Properties

Label 361.2.a.h.1.3
Level $361$
Weight $2$
Character 361.1
Self dual yes
Analytic conductor $2.883$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 361.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.53209 q^{2} +0.652704 q^{3} +4.41147 q^{4} -1.34730 q^{5} +1.65270 q^{6} +1.53209 q^{7} +6.10607 q^{8} -2.57398 q^{9} +O(q^{10})\) \(q+2.53209 q^{2} +0.652704 q^{3} +4.41147 q^{4} -1.34730 q^{5} +1.65270 q^{6} +1.53209 q^{7} +6.10607 q^{8} -2.57398 q^{9} -3.41147 q^{10} -1.18479 q^{11} +2.87939 q^{12} -2.71688 q^{13} +3.87939 q^{14} -0.879385 q^{15} +6.63816 q^{16} +3.87939 q^{17} -6.51754 q^{18} -5.94356 q^{20} +1.00000 q^{21} -3.00000 q^{22} -5.06418 q^{23} +3.98545 q^{24} -3.18479 q^{25} -6.87939 q^{26} -3.63816 q^{27} +6.75877 q^{28} +4.65270 q^{29} -2.22668 q^{30} +3.83750 q^{31} +4.59627 q^{32} -0.773318 q^{33} +9.82295 q^{34} -2.06418 q^{35} -11.3550 q^{36} +4.10607 q^{37} -1.77332 q^{39} -8.22668 q^{40} +9.98545 q^{41} +2.53209 q^{42} -8.70233 q^{43} -5.22668 q^{44} +3.46791 q^{45} -12.8229 q^{46} +0.573978 q^{47} +4.33275 q^{48} -4.65270 q^{49} -8.06418 q^{50} +2.53209 q^{51} -11.9855 q^{52} -2.94356 q^{53} -9.21213 q^{54} +1.59627 q^{55} +9.35504 q^{56} +11.7811 q^{58} +3.93582 q^{59} -3.87939 q^{60} -4.51754 q^{61} +9.71688 q^{62} -3.94356 q^{63} -1.63816 q^{64} +3.66044 q^{65} -1.95811 q^{66} +3.88713 q^{67} +17.1138 q^{68} -3.30541 q^{69} -5.22668 q^{70} +6.93582 q^{71} -15.7169 q^{72} +6.12836 q^{73} +10.3969 q^{74} -2.07873 q^{75} -1.81521 q^{77} -4.49020 q^{78} -9.80840 q^{79} -8.94356 q^{80} +5.34730 q^{81} +25.2841 q^{82} +12.3182 q^{83} +4.41147 q^{84} -5.22668 q^{85} -22.0351 q^{86} +3.03684 q^{87} -7.23442 q^{88} +2.42602 q^{89} +8.78106 q^{90} -4.16250 q^{91} -22.3405 q^{92} +2.50475 q^{93} +1.45336 q^{94} +3.00000 q^{96} -7.36959 q^{97} -11.7811 q^{98} +3.04963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 6 q^{6} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{5} + 6 q^{6} + 6 q^{8} + 3 q^{12} + 6 q^{14} + 3 q^{15} + 3 q^{16} + 6 q^{17} + 3 q^{18} - 3 q^{20} + 3 q^{21} - 9 q^{22} - 6 q^{23} - 6 q^{24} - 6 q^{25} - 15 q^{26} + 6 q^{27} + 9 q^{28} + 15 q^{29} + 9 q^{31} - 9 q^{33} + 9 q^{34} + 3 q^{35} - 9 q^{36} - 12 q^{39} - 18 q^{40} + 12 q^{41} + 3 q^{42} - 9 q^{44} + 15 q^{45} - 18 q^{46} - 6 q^{47} - 6 q^{48} - 15 q^{49} - 15 q^{50} + 3 q^{51} - 18 q^{52} + 6 q^{53} - 3 q^{54} - 9 q^{55} + 3 q^{56} + 18 q^{58} + 21 q^{59} - 6 q^{60} + 9 q^{61} + 21 q^{62} + 3 q^{63} + 12 q^{64} - 12 q^{65} - 9 q^{66} - 18 q^{67} + 15 q^{68} - 12 q^{69} - 9 q^{70} + 30 q^{71} - 39 q^{72} + 3 q^{74} - 15 q^{75} - 9 q^{77} - 12 q^{78} + 9 q^{79} - 12 q^{80} + 15 q^{81} + 18 q^{82} + 3 q^{84} - 9 q^{85} - 21 q^{86} + 21 q^{87} + 9 q^{88} + 15 q^{89} + 9 q^{90} - 15 q^{91} - 24 q^{92} + 24 q^{93} - 9 q^{94} + 9 q^{96} - 15 q^{97} - 18 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.53209 1.79046 0.895229 0.445607i \(-0.147012\pi\)
0.895229 + 0.445607i \(0.147012\pi\)
\(3\) 0.652704 0.376839 0.188419 0.982089i \(-0.439664\pi\)
0.188419 + 0.982089i \(0.439664\pi\)
\(4\) 4.41147 2.20574
\(5\) −1.34730 −0.602529 −0.301265 0.953541i \(-0.597409\pi\)
−0.301265 + 0.953541i \(0.597409\pi\)
\(6\) 1.65270 0.674713
\(7\) 1.53209 0.579075 0.289538 0.957167i \(-0.406498\pi\)
0.289538 + 0.957167i \(0.406498\pi\)
\(8\) 6.10607 2.15882
\(9\) −2.57398 −0.857993
\(10\) −3.41147 −1.07880
\(11\) −1.18479 −0.357228 −0.178614 0.983919i \(-0.557161\pi\)
−0.178614 + 0.983919i \(0.557161\pi\)
\(12\) 2.87939 0.831207
\(13\) −2.71688 −0.753527 −0.376764 0.926309i \(-0.622963\pi\)
−0.376764 + 0.926309i \(0.622963\pi\)
\(14\) 3.87939 1.03681
\(15\) −0.879385 −0.227056
\(16\) 6.63816 1.65954
\(17\) 3.87939 0.940889 0.470445 0.882430i \(-0.344094\pi\)
0.470445 + 0.882430i \(0.344094\pi\)
\(18\) −6.51754 −1.53620
\(19\) 0 0
\(20\) −5.94356 −1.32902
\(21\) 1.00000 0.218218
\(22\) −3.00000 −0.639602
\(23\) −5.06418 −1.05595 −0.527977 0.849259i \(-0.677049\pi\)
−0.527977 + 0.849259i \(0.677049\pi\)
\(24\) 3.98545 0.813527
\(25\) −3.18479 −0.636959
\(26\) −6.87939 −1.34916
\(27\) −3.63816 −0.700163
\(28\) 6.75877 1.27729
\(29\) 4.65270 0.863985 0.431993 0.901877i \(-0.357811\pi\)
0.431993 + 0.901877i \(0.357811\pi\)
\(30\) −2.22668 −0.406535
\(31\) 3.83750 0.689235 0.344617 0.938743i \(-0.388009\pi\)
0.344617 + 0.938743i \(0.388009\pi\)
\(32\) 4.59627 0.812513
\(33\) −0.773318 −0.134617
\(34\) 9.82295 1.68462
\(35\) −2.06418 −0.348910
\(36\) −11.3550 −1.89251
\(37\) 4.10607 0.675033 0.337517 0.941320i \(-0.390413\pi\)
0.337517 + 0.941320i \(0.390413\pi\)
\(38\) 0 0
\(39\) −1.77332 −0.283958
\(40\) −8.22668 −1.30075
\(41\) 9.98545 1.55947 0.779733 0.626112i \(-0.215355\pi\)
0.779733 + 0.626112i \(0.215355\pi\)
\(42\) 2.53209 0.390710
\(43\) −8.70233 −1.32709 −0.663547 0.748135i \(-0.730950\pi\)
−0.663547 + 0.748135i \(0.730950\pi\)
\(44\) −5.22668 −0.787952
\(45\) 3.46791 0.516966
\(46\) −12.8229 −1.89064
\(47\) 0.573978 0.0837233 0.0418616 0.999123i \(-0.486671\pi\)
0.0418616 + 0.999123i \(0.486671\pi\)
\(48\) 4.33275 0.625378
\(49\) −4.65270 −0.664672
\(50\) −8.06418 −1.14045
\(51\) 2.53209 0.354563
\(52\) −11.9855 −1.66208
\(53\) −2.94356 −0.404329 −0.202165 0.979352i \(-0.564798\pi\)
−0.202165 + 0.979352i \(0.564798\pi\)
\(54\) −9.21213 −1.25361
\(55\) 1.59627 0.215241
\(56\) 9.35504 1.25012
\(57\) 0 0
\(58\) 11.7811 1.54693
