.43866 0.376261 0.188131 0.982144i \(-0.439757\pi\)
0.188131 + 0.982144i \(0.439757\pi\)
\(504\) 2.57452 0.114678
\(505\) 15.0884 0.671425
\(506\) 0 0
\(507\) −4.22425 −0.187606
\(508\) −28.6009 −1.26896
\(509\) 1.10299 0.0488890 0.0244445 0.999701i \(-0.492218\pi\)
0.0244445 + 0.999701i \(0.492218\pi\)
\(510\) 0.887166 0.0392844
\(511\) −25.9248 −1.14684
\(512\) 14.3707 0.635103
\(513\) 4.31265 0.190408
\(514\) 1.68735 0.0744258
\(515\) −3.22425 −0.142078
\(516\) −18.2012 −0.801265
\(517\) 0 0
\(518\) 1.29948 0.0570957
\(519\) 8.57452 0.376379
\(520\) 2.27645 0.0998291
\(521\) −12.4485 −0.545379 −0.272690 0.962102i \(-0.587913\pi\)
−0.272690 + 0.962102i \(0.587913\pi\)
\(522\) 0.700523 0.0306611
\(523\) −30.0508 −1.31403 −0.657015 0.753878i \(-0.728181\pi\)
−0.657015 + 0.753878i \(0.728181\pi\)
\(524\) −11.6267 −0.507915
\(525\) −3.35026 −0.146217
\(526\) −2.38313 −0.103910
\(527\) 45.4010 1.97770
\(528\) 0 0
\(529\) 21.8970 0.952044
\(530\) 0.911603 0.0395975
\(531\) 10.7005 0.464363
\(532\) 28.3536 1.22928
\(533\) −12.9986 −0.563031
\(534\) −0.538319 −0.0232953
\(535\) 0.962389 0.0416077
\(536\) −4.55291 −0.196656
\(537\) 14.1768 0.611774
\(538\) −1.13444 −0.0489093
\(539\) 0 0
\(540\) −1.96239 −0.0844478
\(541\) 18.0000 0.773880 0.386940 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(542\) 0.986826 0.0423878
\(543\) −5.22425 −0.224194
\(544\) 10.3803 0.445052
\(545\) −11.4010 −0.488367
\(546\) 1.92478 0.0823729
\(547\) 14.3028 0.611544 0.305772 0.952105i \(-0.401086\pi\)
0.305772 + 0.952105i \(0.401086\pi\)
\(548\) −27.1782 −1.16100
\(549\) 8.70052 0.371329
\(550\) 0 0
\(551\) 15.5778 0.663638
\(552\) 5.14903 0.219157
\(553\) 38.6516 1.64364
\(554\) −0.273624 −0.0116252
\(555\) −2.00000 −0.0848953
\(556\) 26.7123 1.13285
\(557\) 11.7988 0.499930 0.249965 0.968255i \(-0.419581\pi\)
0.249965 + 0.968255i \(0.419581\pi\)
\(558\) 1.92478 0.0814823
\(559\) −27.4763 −1.16212
\(560\) −12.6497 −0.534549
\(561\) 0 0
\(562\) 0.850969 0.0358960
\(563\) −30.4847 −1.28478 −0.642389 0.766379i \(-0.722056\pi\)
−0.642389 + 0.766379i \(0.722056\pi\)
\(564\) 19.4763 0.820099
\(565\) −6.00000 −0.252422
\(566\) −5.15377 −0.216629
\(567\) −3.35026 −0.140698
\(568\) −7.62672 −0.320010
\(569\) 27.0884 1.13560 0.567802 0.823165i \(-0.307794\pi\)
0.567802 + 0.823165i \(0.307794\pi\)
\(570\) 0.836381 0.0350321
\(571\) −7.28489 −0.304863 −0.152432 0.988314i \(-0.548710\pi\)
−0.152432 + 0.988314i \(0.548710\pi\)
\(572\) 0 0
\(573\) −16.6253 −0.694532
\(574\) −2.85097 −0.118997
\(575\) −6.70052 −0.279431
\(576\) −7.11142 −0.296309
\(577\) −31.6239 −1.31652 −0.658260 0.752791i \(-0.728707\pi\)
−0.658260 + 0.752791i \(0.728707\pi\)