\(59\) 3.93582 0.512400 0.256200 0.966624i \(-0.417529\pi\)
0.256200 + 0.966624i \(0.417529\pi\)
\(60\) −3.87939 −0.500826
\(61\) −4.51754 −0.578412 −0.289206 0.957267i \(-0.593391\pi\)
−0.289206 + 0.957267i \(0.593391\pi\)
\(62\) 9.71688 1.23405
\(63\) −3.94356 −0.496842
\(64\) −1.63816 −0.204769
\(65\) 3.66044 0.454022
\(66\) −1.95811 −0.241027
\(67\) 3.88713 0.474888 0.237444 0.971401i \(-0.423690\pi\)
0.237444 + 0.971401i \(0.423690\pi\)
\(68\) 17.1138 2.07535
\(69\) −3.30541 −0.397924
\(70\) −5.22668 −0.624708
\(71\) 6.93582 0.823131 0.411565 0.911380i \(-0.364982\pi\)
0.411565 + 0.911380i \(0.364982\pi\)
\(72\) −15.7169 −1.85225
\(73\) 6.12836 0.717270 0.358635 0.933478i \(-0.383242\pi\)
0.358635 + 0.933478i \(0.383242\pi\)
\(74\) 10.3969 1.20862
\(75\) −2.07873 −0.240031
\(76\) 0 0
\(77\) −1.81521 −0.206862
\(78\) −4.49020 −0.508415
\(79\) −9.80840 −1.10353 −0.551766 0.833999i \(-0.686046\pi\)
−0.551766 + 0.833999i \(0.686046\pi\)
\(80\) −8.94356 −0.999921
\(81\) 5.34730 0.594144
\(82\) 25.2841 2.79216
\(83\) 12.3182 1.35210 0.676049 0.736857i \(-0.263691\pi\)
0.676049 + 0.736857i \(0.263691\pi\)
\(84\) 4.41147 0.481331
\(85\) −5.22668 −0.566913
\(86\) −22.0351 −2.37610
\(87\) 3.03684 0.325583
\(88\) −7.23442 −0.771192
\(89\) 2.42602 0.257158 0.128579 0.991699i \(-0.458958\pi\)
0.128579 + 0.991699i \(0.458958\pi\)
\(90\) 8.78106 0.925605
\(91\) −4.16250 −0.436349
\(92\) −22.3405 −2.32916
\(93\) 2.50475 0.259730
\(94\) 1.45336 0.149903
\(95\) 0 0
\(96\) 3.00000 0.306186
\(97\) −7.36959 −0.748268 −0.374134 0.927375i \(-0.622060\pi\)
−0.374134 + 0.927375i \(0.622060\pi\)
\(98\) −11.7811 −1.19007
\(99\) 3.04963 0.306499
\(100\) −14.0496 −1.40496
\(101\) 2.17024 0.215947 0.107974 0.994154i \(-0.465564\pi\)
0.107974 + 0.994154i \(0.465564\pi\)
\(102\) 6.41147 0.634831
\(103\) 12.4757 1.22926 0.614631 0.788815i \(-0.289305\pi\)
0.614631 + 0.788815i \(0.289305\pi\)
\(104\) −16.5895 −1.62673
\(105\) −1.34730 −0.131483
\(106\) −7.45336 −0.723935
\(107\) 6.68004 0.645784 0.322892 0.946436i \(-0.395345\pi\)
0.322892 + 0.946436i \(0.395345\pi\)
\(108\) −16.0496 −1.54438
\(109\) −9.45336 −0.905468 −0.452734 0.891646i \(-0.649551\pi\)
−0.452734 + 0.891646i \(0.649551\pi\)
\(110\) 4.04189 0.385379
\(111\) 2.68004 0.254379
\(112\) 10.1702 0.960998
\(113\) −1.31046 −0.123278 −0.0616388 0.998099i \(-0.519633\pi\)
−0.0616388 + 0.998099i \(0.519633\pi\)
\(114\) 0 0
\(115\) 6.82295 0.636243
\(116\) 20.5253 1.90572
\(117\) 6.99319 0.646521
\(118\) 9.96585 0.917431
\(119\) 5.94356 0.544846
\(120\) −5.36959 −0.490174
\(121\) −9.59627 −0.872388
\(122\) −11.4388 −1.03562
\(123\) 6.51754 0.587667
\(124\) 16.9290 1.52027
\(125\) 11.0273 0.986315
\(126\) −9.98545 −0.889575
\(127\) 14.5030 1.28693 0.643466 0.765474i \(-0.277496\pi\)
0.643466 + 0.765474i \(0.277496\pi\)
\(128\) −13.3405 −1.17914
\(129\) −5.68004 −0.500100
\(130\) 9.26857 0.812907
\(131\) −19.8084 −1.73067 −0.865334 0.501196i \(-0.832894\pi\)
−0.865334 + 0.501196i \(0.832894\pi\)
\(132\) −3.41147 −0.296931
\(133\) 0 0
\(134\) 9.84255 0.850267
\(135\) 4.90167 0.421869
\(136\) 23.6878 2.03121
\(137\) 10.2044 0.871820 0.435910 0.899990i \(-0.356427\pi\)
0.435910 + 0.899990i \(0.356427\pi\)
\(138\) −8.36959 −0.712466
\(139\) −1.66044 −0.140837 −0.0704185 0.997518i \(-0.522433\pi\)
−0.0704185 + 0.997518i \(0.522433\pi\)
\(140\) −9.10607 −0.769603
\(141\) 0.374638 0.0315502
\(142\) 17.5621 1.47378
\(143\) 3.21894 0.269181
\(144\) −17.0865 −1.42387
\(145\) −6.26857 −0.520576
\(146\) 15.5175 1.28424
\(147\) −3.03684 −0.250474
\(148\) 18.1138 1.48895
\(149\) 11.2071 0.918120 0.459060 0.888405i \(-0.348186\pi\)
0.459060 + 0.888405i \(0.348186\pi\)
\(150\) −5.26352 −0.429764
\(151\) 11.0419 0.898576 0.449288 0.893387i \(-0.351678\pi\)
0.449288 + 0.893387i \(0.351678\pi\)
\(152\) 0 0
\(153\) −9.98545 −0.807276
\(154\) −4.59627 −0.370378
\(155\) −5.17024 −0.415284
\(156\) −7.82295 −0.626337
\(157\) 10.9932 0.877352 0.438676 0.898645i \(-0.355448\pi\)
0.438676 + 0.898645i \(0.355448\pi\)
\(158\) −24.8357 −1.97583
\(159\) −1.92127 −0.152367
\(160\) −6.19253 −0.489563
\(161\) −7.75877 −0.611477
\(162\) 13.5398 1.06379
\(163\) −6.33275 −0.496019 −0.248010 0.968758i \(-0.579776\pi\)
−0.248010 + 0.968758i \(0.579776\pi\)
\(164\) 44.0506 3.43977
\(165\) 1.04189 0.0811110
\(166\) 31.1908 2.42087
\(167\) 13.7784 1.06620 0.533101 0.846051i \(-0.321027\pi\)
0.533101 + 0.846051i \(0.321027\pi\)
\(168\) 6.10607 0.471093
\(169\) −5.61856 −0.432197
\(170\) −13.2344 −1.01503
\(171\) 0 0
\(172\) −38.3901 −2.92722
\(173\) −25.2472 −1.91951 −0.959755 0.280838i \(-0.909388\pi\)
−0.959755 + 0.280838i \(0.909388\pi\)
\(174\) 7.68954 0.582943
\(175\) −4.87939 −0.368847
\(176\) −7.86484 −0.592834
\(177\) 2.56893 0.193092
\(178\) 6.14290 0.460430
\(179\) −5.83069 −0.435806 −0.217903 0.975970i \(-0.569922\pi\)
−0.217903 + 0.975970i \(0.569922\pi\)
\(180\) 15.2986 1.14029
\(181\) −13.5621 −1.00806 −0.504032 0.863685i \(-0.668151\pi\)
−0.504032 + 0.863685i \(0.668151\pi\)
\(182\) −10.5398 −0.781264
\(183\) −2.94862 −0.217968
\(184\) −30.9222 −2.27962
\(185\) −5.53209 −0.406727
\(186\) 6.34224 0.465036
\(187\) −4.59627 −0.336112
\(188\) 2.53209 0.184672
\(189\) −5.57398 −0.405447
\(190\) 0 0
\(191\) −10.2841 −0.744128 −0.372064 0.928207i \(-0.621350\pi\)
−0.372064 + 0.928207i \(0.621350\pi\)
\(192\) −1.06923 −0.0771650
\(193\) −13.8007 −0.993393 −0.496697 0.867924i \(-0.665454\pi\)
−0.496697 + 0.867924i \(0.665454\pi\)
\(194\) −18.6604 −1.33974
\(195\) 2.38919 0.171093
\(196\) −20.5253 −1.46609
\(197\) −7.94087 −0.565764 −0.282882 0.959155i \(-0.591290\pi\)
−0.282882 + 0.959155i \(0.591290\pi\)
\(198\) 7.72193 0.548774
\(199\) 27.0351 1.91647 0.958233 0.285988i \(-0.0923219\pi\)
0.958233 + 0.285988i \(0.0923219\pi\)
\(200\) −19.4466 −1.37508
\(201\) 2.53714 0.178956
\(202\) 5.49525 0.386645
\(203\) 7.12836 0.500312
\(204\) 11.1702 0.782074
\(205\) −13.4534 −0.939624
\(206\) 31.5895 2.20094
\(207\) 13.0351 0.906001
\(208\) −18.0351 −1.25051
\(209\) 0 0
\(210\) −3.41147 −0.235414
\(211\) 8.07192 0.555694 0.277847 0.960625i \(-0.410379\pi\)
0.277847 + 0.960625i \(0.410379\pi\)
\(212\) −12.9855 −0.891845
\(213\) 4.52704 0.310187
\(214\) 16.9145 1.15625
\(215\) 11.7246 0.799613
\(216\) −22.2148 −1.51153
\(217\) 5.87939 0.399119
\(218\) −23.9368 −1.62120
\(219\) 4.00000 0.270295
\(220\) 7.04189 0.474764
\(221\) −10.5398 −0.708986
\(222\) 6.78611 0.455454
\(223\) 15.4757 1.03633 0.518163 0.855282i \(-0.326616\pi\)
0.518163 + 0.855282i \(0.326616\pi\)
\(224\) 7.04189 0.470506
\(225\) 8.19759 0.546506
\(226\) −3.31820 −0.220723
\(227\) −9.87258 −0.655266 −0.327633 0.944805i \(-0.606251\pi\)
−0.327633 + 0.944805i \(0.606251\pi\)
\(228\) 0 0
\(229\) 20.1189 1.32949 0.664746 0.747070i \(-0.268540\pi\)
0.664746 + 0.747070i \(0.268540\pi\)
\(230\) 17.2763 1.13917
\(231\) −1.18479 −0.0779536
\(232\) 28.4097 1.86519
\(233\) 3.53478 0.231571 0.115785 0.993274i \(-0.463061\pi\)
0.115785 + 0.993274i \(0.463061\pi\)
\(234\) 17.7074 1.15757
\(235\) −0.773318 −0.0504457
\(236\) 17.3628 1.13022
\(237\) −6.40198 −0.415853
\(238\) 15.0496 0.975523
\(239\) 11.9736 0.774507 0.387254 0.921973i \(-0.373424\pi\)
0.387254 + 0.921973i \(0.373424\pi\)
\(240\) −5.83750 −0.376809
\(241\) −12.9017 −0.831070 −0.415535 0.909577i \(-0.636406\pi\)
−0.415535 + 0.909577i \(0.636406\pi\)
\(242\) −24.2986 −1.56197
\(243\) 14.4047 0.924060
\(244\) −19.9290 −1.27582
\(245\) 6.26857 0.400484
\(246\) 16.5030 1.05219
\(247\) 0 0
\(248\) 23.4320 1.48793
\(249\) 8.04013 0.509523
\(250\) 27.9222 1.76596
\(251\) 14.3628 0.906571 0.453285 0.891366i \(-0.350252\pi\)
0.453285 + 0.891366i \(0.350252\pi\)
\(252\) −17.3969 −1.09590
\(253\) 6.00000 0.377217
\(254\) 36.7229 2.30420
\(255\) −3.41147 −0.213635
\(256\) −30.5030 −1.90644
\(257\) −4.97771 −0.310501 −0.155251 0.987875i \(-0.549618\pi\)
−0.155251 + 0.987875i \(0.549618\pi\)
\(258\) −14.3824 −0.895408
\(259\) 6.29086 0.390895
\(260\) 16.1480 1.00145
\(261\) −11.9760 −0.741293
\(262\) −50.1566 −3.09869
\(263\) −24.0428 −1.48254 −0.741272 0.671205i \(-0.765777\pi\)
−0.741272 + 0.671205i \(0.765777\pi\)
\(264\) −4.72193 −0.290615
\(265\) 3.96585 0.243620
\(266\) 0 0
\(267\) 1.58347 0.0969070
\(268\) 17.1480 1.04748
\(269\) 13.1111 0.799399 0.399700 0.916646i \(-0.369114\pi\)
0.399700 + 0.916646i \(0.369114\pi\)
\(270\) 12.4115 0.755338
\(271\) −26.5699 −1.61400 −0.807002 0.590549i \(-0.798911\pi\)
−0.807002 + 0.590549i \(0.798911\pi\)
\(272\) 25.7520 1.56144
\(273\) −2.71688 −0.164433
\(274\) 25.8384 1.56096
\(275\) 3.77332 0.227540
\(276\) −14.5817 −0.877716
\(277\) −16.5107 −0.992034 −0.496017 0.868313i \(-0.665205\pi\)
−0.496017 + 0.868313i \(0.665205\pi\)
\(278\) −4.20439 −0.252163
\(279\) −9.87763 −0.591358
\(280\) −12.6040 −0.753234
\(281\) 19.3901 1.15672 0.578359 0.815783i \(-0.303693\pi\)
0.578359 + 0.815783i \(0.303693\pi\)
\(282\) 0.948615 0.0564892
\(283\) −11.3105 −0.672337 −0.336169 0.941802i \(-0.609131\pi\)
−0.336169 + 0.941802i \(0.609131\pi\)
\(284\) 30.5972 1.81561
\(285\) 0 0
\(286\) 8.15064 0.481958
\(287\) 15.2986 0.903048
\(288\) −11.8307 −0.697130
\(289\) −1.95037 −0.114728
\(290\) −15.8726 −0.932070
\(291\) −4.81016 −0.281976
\(292\) 27.0351 1.58211
\(293\) −3.89899 −0.227781 −0.113891 0.993493i \(-0.536331\pi\)
−0.113891 + 0.993493i \(0.536331\pi\)
\(294\) −7.68954 −0.448463
\(295\) −5.30272 −0.308736
\(296\) 25.0719 1.45728
\(297\) 4.31046 0.250118
\(298\) 28.3773 1.64385
\(299\) 13.7588 0.795690
\(300\) −9.17024 −0.529444
\(301\) −13.3327 −0.768487
\(302\) 27.9590 1.60886
\(303\) 1.41653 0.0813773
\(304\) 0 0
\(305\) 6.08647 0.348510
\(306\) −25.2841 −1.44539
\(307\) −23.1753 −1.32268 −0.661342 0.750084i \(-0.730013\pi\)
−0.661342 + 0.750084i \(0.730013\pi\)
\(308\) −8.00774 −0.456283
\(309\) 8.14290 0.463234
\(310\) −13.0915 −0.743548
\(311\) −3.46110 −0.196261 −0.0981306 0.995174i \(-0.531286\pi\)
−0.0981306 + 0.995174i \(0.531286\pi\)
\(312\) −10.8280 −0.613015
\(313\) −22.8898 −1.29381 −0.646904 0.762571i \(-0.723937\pi\)
−0.646904 + 0.762571i \(0.723937\pi\)
\(314\) 27.8357 1.57086
\(315\) 5.31315 0.299362
\(316\) −43.2695 −2.43410
\(317\) −26.1206 −1.46708 −0.733540 0.679646i \(-0.762133\pi\)
−0.733540 + 0.679646i \(0.762133\pi\)
\(318\) −4.86484 −0.272807
\(319\) −5.51249 −0.308640
\(320\) 2.20708 0.123380
\(321\) 4.36009 0.243356
\(322\) −19.6459 −1.09482
\(323\) 0 0
\(324\) 23.5895 1.31053
\(325\) 8.65270 0.479966
\(326\) −16.0351 −0.888101
\(327\) −6.17024 −0.341215
\(328\) 60.9718 3.36661
\(329\) 0.879385 0.0484821
\(330\) 2.63816 0.145226
\(331\) −19.0446 −1.04678 −0.523392 0.852092i \(-0.675334\pi\)
−0.523392 + 0.852092i \(0.675334\pi\)
\(332\) 54.3414 2.98237
\(333\) −10.5689 −0.579174
\(334\) 34.8881 1.90899
\(335\) −5.23711 −0.286134
\(336\) 6.63816 0.362141
\(337\) −1.70140 −0.0926812 −0.0463406 0.998926i \(-0.514756\pi\)