\(578\) 0.761432 0.0316714
\(579\) 16.3634 0.680041
\(580\) −7.08840 −0.294330
\(581\) 36.4749 1.51323
\(582\) 0.0145884 0.000604711 0
\(583\) 0 0
\(584\) −5.94639 −0.246063
\(585\) −2.96239 −0.122480
\(586\) 0.664327 0.0274431
\(587\) 33.1490 1.36821 0.684103 0.729385i \(-0.260194\pi\)
0.684103 + 0.729385i \(0.260194\pi\)
\(588\) −8.28963 −0.341858
\(589\) 42.8021 1.76363
\(590\) 2.07522 0.0854356
\(591\) 20.4241 0.840134
\(592\) −7.55149 −0.310364
\(593\) −34.4993 −1.41672 −0.708358 0.705853i \(-0.750564\pi\)
−0.708358 + 0.705853i \(0.750564\pi\)
\(594\) 0 0
\(595\) −15.3258 −0.628298
\(596\) 3.01600 0.123540
\(597\) −8.62530 −0.353010
\(598\) 3.84955 0.157420
\(599\) −14.4485 −0.590350 −0.295175 0.955443i \(-0.595378\pi\)
−0.295175 + 0.955443i \(0.595378\pi\)
\(600\) −0.768452 −0.0313719
\(601\) 15.9248 0.649585 0.324793 0.945785i \(-0.394705\pi\)
0.324793 + 0.945785i \(0.394705\pi\)
\(602\) −6.02635 −0.245616
\(603\) 5.92478 0.241276
\(604\) −13.2680 −0.539868
\(605\) 0 0
\(606\) 2.92619 0.118868
\(607\) 14.5745 0.591561 0.295781 0.955256i \(-0.404420\pi\)
0.295781 + 0.955256i \(0.404420\pi\)
\(608\) 9.78609 0.396878
\(609\) −12.1016 −0.490380
\(610\) 1.68735 0.0683188
\(611\) 29.4010 1.18944
\(612\) −8.97698 −0.362873
\(613\) −16.4123 −0.662887 −0.331443 0.943475i \(-0.607536\pi\)
−0.331443 + 0.943475i \(0.607536\pi\)
\(614\) 3.22899 0.130312
\(615\) 4.38787 0.176936
\(616\) 0 0
\(617\) −17.8496 −0.718596 −0.359298 0.933223i \(-0.616984\pi\)
−0.359298 + 0.933223i \(0.616984\pi\)
\(618\) −0.625301 −0.0251533
\(619\) −0.402462 −0.0161763 −0.00808815 0.999967i \(-0.502575\pi\)
−0.00808815 + 0.999967i \(0.502575\pi\)
\(620\) −19.4763 −0.782186
\(621\) −6.70052 −0.268883
\(622\) 6.39963 0.256602
\(623\) 9.29948 0.372576
\(624\) −11.1852 −0.447767
\(625\) 1.00000 0.0400000
\(626\) 2.98683 0.119378
\(627\) 0 0
\(628\) 10.7466 0.428835
\(629\) −9.14903 −0.364796
\(630\) −0.649738 −0.0258862
\(631\) −38.0263 −1.51380 −0.756902 0.653528i \(-0.773288\pi\)
−0.756902 + 0.653528i \(0.773288\pi\)
\(632\) 8.86556 0.352653
\(633\) −9.08840 −0.361231
\(634\) 0.417050 0.0165632
\(635\) 14.5745 0.578372
\(636\) −9.22425 −0.365765
\(637\) −12.5139 −0.495818
\(638\) 0 0
\(639\) 9.92478 0.392618
\(640\) −5.91748 −0.233909
\(641\) −28.0263 −1.10697 −0.553487 0.832858i \(-0.686703\pi\)
−0.553487 + 0.832858i \(0.686703\pi\)
\(642\) 0.186642 0.00736619
\(643\) −4.62530 −0.182404 −0.0912020 0.995832i \(-0.529071\pi\)
−0.0912020 + 0.995832i \(0.529071\pi\)
\(644\) −44.0527 −1.73592
\(645\) 9.27504 0.365204
\(646\) 3.82604 0.150533
\(647\) 23.5778 0.926941 0.463470 0.886112i \(-0.346604\pi\)
0.463470 + 0.886112i \(0.346604\pi\)
\(648\) −0.768452 −0.0301876