−0.0463406 + 0.998926i \(0.514756\pi\)
\(338\) −14.2267 −0.773829
\(339\) −0.855342 −0.0464558
\(340\) −23.0574 −1.25046
\(341\) −4.54664 −0.246214
\(342\) 0 0
\(343\) −17.8530 −0.963970
\(344\) −53.1370 −2.86496
\(345\) 4.45336 0.239761
\(346\) −63.9282 −3.43680
\(347\) −4.90167 −0.263136 −0.131568 0.991307i \(-0.542001\pi\)
−0.131568 + 0.991307i \(0.542001\pi\)
\(348\) 13.3969 0.718151
\(349\) −28.1293 −1.50573 −0.752863 0.658177i \(-0.771328\pi\)
−0.752863 + 0.658177i \(0.771328\pi\)
\(350\) −12.3550 −0.660405
\(351\) 9.88444 0.527592
\(352\) −5.44562 −0.290253
\(353\) −8.31996 −0.442827 −0.221413 0.975180i \(-0.571067\pi\)
−0.221413 + 0.975180i \(0.571067\pi\)
\(354\) 6.50475 0.345723
\(355\) −9.34461 −0.495960
\(356\) 10.7023 0.567223
\(357\) 3.87939 0.205319
\(358\) −14.7638 −0.780292
\(359\) 24.9290 1.31570 0.657852 0.753148i \(-0.271465\pi\)
0.657852 + 0.753148i \(0.271465\pi\)
\(360\) 21.1753 1.11604
\(361\) 0 0
\(362\) −34.3405 −1.80490
\(363\) −6.26352 −0.328749
\(364\) −18.3628 −0.962471
\(365\) −8.25671 −0.432176
\(366\) −7.46616 −0.390262
\(367\) −2.58584 −0.134980 −0.0674898 0.997720i \(-0.521499\pi\)
−0.0674898 + 0.997720i \(0.521499\pi\)
\(368\) −33.6168 −1.75240
\(369\) −25.7023 −1.33801
\(370\) −14.0077 −0.728228
\(371\) −4.50980 −0.234137
\(372\) 11.0496 0.572897
\(373\) 23.3833 1.21074 0.605371 0.795943i \(-0.293025\pi\)
0.605371 + 0.795943i \(0.293025\pi\)
\(374\) −11.6382 −0.601795
\(375\) 7.19759 0.371682
\(376\) 3.50475 0.180744
\(377\) −12.6408 −0.651037
\(378\) −14.1138 −0.725936
\(379\) −25.4388 −1.30670 −0.653352 0.757054i \(-0.726638\pi\)
−0.653352 + 0.757054i \(0.726638\pi\)
\(380\) 0 0
\(381\) 9.46616 0.484966
\(382\) −26.0401 −1.33233
\(383\) 27.4807 1.40420 0.702099 0.712079i \(-0.252246\pi\)
0.702099 + 0.712079i \(0.252246\pi\)
\(384\) −8.70739 −0.444347
\(385\) 2.44562 0.124640
\(386\) −34.9445 −1.77863
\(387\) 22.3996 1.13864
\(388\) −32.5107 −1.65048
\(389\) −3.34224 −0.169458 −0.0847292 0.996404i \(-0.527003\pi\)
−0.0847292 + 0.996404i \(0.527003\pi\)
\(390\) 6.04963 0.306335
\(391\) −19.6459 −0.993536
\(392\) −28.4097 −1.43491
\(393\) −12.9290 −0.652183
\(394\) −20.1070 −1.01298
\(395\) 13.2148 0.664910
\(396\) 13.4534 0.676057
\(397\) −13.1233 −0.658640 −0.329320 0.944218i \(-0.606819\pi\)
−0.329320 + 0.944218i \(0.606819\pi\)
\(398\) 68.4552 3.43135
\(399\) 0 0
\(400\) −21.1411 −1.05706
\(401\) −17.1138 −0.854623 −0.427311 0.904105i \(-0.640539\pi\)
−0.427311 + 0.904105i \(0.640539\pi\)
\(402\) 6.42427 0.320413
\(403\) −10.4260 −0.519357
\(404\) 9.57398 0.476323
\(405\) −7.20439 −0.357989
\(406\) 18.0496 0.895788
\(407\) −4.86484 −0.241141
\(408\) 15.4611 0.765439
\(409\) 8.79797 0.435032 0.217516 0.976057i \(-0.430205\pi\)
0.217516 + 0.976057i \(0.430205\pi\)
\(410\) −34.0651 −1.68236
\(411\) 6.66044 0.328535
\(412\) 55.0360 2.71143
\(413\) 6.03003 0.296718
\(414\) 33.0060 1.62216
\(415\) −16.5963 −0.814679
\(416\) −12.4875 −0.612251
\(417\) −1.08378 −0.0530728
\(418\) 0 0
\(419\) 6.84018 0.334165 0.167082 0.985943i \(-0.446565\pi\)
0.167082 + 0.985943i \(0.446565\pi\)
\(420\) −5.94356 −0.290016
\(421\) 4.82295 0.235056 0.117528 0.993070i \(-0.462503\pi\)
0.117528 + 0.993070i \(0.462503\pi\)
\(422\) 20.4388 0.994946
\(423\) −1.47741 −0.0718340
\(424\) −17.9736 −0.872875
\(425\) −12.3550 −0.599307
\(426\) 11.4629 0.555377
\(427\) −6.92127 −0.334944
\(428\) 29.4688 1.42443
\(429\) 2.10101 0.101438
\(430\) 29.6878 1.43167
\(431\) −1.30365 −0.0627947 −0.0313974 0.999507i \(-0.509996\pi\)
−0.0313974 + 0.999507i \(0.509996\pi\)
\(432\) −24.1506 −1.16195
\(433\) −19.8239 −0.952675 −0.476337 0.879263i \(-0.658036\pi\)
−0.476337 + 0.879263i \(0.658036\pi\)
\(434\) 14.8871 0.714605
\(435\) −4.09152 −0.196173
\(436\) −41.7033 −1.99722
\(437\) 0 0
\(438\) 10.1284 0.483952
\(439\) 34.5672 1.64980 0.824901 0.565278i \(-0.191231\pi\)
0.824901 + 0.565278i \(0.191231\pi\)
\(440\) 9.74691 0.464666
\(441\) 11.9760 0.570284
\(442\) −26.6878 −1.26941
\(443\) 17.0101 0.808174 0.404087 0.914720i \(-0.367589\pi\)
0.404087 + 0.914720i \(0.367589\pi\)
\(444\) 11.8229 0.561092
\(445\) −3.26857 −0.154945
\(446\) 39.1857 1.85550
\(447\) 7.31490 0.345983
\(448\) −2.50980 −0.118577
\(449\) −37.4097 −1.76547 −0.882737 0.469868i \(-0.844302\pi\)
−0.882737 + 0.469868i \(0.844302\pi\)
\(450\) 20.7570 0.978495
\(451\) −11.8307 −0.557085
\(452\) −5.78106 −0.271918
\(453\) 7.20708 0.338618
\(454\) −24.9982 −1.17323
\(455\) 5.60813 0.262913
\(456\) 0 0
\(457\) 9.11112 0.426200 0.213100 0.977030i \(-0.431644\pi\)
0.213100 + 0.977030i \(0.431644\pi\)
\(458\) 50.9427 2.38040
\(459\) −14.1138 −0.658776
\(460\) 30.0993 1.40339
\(461\) −24.4483 −1.13867 −0.569336 0.822105i \(-0.692800\pi\)
−0.569336 + 0.822105i \(0.692800\pi\)
\(462\) −3.00000 −0.139573
\(463\) −0.250725 −0.0116522 −0.00582609 0.999983i \(-0.501855\pi\)
−0.00582609 + 0.999983i \(0.501855\pi\)
\(464\) 30.8854 1.43382
\(465\) −3.37464 −0.156495
\(466\) 8.95037 0.414618
\(467\) 15.3618 0.710861 0.355431 0.934703i \(-0.384334\pi\)
0.355431 + 0.934703i \(0.384334\pi\)
\(468\) 30.8503 1.42606
\(469\) 5.95542 0.274996
\(470\) −1.95811 −0.0903209
\(471\) 7.17530 0.330620
\(472\) 24.0324 1.10618
\(473\) 10.3105 0.474075
\(474\) −16.2104 −0.744567
\(475\) 0 0
\(476\) 26.2199 1.20179
\(477\) 7.57667 0.346912
\(478\) 30.3182 1.38672
\(479\) −0.719246 −0.0328632 −0.0164316 0.999865i \(-0.505231\pi\)
−0.0164316 + 0.999865i \(0.505231\pi\)
\(480\) −4.04189 −0.184486
\(481\) −11.1557 −0.508656
\(482\) −32.6682 −1.48800
\(483\) −5.06418 −0.230428
\(484\) −42.3337 −1.92426
\(485\) 9.92902 0.450853
\(486\) 36.4739 1.65449
\(487\) −11.7469 −0.532303 −0.266152 0.963931i \(-0.585752\pi\)
−0.266152 + 0.963931i \(0.585752\pi\)
\(488\) −27.5844 −1.24869
\(489\) −4.13341 −0.186919
\(490\) 15.8726 0.717050
\(491\) 0.0888306 0.00400887 0.00200443 0.999998i \(-0.499362\pi\)
0.00200443 + 0.999998i \(0.499362\pi\)
\(492\) 28.7520 1.29624
\(493\) 18.0496 0.812914
\(494\) 0 0
\(495\) −4.10876 −0.184675
\(496\) 25.4739 1.14381
\(497\) 10.6263 0.476655
\(498\) 20.3583 0.912279
\(499\) 14.6905 0.657636 0.328818 0.944393i \(-0.393350\pi\)
0.328818 + 0.944393i \(0.393350\pi\)
\(500\) 48.6468 2.17555
\(501\) 8.99319 0.401786
\(502\) 36.3678 1.62318
\(503\) −4.90404 −0.218660 −0.109330 0.994005i \(-0.534871\pi\)
−0.109330 + 0.994005i \(0.534871\pi\)
\(504\) −24.0797 −1.07259
\(505\) −2.92396 −0.130115
\(506\) 15.1925 0.675390
\(507\) −3.66725 −0.162868
\(508\) 63.9796 2.83863
\(509\) −6.41384 −0.284288 −0.142144 0.989846i \(-0.545400\pi\)
−0.142144 + 0.989846i \(0.545400\pi\)
\(510\) −8.63816 −0.382504
\(511\) 9.38919 0.415353
\(512\) −50.5553 −2.23425
\(513\) 0 0
\(514\) −12.6040 −0.555939
\(515\) −16.8084 −0.740667
\(516\) −25.0574 −1.10309
\(517\) −0.680045 −0.0299083
\(518\) 15.9290 0.699881
\(519\) −16.4789 −0.723346
\(520\) 22.3509 0.980153
\(521\) 35.8135 1.56902 0.784508 0.620119i \(-0.212916\pi\)
0.784508 + 0.620119i \(0.212916\pi\)
\(522\) −30.3242 −1.32725
\(523\) −38.7725 −1.69540 −0.847701 0.530474i \(-0.822014\pi\)
−0.847701 + 0.530474i \(0.822014\pi\)
\(524\) −87.3842 −3.81740
\(525\) −3.18479 −0.138996
\(526\) −60.8786 −2.65443
\(527\) 14.8871 0.648493
\(528\) −5.13341 −0.223403
\(529\) 2.64590 0.115039
\(530\) 10.0419 0.436192
\(531\) −10.1307 −0.439636
\(532\) 0 0
\(533\) −27.1293 −1.17510
\(534\) 4.00950 0.173508
\(535\) −9.00000 −0.389104
\(536\) 23.7351 1.02520
\(537\) −3.80571 −0.164229
\(538\) 33.1985 1.43129
\(539\) 5.51249 0.237440
\(540\) 21.6236 0.930532
\(541\) 9.49020 0.408016 0.204008 0.978969i \(-0.434603\pi\)
0.204008 + 0.978969i \(0.434603\pi\)
\(542\) −67.2772 −2.88981
\(543\) −8.85204 −0.379878
\(544\) 17.8307 0.764484
\(545\) 12.7365 0.545571
\(546\) −6.87939 −0.294411
\(547\) −14.2121 −0.607667 −0.303833 0.952725i \(-0.598267\pi\)
−0.303833 + 0.952725i \(0.598267\pi\)
\(548\) 45.0164 1.92301
\(549\) 11.6281 0.496273
\(550\) 9.55438 0.407400
\(551\) 0 0
\(552\) −20.1830 −0.859047
\(553\) −15.0273 −0.639028
\(554\) −41.8066 −1.77619
\(555\) −3.61081 −0.153271
\(556\) −7.32501 −0.310650
\(557\) 22.5398 0.955043 0.477522 0.878620i \(-0.341535\pi\)
0.477522 + 0.878620i \(0.341535\pi\)
\(558\) −25.0110 −1.05880
\(559\) 23.6432 1.00000
\(560\) −13.7023 −0.579029
\(561\) −3.00000 −0.126660
\(562\) 49.0975 2.07105
\(563\) 42.9718 1.81105 0.905524 0.424296i \(-0.139478\pi\)
0.905524 + 0.424296i \(0.139478\pi\)
\(564\) 1.65270 0.0695914
\(565\) 1.76558 0.0742784
\(566\) −28.6391 −1.20379
\(567\) 8.19253 0.344054
\(568\) 42.3506 1.77699
\(569\) −7.42696 −0.311354 −0.155677 0.987808i \(-0.549756\pi\)
−0.155677 + 0.987808i \(0.549756\pi\)
\(570\) 0 0
\(571\) 4.04458 0.169260 0.0846301 0.996412i \(-0.473029\pi\)
0.0846301 + 0.996412i \(0.473029\pi\)
\(572\) 14.2003 0.593743
\(573\) −6.71244 −0.280416
\(574\) 38.7374 1.61687
\(575\) 16.1284 0.672599
\(576\) 4.21658 0.175691
\(577\) 3.23442 0.134651 0.0673254 0.997731i \(-0.478553\pi\)
0.0673254 + 0.997731i \(0.478553\pi\)
\(578\) −4.93851 −0.205415
\(579\) −9.00774 −0.374349
\(580\) −27.6536 −1.14825
\(581\) 18.8726 0.782966
\(582\) −12.1797 −0.504866
\(583\) 3.48751 0.144438
\(584\) 37.4201 1.54846
\(585\) −9.42190 −0.389548
\(586\) −9.87258 −0.407832
\(587\) −40.8084 −1.68434 −0.842171 0.539210i \(-0.818723\pi\)
−0.842171 + 0.539210i \(0.818723\pi\)
\(588\) −13.3969 −0.552480
\(589\) 0 0
\(590\) −13.4270 −0.552779
\(591\) −5.18304 −0.213202
\(592\) 27.2567 1.12024
\(593\) −11.0642 −0.454351 −0.227176 0.973854i \(-0.572949\pi\)
−0.227176 + 0.973854i \(0.572949\pi\)
\(594\) 10.9145 0.447826
\(595\) −8.00774 −0.328285
\(596\) 49.4397 2.02513
\(597\) 17.6459 0.722198
\(598\) 34.8384 1.42465
\(599\) 44.5577 1.82058 0.910289 0.413974i \(-0.135860\pi\)
0.910289 + 0.413974i \(0.135860\pi\)
\(600\) −12.6928 −0.518183
\(601\) 4.99907 0.203916 0.101958 0.994789i \(-0.467489\pi\)
0.101958 + 0.994789i \(0.467489\pi\)
\(602\) −33.7597 −1.37594
\(603\) −10.0054 −0.407450
\(604\) 48.7110 1.98202
\(605\) 12.9290 0.525639
\(606\) 3.58677 0.145703
\(607\) 31.1881 1.26589 0.632943 0.774199i \(-0.281847\pi\)
0.632943 + 0.774199i \(0.281847\pi\)
\(608\) 0 0
\(609\) 4.65270 0.188537
\(610\) 15.4115 0.623992
\(611\) −1.55943 −0.0630878
\(612\) −44.0506 −1.78064
\(613\) 16.3696 0.661161 0.330581 0.943778i \(-0.392755\pi\)
0.330581 + 0.943778i \(0.392755\pi\)
\(614\) −58.6819 −2.36821
\(615\) −8.78106 −0.354086
\(616\) −11.0838 −0.446578
\(617\) 16.0583 0.646483 0.323241 0.946317i \(-0.395227\pi\)
0.323241 + 0.946317i \(0.395227\pi\)
\(618\) 20.6186 0.829400
\(619\) 23.8425 0.958313 0.479156 0.877730i \(-0.340943\pi\)
0.479156 + 0.877730i \(0.340943\pi\)
\(620\) −22.8084 −0.916007
\(621\) 18.4243 0.739340
\(622\) −8.76382 −0.351397
\(623\) 3.71688 0.148914
\(624\) −11.7716 −0.471240
\(625\) 1.06687 0.0426746