\(649\) 0 0
\(650\) −0.574515 −0.0225344
\(651\) −33.2506 −1.30319
\(652\) −24.7757 −0.970293
\(653\) −2.25202 −0.0881282 −0.0440641 0.999029i \(-0.514031\pi\)
−0.0440641 + 0.999029i \(0.514031\pi\)
\(654\) −2.21108 −0.0864601
\(655\) 5.92478 0.231500
\(656\) 16.5675 0.646852
\(657\) 7.73813 0.301893
\(658\) 6.44851 0.251389
\(659\) 41.4010 1.61276 0.806378 0.591401i \(-0.201425\pi\)
0.806378 + 0.591401i \(0.201425\pi\)
\(660\) 0 0
\(661\) 3.40105 0.132285 0.0661427 0.997810i \(-0.478931\pi\)
0.0661427 + 0.997810i \(0.478931\pi\)
\(662\) −2.82179 −0.109672
\(663\) −13.5515 −0.526296
\(664\) 8.36626 0.324674
\(665\) −14.4485 −0.560289
\(666\) −0.387873 −0.0150298
\(667\) −24.2031 −0.937149
\(668\) 36.0362 1.39428
\(669\) −6.70052 −0.259057
\(670\) 1.14903 0.0443909
\(671\) 0 0
\(672\) −7.60228 −0.293264
\(673\) −0.887166 −0.0341977 −0.0170989 0.999854i \(-0.505443\pi\)
−0.0170989 + 0.999854i \(0.505443\pi\)
\(674\) −3.15377 −0.121479
\(675\) 1.00000 0.0384900
\(676\) 8.28963 0.318832
\(677\) −18.9018 −0.726453 −0.363227 0.931701i \(-0.618325\pi\)
−0.363227 + 0.931701i \(0.618325\pi\)
\(678\) −1.16362 −0.0446885
\(679\) −0.252016 −0.00967149
\(680\) −3.51530 −0.134805
\(681\) −16.9624 −0.650000
\(682\) 0 0
\(683\) −20.8773 −0.798848 −0.399424 0.916766i \(-0.630790\pi\)
−0.399424 + 0.916766i \(0.630790\pi\)
\(684\) −8.46310 −0.323595
\(685\) 13.8496 0.529164
\(686\) 1.80351 0.0688583
\(687\) 25.8496 0.986222
\(688\) 35.0202 1.33513
\(689\) −13.9248 −0.530492
\(690\) −1.29948 −0.0494702
\(691\) −2.44851 −0.0931456 −0.0465728 0.998915i \(-0.514830\pi\)
−0.0465728 + 0.998915i \(0.514830\pi\)
\(692\) −16.8265 −0.639649
\(693\) 0 0
\(694\) 0.186642 0.00708485
\(695\) −13.6121 −0.516337
\(696\) −2.77575 −0.105214
\(697\) 20.0724 0.760296
\(698\) −4.01459 −0.151954
\(699\) 19.2750 0.729049
\(700\) 6.57452 0.248493
\(701\) 2.98683 0.112811 0.0564054 0.998408i \(-0.482036\pi\)
0.0564054 + 0.998408i \(0.482036\pi\)
\(702\) −0.574515 −0.0216837
\(703\) −8.62530 −0.325309
\(704\) 0 0
\(705\) −9.92478 −0.373789
\(706\) 3.98541 0.149993
\(707\) −50.5501 −1.90113
\(708\) −20.9986 −0.789175
\(709\) 24.1768 0.907979 0.453989 0.891007i \(-0.350000\pi\)
0.453989 + 0.891007i \(0.350000\pi\)
\(710\) 1.92478 0.0722356
\(711\) −11.5369 −0.432668
\(712\) 2.13303 0.0799386
\(713\) −66.5012 −2.49049
\(714\) −2.97224 −0.111233
\(715\) 0 0
\(716\) −27.8204 −1.03970
\(717\) −26.5501 −0.991531
\(718\) −3.47627 −0.129733
\(719\) −30.0263 −1.11979 −0.559897 0.828562i \(-0.689159\pi\)
−0.559897 + 0.828562i \(0.689159\pi\)
\(720\) 3.77575 0.140714
\(721\) 10.8021 0.402291
\(722\) −0.0777777 −0.00289459
\(723\) −28.5501 −1.06179
\(724\) 10.2520 0.381013
\(725\) 3.61213 0.134151
\(726\) 0 0
\(727\) 14.9525 0.554559 0.277279 0.960789i \(-0.410567\pi\)
0.277279 + 0.960789i \(0.410567\pi\)
\(728\) −7.62672 −0.282665
\(729\) 1.00000 0.0370370
\(730\) 1.50071 0.0555437
\(731\) 42.4288 1.56929
\(732\) −17.0738 −0.631066
\(733\) 19.1128 0.705949 0.352974 0.935633i \(-0.385170\pi\)
0.352974 + 0.935633i \(0.385170\pi\)
\(734\) −5.75081 −0.212266
\(735\) 4.22425 0.155814
\(736\) −15.2046 −0.560447
\(737\) 0 0
\(738\) 0.850969 0.0313246
\(739\) 3.31406 0.121910 0.0609549 0.998141i \(-0.480585\pi\)
0.0609549 + 0.998141i \(0.480585\pi\)
\(740\) 3.92478 0.144278
\(741\) −12.7757 −0.469329
\(742\) −3.05411 −0.112120
\(743\) 34.9887 1.28361 0.641806 0.766867i \(-0.278185\pi\)
0.641806 + 0.766867i \(0.278185\pi\)
\(744\) −7.62672 −0.279609
\(745\) −1.53690 −0.0563078
\(746\) 1.77242 0.0648930
\(747\) −10.8872 −0.398341
\(748\) 0 0
\(749\) −3.22425 −0.117812
\(750\) 0.193937 0.00708156
\(751\) −26.9234 −0.982447 −0.491224 0.871033i \(-0.663450\pi\)
−0.491224 + 0.871033i \(0.663450\pi\)
\(752\) −37.4734 −1.36652
\(753\) 29.9248 1.09052
\(754\) −2.07522 −0.0755752
\(755\) 6.76116 0.246064
\(756\) 6.57452 0.239113
\(757\) 15.9248 0.578796 0.289398 0.957209i \(-0.406545\pi\)
0.289398 + 0.957209i \(0.406545\pi\)
\(758\) −3.87873 −0.140882
\(759\) 0 0
\(760\) −3.31406 −0.120214
\(761\) 30.9380 1.12150 0.560750 0.827985i \(-0.310513\pi\)
0.560750 + 0.827985i \(0.310513\pi\)
\(762\) 2.82653 0.102394
\(763\) 38.1965 1.38281
\(764\) 32.6253 1.18034
\(765\) 4.57452 0.165392
\(766\) −6.77292 −0.244715
\(767\) −31.6991 −1.14459
\(768\) 13.0752 0.471811
\(769\) −9.32582 −0.336298 −0.168149 0.985762i \(-0.553779\pi\)
−0.168149 + 0.985762i \(0.553779\pi\)
\(770\) 0 0
\(771\) 8.70052 0.313342
\(772\) −32.1114 −1.15572
\(773\) 44.7005 1.60777 0.803883 0.594787i \(-0.202764\pi\)
0.803883 + 0.594787i \(0.202764\pi\)
\(774\) 1.79877 0.0646554
\(775\) 9.92478 0.356509
\(776\) −0.0578051 −0.00207508
\(777\) 6.70052 0.240380
\(778\) 0.538319 0.0192997
\(779\) 18.9234 0.678000
\(780\) 5.81336 0.208152
\(781\) 0 0
\(782\) −5.94448 −0.212574
\(783\) 3.61213 0.129087
\(784\) 15.9497 0.569633
\(785\) −5.47627 −0.195456
\(786\) 1.14903 0.0409846
\(787\) −21.6775 −0.772719 −0.386360 0.922348i \(-0.626268\pi\)
−0.386360 + 0.922348i \(0.626268\pi\)
\(788\) −40.0800 −1.42779
\(789\) −12.2882 −0.437472
\(790\) −2.23743 −0.0796041
\(791\) 20.1016 0.714730
\(792\) 0 0
\(793\) −25.7743 −0.915273
\(794\) −3.86414 −0.137133
\(795\) 4.70052 0.166710
\(796\) 16.9262 0.599933
\(797\) 22.7466 0.805725 0.402862 0.915261i \(-0.368015\pi\)
0.402862 + 0.915261i \(0.368015\pi\)
\(798\) −2.80209 −0.0991930
\(799\) −45.4010 −1.60617