\(626\) −57.9590 −2.31651
\(627\) 0 0
\(628\) 48.4962 1.93521
\(629\) 15.9290 0.635131
\(630\) 13.4534 0.535995
\(631\) 21.4730 0.854825 0.427413 0.904057i \(-0.359425\pi\)
0.427413 + 0.904057i \(0.359425\pi\)
\(632\) −59.8907 −2.38233
\(633\) 5.26857 0.209407
\(634\) −66.1397 −2.62674
\(635\) −19.5398 −0.775414
\(636\) −8.47565 −0.336081
\(637\) 12.6408 0.500848
\(638\) −13.9581 −0.552607
\(639\) −17.8527 −0.706240
\(640\) 17.9736 0.710469
\(641\) −12.7537 −0.503742 −0.251871 0.967761i \(-0.581046\pi\)
−0.251871 + 0.967761i \(0.581046\pi\)
\(642\) 11.0401 0.435719
\(643\) 28.6081 1.12819 0.564097 0.825708i \(-0.309224\pi\)
0.564097 + 0.825708i \(0.309224\pi\)
\(644\) −34.2276 −1.34876
\(645\) 7.65270 0.301325
\(646\) 0 0
\(647\) 16.7128 0.657046 0.328523 0.944496i \(-0.393449\pi\)
0.328523 + 0.944496i \(0.393449\pi\)
\(648\) 32.6509 1.28265
\(649\) −4.66313 −0.183044
\(650\) 21.9094 0.859358
\(651\) 3.83750 0.150403
\(652\) −27.9368 −1.09409
\(653\) 27.0000 1.05659 0.528296 0.849060i \(-0.322831\pi\)
0.528296 + 0.849060i \(0.322831\pi\)
\(654\) −15.6236 −0.610931
\(655\) 26.6878 1.04278
\(656\) 66.2850 2.58799
\(657\) −15.7743 −0.615412
\(658\) 2.22668 0.0868051
\(659\) 43.9009 1.71013 0.855067 0.518517i \(-0.173516\pi\)
0.855067 + 0.518517i \(0.173516\pi\)
\(660\) 4.59627 0.178909
\(661\) 10.7561 0.418363 0.209182 0.977877i \(-0.432920\pi\)
0.209182 + 0.977877i \(0.432920\pi\)
\(662\) −48.2226 −1.87422
\(663\) −6.87939 −0.267173
\(664\) 75.2158 2.91894
\(665\) 0 0
\(666\) −26.7615 −1.03699
\(667\) −23.5621 −0.912329
\(668\) 60.7829 2.35176
\(669\) 10.1010 0.390528
\(670\) −13.2608 −0.512311
\(671\) 5.35235 0.206625
\(672\) 4.59627 0.177305
\(673\) −4.65776 −0.179543 −0.0897717 0.995962i \(-0.528614\pi\)
−0.0897717 + 0.995962i \(0.528614\pi\)
\(674\) −4.30810 −0.165942
\(675\) 11.5868 0.445975
\(676\) −24.7861 −0.953312
\(677\) −3.26857 −0.125621 −0.0628107 0.998025i \(-0.520006\pi\)
−0.0628107 + 0.998025i \(0.520006\pi\)
\(678\) −2.16580 −0.0831771
\(679\) −11.2909 −0.433303
\(680\) −31.9145 −1.22386
\(681\) −6.44387 −0.246930
\(682\) −11.5125 −0.440836
\(683\) −6.21894 −0.237961 −0.118981 0.992897i \(-0.537963\pi\)
−0.118981 + 0.992897i \(0.537963\pi\)
\(684\) 0 0
\(685\) −13.7483 −0.525297
\(686\) −45.2053 −1.72595
\(687\) 13.1317 0.501004
\(688\) −57.7674 −2.20236
\(689\) 7.99731 0.304673
\(690\) 11.2763 0.429282
\(691\) 22.2175 0.845194 0.422597 0.906318i \(-0.361119\pi\)
0.422597 + 0.906318i \(0.361119\pi\)
\(692\) −111.377 −4.23393
\(693\) 4.67230 0.177486
\(694\) −12.4115 −0.471133
\(695\) 2.23711 0.0848584
\(696\) 18.5431 0.702875
\(697\) 38.7374 1.46728
\(698\) −71.2259 −2.69594
\(699\) 2.30716 0.0872649
\(700\) −21.5253 −0.813579
\(701\) −27.7725 −1.04895 −0.524476 0.851425i \(-0.675739\pi\)
−0.524476 + 0.851425i \(0.675739\pi\)
\(702\) 25.0283 0.944631
\(703\) 0 0
\(704\) 1.94087 0.0731495
\(705\) −0.504748 −0.0190099
\(706\) −21.0669 −0.792862
\(707\) 3.32501 0.125050
\(708\) 11.3327 0.425911
\(709\) −6.10782 −0.229384 −0.114692 0.993401i \(-0.536588\pi\)
−0.114692 + 0.993401i \(0.536588\pi\)
\(710\) −23.6614 −0.887996
\(711\) 25.2466 0.946822
\(712\) 14.8135 0.555158
\(713\) −19.4338 −0.727800
\(714\) 9.82295 0.367615
\(715\) −4.33687 −0.162190
\(716\) −25.7219 −0.961274
\(717\) 7.81521 0.291864
\(718\) 63.1225 2.35571
\(719\) 38.7238 1.44415 0.722077 0.691813i \(-0.243188\pi\)
0.722077 + 0.691813i \(0.243188\pi\)
\(720\) 23.0205 0.857925
\(721\) 19.1138 0.711835
\(722\) 0 0
\(723\) −8.42097 −0.313179
\(724\) −59.8289 −2.22352
\(725\) −14.8179 −0.550323
\(726\) −15.8598 −0.588612
\(727\) −11.0779 −0.410857 −0.205428 0.978672i \(-0.565859\pi\)
−0.205428 + 0.978672i \(0.565859\pi\)
\(728\) −25.4165 −0.941999
\(729\) −6.63991 −0.245923
\(730\) −20.9067 −0.773793
\(731\) −33.7597 −1.24865
\(732\) −13.0077 −0.480780
\(733\) −15.8075 −0.583862 −0.291931 0.956439i \(-0.594298\pi\)
−0.291931 + 0.956439i \(0.594298\pi\)
\(734\) −6.54757 −0.241675
\(735\) 4.09152 0.150918
\(736\) −23.2763 −0.857976
\(737\) −4.60544 −0.169643
\(738\) −65.0806 −2.39565
\(739\) 1.54933 0.0569928 0.0284964 0.999594i \(-0.490928\pi\)
0.0284964 + 0.999594i \(0.490928\pi\)
\(740\) −24.4047 −0.897133
\(741\) 0 0
\(742\) −11.4192 −0.419213
\(743\) −38.1634 −1.40008 −0.700040 0.714103i \(-0.746835\pi\)
−0.700040 + 0.714103i \(0.746835\pi\)
\(744\) 15.2942 0.560711
\(745\) −15.0993 −0.553194
\(746\) 59.2086 2.16778
\(747\) −31.7068 −1.16009
\(748\) −20.2763 −0.741375
\(749\) 10.2344 0.373958
\(750\) 18.2249 0.665480
\(751\) 25.3482 0.924970 0.462485 0.886627i \(-0.346958\pi\)
0.462485 + 0.886627i \(0.346958\pi\)
\(752\) 3.81016 0.138942
\(753\) 9.37464 0.341631
\(754\) −32.0077 −1.16565
\(755\) −14.8767 −0.541418
\(756\) −24.5895 −0.894310
\(757\) 42.3705 1.53998 0.769991 0.638054i \(-0.220260\pi\)
0.769991 + 0.638054i \(0.220260\pi\)
\(758\) −64.4133 −2.33960
\(759\) 3.91622 0.142150
\(760\) 0 0
\(761\) −2.85710 −0.103570 −0.0517848 0.998658i \(-0.516491\pi\)
−0.0517848 + 0.998658i \(0.516491\pi\)
\(762\) 23.9691 0.868311
\(763\) −14.4834 −0.524334
\(764\) −45.3678 −1.64135
\(765\) 13.4534 0.486407
\(766\) 69.5836 2.51416
\(767\) −10.6932 −0.386108
\(768\) −19.9094 −0.718419
\(769\) 19.1206 0.689507 0.344754 0.938693i \(-0.387963\pi\)
0.344754 + 0.938693i \(0.387963\pi\)
\(770\) 6.19253 0.223163
\(771\) −3.24897 −0.117009
\(772\) −60.8813 −2.19116
\(773\) −2.51485 −0.0904530 −0.0452265 0.998977i \(-0.514401\pi\)