\(800\) 2.26916 0.0802269
\(801\) −2.77575 −0.0980762
\(802\) 0.387873 0.0136963
\(803\) 0 0
\(804\) −11.6267 −0.410043
\(805\) 22.4485 0.791206
\(806\) −5.70194 −0.200842
\(807\) −5.84955 −0.205914
\(808\) −11.5947 −0.407900
\(809\) 23.6121 0.830158 0.415079 0.909785i \(-0.363754\pi\)
0.415079 + 0.909785i \(0.363754\pi\)
\(810\) 0.193937 0.00681424
\(811\) 26.0870 0.916038 0.458019 0.888942i \(-0.348559\pi\)
0.458019 + 0.888942i \(0.348559\pi\)
\(812\) 23.7480 0.833391
\(813\) 5.08840 0.178458
\(814\) 0 0
\(815\) 12.6253 0.442245
\(816\) 17.2722 0.604648
\(817\) 40.0000 1.39942
\(818\) 2.53549 0.0886513
\(819\) 9.92478 0.346800
\(820\) −8.61071 −0.300699
\(821\) 54.4142 1.89907 0.949535 0.313662i \(-0.101556\pi\)
0.949535 + 0.313662i \(0.101556\pi\)
\(822\) 2.68594 0.0936827
\(823\) 0.121269 0.00422716 0.00211358 0.999998i \(-0.499327\pi\)
0.00211358 + 0.999998i \(0.499327\pi\)
\(824\) 2.47768 0.0863142
\(825\) 0 0
\(826\) −6.95254 −0.241910
\(827\) 18.2130 0.633328 0.316664 0.948538i \(-0.397437\pi\)
0.316664 + 0.948538i \(0.397437\pi\)
\(828\) 13.1490 0.456960
\(829\) 13.0738 0.454072 0.227036 0.973886i \(-0.427096\pi\)
0.227036 + 0.973886i \(0.427096\pi\)
\(830\) −2.11142 −0.0732884
\(831\) −1.41090 −0.0489434
\(832\) 21.0668 0.730359
\(833\) 19.3239 0.669534
\(834\) −2.63989 −0.0914119
\(835\) −18.3634 −0.635493
\(836\) 0 0
\(837\) 9.92478 0.343050
\(838\) 1.40105 0.0483984
\(839\) −26.5501 −0.916610 −0.458305 0.888795i \(-0.651543\pi\)
−0.458305 + 0.888795i \(0.651543\pi\)
\(840\) 2.57452 0.0888292
\(841\) −15.9525 −0.550088
\(842\) 5.93937 0.204684
\(843\) 4.38787 0.151126
\(844\) 17.8350 0.613905
\(845\) −4.22425 −0.145319
\(846\) −1.92478 −0.0661752
\(847\) 0 0
\(848\) 17.7480 0.609468
\(849\) −26.5745 −0.912035
\(850\) 0.887166 0.0304295
\(851\) 13.4010 0.459382
\(852\) −19.4763 −0.667246
\(853\) −40.6155 −1.39065 −0.695323 0.718697i \(-0.744739\pi\)
−0.695323 + 0.718697i \(0.744739\pi\)
\(854\) −5.65306 −0.193444
\(855\) 4.31265 0.147490
\(856\) −0.739549 −0.0252773
\(857\) −20.1721 −0.689064 −0.344532 0.938775i \(-0.611962\pi\)
−0.344532 + 0.938775i \(0.611962\pi\)
\(858\) 0 0
\(859\) 21.8035 0.743926 0.371963 0.928248i \(-0.378685\pi\)
0.371963 + 0.928248i \(0.378685\pi\)
\(860\) −18.2012 −0.620657
\(861\) −14.7005 −0.500993
\(862\) 6.56978 0.223767
\(863\) 35.4274 1.20596 0.602981 0.797755i \(-0.293979\pi\)
0.602981 + 0.797755i \(0.293979\pi\)
\(864\) 2.26916 0.0771984
\(865\) 8.57452 0.291542
\(866\) −1.83780 −0.0624508
\(867\) 3.92619 0.133340
\(868\) 65.2506 2.21475
\(869\) 0 0
\(870\) 0.700523 0.0237500
\(871\) −17.5515 −0.594710
\(872\) 8.76116 0.296690
\(873\) 0.0752228 0.00254591
\(874\) −5.60419 −0.189564
\(875\) −3.35026 −0.113260
\(876\) −15.1852 −0.513061
\(877\) 14.0362 0.473969 0.236984 0.971513i \(-0.423841\pi\)
0.236984 + 0.971513i \(0.423841\pi\)
\(878\) 5.71370 0.192828
\(879\) 3.42548 0.115539
\(880\) 0 0
\(881\) −21.0738 −0.709995 −0.354997 0.934867i \(-0.615518\pi\)
−0.354997 + 0.934867i \(0.615518\pi\)
\(882\) 0.819237 0.0275852
\(883\) −42.1476 −1.41838 −0.709190 0.705017i \(-0.750939\pi\)
−0.709190 + 0.705017i \(0.750939\pi\)
\(884\) 26.5933 0.894429
\(885\) 10.7005 0.359694
\(886\) −3.69911 −0.124274
\(887\) −6.93604 −0.232889 −0.116445 0.993197i \(-0.537150\pi\)
−0.116445 + 0.993197i \(0.537150\pi\)
\(888\) 1.53690 0.0515751
\(889\) −48.8284 −1.63765
\(890\) −0.538319 −0.0180445
\(891\) 0 0
\(892\) 13.1490 0.440262
\(893\) −42.8021 −1.43232
\(894\) −0.298062 −0.00996868
\(895\) 14.1768 0.473878
\(896\) 19.8251 0.662311
\(897\) 19.8496 0.662757
\(898\) 6.95765 0.232180
\(899\) 35.8496 1.19565
\(900\) −1.96239 −0.0654130
\(901\) 21.5026 0.716356
\(902\) 0 0
\(903\) −31.0738 −1.03407
\(904\) 4.61071 0.153350
\(905\) −5.22425 −0.173660
\(906\) 1.31124 0.0435629
\(907\) 53.2017 1.76653 0.883267 0.468870i \(-0.155339\pi\)
0.883267 + 0.468870i \(0.155339\pi\)
\(908\) 33.2868 1.10466
\(909\) 15.0884 0.500451
\(910\) 1.92478 0.0638057
\(911\) 36.4749 1.20847 0.604233 0.796808i \(-0.293480\pi\)
0.604233 + 0.796808i \(0.293480\pi\)
\(912\) 16.2835 0.539200
\(913\) 0 0
\(914\) −1.02585 −0.0339322
\(915\) 8.70052 0.287630
\(916\) −50.7269 −1.67606
\(917\) −19.8496 −0.655490
\(918\) 0.887166 0.0292808
\(919\) −9.73340 −0.321075 −0.160538 0.987030i \(-0.551323\pi\)
−0.160538 + 0.987030i \(0.551323\pi\)
\(920\) 5.14903 0.169759
\(921\) 16.6497 0.548628
\(922\) −7.04746 −0.232096
\(923\) −29.4010 −0.967747
\(924\) 0 0
\(925\) −2.00000 −0.0657596
\(926\) 2.04605 0.0672372
\(927\) −3.22425 −0.105898
\(928\) 8.19649 0.269063
\(929\) −24.1768 −0.793215 −0.396607 0.917988i \(-0.629813\pi\)
−0.396607 + 0.917988i \(0.629813\pi\)
\(930\) 1.92478 0.0631159
\(931\) 18.2177 0.597062
\(932\) −37.8251 −1.23900
\(933\) 32.9986 1.08033
\(934\) 3.62672 0.118670
\(935\) 0 0
\(936\) 2.27645 0.0744082
\(937\) 7.48612 0.244561 0.122280 0.992496i \(-0.460979\pi\)
0.122280 + 0.992496i \(0.460979\pi\)
\(938\) −3.84955 −0.125692
\(939\) 15.4010 0.502594
\(940\) 19.4763 0.635246
\(941\) 21.2360 0.692274 0.346137 0.938184i \(-0.387493\pi\)
0.346137 + 0.938184i \(0.387493\pi\)
\(942\) −1.06205 −0.0346034
\(943\) −29.4010 −0.957430
\(944\) 40.4025 1.31499
\(945\) −3.35026 −0.108984
\(946\) 0 0
\(947\) −15.4763 −0.502911 −0.251456 0.967869i \(-0.580909\pi\)
−0.251456 + 0.967869i \(0.580909\pi\)
\(948\) 22.6399 0.735309
\(949\) −22.9234 −0.744124