−0.0452265 + 0.998977i \(0.514401\pi\)
\(774\) 56.7178 2.03868
\(775\) −12.2216 −0.439014
\(776\) −44.9992 −1.61538
\(777\) 4.10607 0.147304
\(778\) −8.46286 −0.303408
\(779\) 0 0
\(780\) 10.5398 0.377386
\(781\) −8.21751 −0.294046
\(782\) −49.7452 −1.77888
\(783\) −16.9273 −0.604931
\(784\) −30.8854 −1.10305
\(785\) −14.8111 −0.528630
\(786\) −32.7374 −1.16770
\(787\) −2.72605 −0.0971733 −0.0485866 0.998819i \(-0.515472\pi\)
−0.0485866 + 0.998819i \(0.515472\pi\)
\(788\) −35.0310 −1.24793
\(789\) −15.6928 −0.558680
\(790\) 33.4611 1.19049
\(791\) −2.00774 −0.0713870
\(792\) 18.6212 0.661677
\(793\) 12.2736 0.435849
\(794\) −33.2294 −1.17927
\(795\) 2.58853 0.0918056
\(796\) 119.265 4.22722
\(797\) −22.0327 −0.780439 −0.390219 0.920722i \(-0.627601\pi\)
−0.390219 + 0.920722i \(0.627601\pi\)
\(798\) 0 0
\(799\) 2.22668 0.0787743
\(800\) −14.6382 −0.517537
\(801\) −6.24453 −0.220640
\(802\) −43.3337 −1.53017
\(803\) −7.26083 −0.256229
\(804\) 11.1925 0.394730
\(805\) 10.4534 0.368433
\(806\) −26.3996 −0.929887
\(807\) 8.55768 0.301244
\(808\) 13.2517 0.466192
\(809\) 54.7205 1.92387 0.961935 0.273278i \(-0.0881077\pi\)
0.961935 + 0.273278i \(0.0881077\pi\)
\(810\) −18.2422 −0.640964
\(811\) 2.31046 0.0811312 0.0405656 0.999177i \(-0.487084\pi\)
0.0405656 + 0.999177i \(0.487084\pi\)
\(812\) 31.4466 1.10356
\(813\) −17.3422 −0.608219
\(814\) −12.3182 −0.431753
\(815\) 8.53209 0.298866
\(816\) 16.8084 0.588412
\(817\) 0 0
\(818\) 22.2772 0.778906
\(819\) 10.7142 0.374384
\(820\) −59.3492 −2.07256
\(821\) 1.11112 0.0387783 0.0193892 0.999812i \(-0.493828\pi\)
0.0193892 + 0.999812i \(0.493828\pi\)
\(822\) 16.8648 0.588229
\(823\) 20.6477 0.719732 0.359866 0.933004i \(-0.382822\pi\)
0.359866 + 0.933004i \(0.382822\pi\)
\(824\) 76.1772 2.65376
\(825\) 2.46286 0.0857457
\(826\) 15.2686 0.531262
\(827\) −36.3054 −1.26246 −0.631231 0.775595i \(-0.717450\pi\)
−0.631231 + 0.775595i \(0.717450\pi\)
\(828\) 57.5039 1.99840
\(829\) 7.14971 0.248320 0.124160 0.992262i \(-0.460376\pi\)
0.124160 + 0.992262i \(0.460376\pi\)
\(830\) −42.0232 −1.45865
\(831\) −10.7766 −0.373837
\(832\) 4.45067 0.154299
\(833\) −18.0496 −0.625383
\(834\) −2.74422 −0.0950247
\(835\) −18.5635 −0.642418
\(836\) 0 0
\(837\) −13.9614 −0.482577
\(838\) 17.3200 0.598308
\(839\) 34.6067 1.19476 0.597378 0.801960i \(-0.296209\pi\)
0.597378 + 0.801960i \(0.296209\pi\)
\(840\) −8.22668 −0.283847
\(841\) −7.35235 −0.253529
\(842\) 12.2121 0.420858
\(843\) 12.6560 0.435896
\(844\) 35.6091 1.22571
\(845\) 7.56986 0.260411
\(846\) −3.74092 −0.128616
\(847\) −14.7023 −0.505178
\(848\) −19.5398 −0.671001
\(849\) −7.38238 −0.253363
\(850\) −31.2841 −1.07303
\(851\) −20.7939 −0.712804
\(852\) 19.9709 0.684192
\(853\) 33.2508 1.13849 0.569243 0.822169i \(-0.307236\pi\)
0.569243 + 0.822169i \(0.307236\pi\)
\(854\) −17.5253 −0.599703
\(855\) 0 0
\(856\) 40.7888 1.39413
\(857\) 3.88619 0.132750 0.0663749 0.997795i \(-0.478857\pi\)
0.0663749 + 0.997795i \(0.478857\pi\)
\(858\) 5.31996 0.181620
\(859\) 1.65776 0.0565619 0.0282810 0.999600i \(-0.490997\pi\)
0.0282810 + 0.999600i \(0.490997\pi\)
\(860\) 51.7229 1.76374
\(861\) 9.98545 0.340303
\(862\) −3.30096 −0.112431
\(863\) 52.7187 1.79457 0.897284 0.441455i \(-0.145537\pi\)
0.897284 + 0.441455i \(0.145537\pi\)
\(864\) −16.7219 −0.568892
\(865\) 34.0155 1.15656
\(866\) −50.1958 −1.70572
\(867\) −1.27301 −0.0432338
\(868\) 25.9368 0.880351
\(869\) 11.6209 0.394213
\(870\) −10.3601 −0.351240
\(871\) −10.5609 −0.357841
\(872\) −57.7229 −1.95474
\(873\) 18.9691 0.642008
\(874\) 0 0
\(875\) 16.8949 0.571151
\(876\) 17.6459 0.596200
\(877\) 21.1898 0.715530 0.357765 0.933812i \(-0.383539\pi\)
0.357765 + 0.933812i \(0.383539\pi\)
\(878\) 87.5271 2.95390
\(879\) −2.54488 −0.0858367
\(880\) 10.5963 0.357200
\(881\) 32.1010 1.08151 0.540755 0.841180i \(-0.318138\pi\)
0.540755 + 0.841180i \(0.318138\pi\)
\(882\) 30.3242 1.02107
\(883\) −47.2968 −1.59167 −0.795833 0.605516i \(-0.792967\pi\)
−0.795833 + 0.605516i \(0.792967\pi\)
\(884\) −46.4962 −1.56384
\(885\) −3.46110 −0.116344
\(886\) 43.0711 1.44700
\(887\) −10.5631 −0.354673 −0.177336 0.984150i \(-0.556748\pi\)
−0.177336 + 0.984150i \(0.556748\pi\)
\(888\) 16.3645 0.549158
\(889\) 22.2199 0.745231
\(890\) −8.27631 −0.277423
\(891\) −6.33544 −0.212245
\(892\) 68.2704 2.28586
\(893\) 0 0
\(894\) 18.5220 0.619468
\(895\) 7.85567 0.262586
\(896\) −20.4388 −0.682813
\(897\) 8.98040 0.299847
\(898\) −94.7247 −3.16101
\(899\) 17.8547 0.595489
\(900\) 36.1634 1.20545
\(901\) −11.4192 −0.380429
\(902\) −29.9564 −0.997438
\(903\) −8.70233 −0.289596
\(904\) −8.00175 −0.266134
\(905\) 18.2722 0.607388
\(906\) 18.2490 0.606281
\(907\) −42.9205 −1.42515 −0.712575 0.701596i \(-0.752471\pi\)
−0.712575 + 0.701596i \(0.752471\pi\)
\(908\) −43.5526 −1.44534
\(909\) −5.58616 −0.185281
\(910\) 14.2003 0.470735
\(911\) 55.1411 1.82691 0.913454 0.406942i \(-0.133405\pi\)
0.913454 + 0.406942i \(0.133405\pi\)
\(912\) 0 0
\(913\) −14.5945 −0.483008
\(914\) 23.0702 0.763093
\(915\) 3.97266 0.131332
\(916\) 88.7538 2.93251
\(917\) −30.3482 −1.00219
\(918\) −35.7374 −1.17951
\(919\) −24.5577 −0.810083 −0.405041 0.914298i \(-0.632743\pi\)
−0.405041 + 0.914298i \(0.632743\pi\)
\(920\) 41.6614 1.37353
\(921\) −15.1266 −0.498438
\(922\) −61.9053 −2.03874
\(923\) −18.8438 −0.620251
\(924\) −5.22668 −0.171945
\(925\) −13.0770 −0.429968