\(950\) 0.836381 0.0271358
\(951\) 2.15045 0.0697330
\(952\) 11.7772 0.381700
\(953\) 32.0508 1.03823 0.519113 0.854705i \(-0.326262\pi\)
0.519113 + 0.854705i \(0.326262\pi\)
\(954\) 0.911603 0.0295142
\(955\) −16.6253 −0.537982
\(956\) 52.1016 1.68509
\(957\) 0 0
\(958\) 1.80351 0.0582687
\(959\) −46.3996 −1.49832
\(960\) −7.11142 −0.229520
\(961\) 67.5012 2.17746
\(962\) 1.14903 0.0370462
\(963\) 0.962389 0.0310125
\(964\) 56.0263 1.80449
\(965\) 16.3634 0.526758
\(966\) 4.35359 0.140074
\(967\) 17.3766 0.558794 0.279397 0.960176i \(-0.409865\pi\)
0.279397 + 0.960176i \(0.409865\pi\)
\(968\) 0 0
\(969\) 19.7283 0.633764
\(970\) 0.0145884 0.000468407 0
\(971\) 36.2031 1.16181 0.580907 0.813970i \(-0.302698\pi\)
0.580907 + 0.813970i \(0.302698\pi\)
\(972\) −1.96239 −0.0629436
\(973\) 45.6042 1.46200
\(974\) −6.88015 −0.220454
\(975\) −2.96239 −0.0948724
\(976\) 32.8510 1.05153
\(977\) −28.1476 −0.900522 −0.450261 0.892897i \(-0.648669\pi\)
−0.450261 + 0.892897i \(0.648669\pi\)
\(978\) 2.44851 0.0782946
\(979\) 0 0
\(980\) −8.28963 −0.264802
\(981\) −11.4010 −0.364007
\(982\) −4.80492 −0.153331
\(983\) 7.07381 0.225619 0.112810 0.993617i \(-0.464015\pi\)
0.112810 + 0.993617i \(0.464015\pi\)
\(984\) −3.37187 −0.107491
\(985\) 20.4241 0.650765
\(986\) 3.20456 0.102054
\(987\) 33.2506 1.05838
\(988\) 25.0710 0.797614
\(989\) −62.1476 −1.97618
\(990\) 0 0
\(991\) 44.4260 1.41124 0.705619 0.708592i \(-0.250669\pi\)
0.705619 + 0.708592i \(0.250669\pi\)
\(992\) 22.5209 0.715039
\(993\) −14.5501 −0.461733
\(994\) −6.44851 −0.204534
\(995\) −8.62530 −0.273440
\(996\) 21.3649 0.676971
\(997\) 28.4847 0.902120 0.451060 0.892494i \(-0.351046\pi\)
0.451060 + 0.892494i \(0.351046\pi\)
\(998\) 2.74940 0.0870307
\(999\) −2.00000 −0.0632772
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 1815.2.a.m.1.2 3
3.2 odd 2 5445.2.a.z.1.2 3
5.4 even 2 9075.2.a.cf.1.2 3
11.10 odd 2 165.2.a.c.1.2 3
33.32 even 2 495.2.a.e.1.2 3
44.43 even 2 2640.2.a.be.1.1 3
55.32 even 4 825.2.c.g.199.3 6
55.43 even 4 825.2.c.g.199.4 6
55.54 odd 2 825.2.a.k.1.2 3
77.76 even 2 8085.2.a.bk.1.2 3
132.131 odd 2 7920.2.a.cj.1.1 3
165.32 odd 4 2475.2.c.r.199.4 6
165.98 odd 4 2475.2.c.r.199.3 6
165.164 even 2 2475.2.a.bb.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 11.10 odd 2
495.2.a.e.1.2 3 33.32 even 2
825.2.a.k.1.2 3 55.54 odd 2
825.2.c.g.199.3 6 55.32 even 4
825.2.c.g.199.4 6 55.43 even 4
1815.2.a.m.1.2 3 1.1 even 1 trivial
2475.2.a.bb.1.2 3 165.164 even 2
2475.2.c.r.199.3 6 165.98 odd 4
2475.2.c.r.199.4 6 165.32 odd 4
2640.2.a.be.1.1 3 44.43 even 2
5445.2.a.z.1.2 3 3.2 odd 2
7920.2.a.cj.1.1 3 132.131 odd 2
8085.2.a.bk.1.2 3 77.76 even 2
9075.2.a.cf.1.2 3 5.4 even 2