\(926\) −0.634858 −0.0208627
\(927\) −32.1121 −1.05470
\(928\) 21.3851 0.701999
\(929\) −22.2772 −0.730893 −0.365446 0.930832i \(-0.619084\pi\)
−0.365446 + 0.930832i \(0.619084\pi\)
\(930\) −8.54488 −0.280198
\(931\) 0 0
\(932\) 15.5936 0.510785
\(933\) −2.25908 −0.0739588
\(934\) 38.8976 1.27277
\(935\) 6.19253 0.202517
\(936\) 42.7009 1.39572
\(937\) −9.55169 −0.312040 −0.156020 0.987754i \(-0.549866\pi\)
−0.156020 + 0.987754i \(0.549866\pi\)
\(938\) 15.0797 0.492368
\(939\) −14.9403 −0.487557
\(940\) −3.41147 −0.111270
\(941\) 55.7256 1.81660 0.908301 0.418318i \(-0.137380\pi\)
0.908301 + 0.418318i \(0.137380\pi\)
\(942\) 18.1685 0.591961
\(943\) −50.5681 −1.64672
\(944\) 26.1266 0.850348
\(945\) 7.50980 0.244294
\(946\) 26.1070 0.848812
\(947\) −27.0428 −0.878774 −0.439387 0.898298i \(-0.644804\pi\)
−0.439387 + 0.898298i \(0.644804\pi\)
\(948\) −28.2422 −0.917263
\(949\) −16.6500 −0.540482
\(950\) 0 0
\(951\) −17.0490 −0.552852
\(952\) 36.2918 1.17622
\(953\) −23.1310 −0.749288 −0.374644 0.927169i \(-0.622235\pi\)
−0.374644 + 0.927169i \(0.622235\pi\)
\(954\) 19.1848 0.621131
\(955\) 13.8557 0.448359
\(956\) 52.8212 1.70836
\(957\) −3.59802 −0.116308
\(958\) −1.82119 −0.0588401
\(959\) 15.6340 0.504849
\(960\) 1.44057 0.0464942
\(961\) −16.2736 −0.524956
\(962\) −28.2472 −0.910727
\(963\) −17.1943 −0.554078
\(964\) −56.9154 −1.83312
\(965\) 18.5936 0.598548
\(966\) −12.8229 −0.412572
\(967\) 39.0351 1.25528 0.627642 0.778502i \(-0.284020\pi\)
0.627642 + 0.778502i \(0.284020\pi\)
\(968\) −58.5954 −1.88333
\(969\) 0 0
\(970\) 25.1411 0.807234
\(971\) 41.2026 1.32226 0.661128 0.750273i \(-0.270078\pi\)
0.661128 + 0.750273i \(0.270078\pi\)
\(972\) 63.5458 2.03823
\(973\) −2.54395 −0.0815552
\(974\) −29.7442 −0.953066
\(975\) 5.64765 0.180870
\(976\) −29.9881 −0.959897
\(977\) −22.4938 −0.719641 −0.359821 0.933022i \(-0.617162\pi\)
−0.359821 + 0.933022i \(0.617162\pi\)
\(978\) −10.4662 −0.334671
\(979\) −2.87433 −0.0918641
\(980\) 27.6536 0.883363
\(981\) 24.3327 0.776885
\(982\) 0.224927 0.00717771
\(983\) 44.5461 1.42080 0.710401 0.703798i \(-0.248514\pi\)
0.710401 + 0.703798i \(0.248514\pi\)
\(984\) 39.7965 1.26867
\(985\) 10.6987 0.340889
\(986\) 45.7033 1.45549
\(987\) 0.573978 0.0182699
\(988\) 0 0
\(989\) 44.0702 1.40135
\(990\) −10.4037 −0.330652
\(991\) 45.3296 1.43994 0.719971 0.694005i \(-0.244155\pi\)
0.719971 + 0.694005i \(0.244155\pi\)
\(992\) 17.6382 0.560012
\(993\) −12.4305 −0.394469
\(994\) 26.9067 0.853430
\(995\) −36.4243 −1.15473
\(996\) 35.4688 1.12387
\(997\) −10.4911 −0.332258 −0.166129 0.986104i \(-0.553127\pi\)
−0.166129 + 0.986104i \(0.553127\pi\)
\(998\) 37.1976 1.17747
\(999\) −14.9385 −0.472634
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 361.2.a.h.1.3 3
3.2 odd 2 3249.2.a.s.1.1 3
4.3 odd 2 5776.2.a.bi.1.2 3
5.4 even 2 9025.2.a.x.1.1 3
19.2 odd 18 361.2.e.g.99.1 6
19.3 odd 18 361.2.e.f.28.1 6
19.4 even 9 361.2.e.h.54.1 6
19.5 even 9 361.2.e.h.234.1 6
19.6 even 9 361.2.e.b.245.1 6
19.7 even 3 361.2.c.h.68.1 6
19.8 odd 6 361.2.c.i.292.3 6
19.9 even 9 361.2.e.a.62.1 6
19.10 odd 18 361.2.e.g.62.1 6
19.11 even 3 361.2.c.h.292.1 6
19.12 odd 6 361.2.c.i.68.3 6
19.13 odd 18 361.2.e.f.245.1 6
19.14 odd 18 19.2.e.a.6.1 6
19.15 odd 18 19.2.e.a.16.1 yes 6
19.16 even 9 361.2.e.b.28.1 6
19.17 even 9 361.2.e.a.99.1 6
19.18 odd 2 361.2.a.g.1.1 3
57.14 even 18 171.2.u.c.82.1 6
57.53 even 18 171.2.u.c.73.1 6
57.56 even 2 3249.2.a.z.1.3 3
76.15 even 18 304.2.u.b.225.1 6
76.71 even 18 304.2.u.b.177.1 6
76.75 even 2 5776.2.a.br.1.2 3
95.14 odd 18 475.2.l.a.101.1 6
95.33 even 36 475.2.u.a.424.2 12
95.34 odd 18 475.2.l.a.301.1 6
95.52 even 36 475.2.u.a.424.1 12
95.53 even 36 475.2.u.a.149.1 12
95.72 even 36 475.2.u.a.149.2 12
95.94 odd 2 9025.2.a.bd.1.3 3
133.33 even 18 931.2.x.b.557.1 6
133.34 even 18 931.2.w.a.491.1 6
133.52 even 18 931.2.v.a.177.1 6
133.53 odd 18 931.2.x.a.814.1 6
133.72 odd 18 931.2.v.b.263.1 6
133.90 even 18 931.2.w.a.785.1 6
133.109 odd 18 931.2.v.b.177.1 6
133.110 even 18 931.2.v.a.263.1 6
133.128 odd 18 931.2.x.a.557.1 6
133.129 even 18 931.2.x.b.814.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
19.2.e.a.6.1 6 19.14 odd 18
19.2.e.a.16.1 yes 6 19.15 odd 18
171.2.u.c.73.1 6 57.53 even 18
171.2.u.c.82.1 6 57.14 even 18
304.2.u.b.177.1 6 76.71 even 18
304.2.u.b.225.1 6 76.15 even 18
361.2.a.g.1.1 3 19.18 odd 2
361.2.a.h.1.3 3 1.1 even 1 trivial
361.2.c.h.68.1 6 19.7 even 3
361.2.c.h.292.1 6 19.11 even 3
361.2.c.i.68.3 6 19.12 odd 6
361.2.c.i.292.3 6 19.8 odd 6
361.2.e.a.62.1 6 19.9 even 9
361.2.e.a.99.1 6 19.17 even 9
361.2.e.b.28.1 6 19.16 even 9
361.2.e.b.245.1 6 19.6 even 9
361.2.e.f.28.1 6 19.3 odd 18
361.2.e.f.245.1 6 19.13 odd 18
361.2.e.g.62.1 6 19.10 odd 18
361.2.e.g.99.1 6 19.2 odd 18
361.2.e.h.54.1 6 19.4 even 9
361.2.e.h.234.1 6 19.5 even 9
475.2.l.a.101.1 6 95.14 odd 18
475.2.l.a.301.1 6 95.34 odd 18
475.2.u.a.149.1 12 95.53 even 36
475.2.u.a.149.2 12 95.72 even 36
475.2.u.a.424.1 12 95.52 even 36
475.2.u.a.424.2 12 95.33 even 36
931.2.v.a.177.1 6 133.52 even 18
931.2.v.a.263.1 6 133.110 even 18
931.2.v.b.177.1 6 133.109 odd 18
931.2.v.b.263.1 6 133.72 odd 18
931.2.w.a.491.1 6 133.34 even 18
931.2.w.a.785.1 6 133.90 even 18
931.2.x.a.557.1 6 133.128 odd 18
931.2.x.a.814.1 6 133.53 odd 18
931.2.x.b.557.1 6 133.33 even 18
931.2.x.b.814.1 6 133.129 even 18
3249.2.a.s.1.1 3 3.2 odd 2
3249.2.a.z.1.3 3 57.56 even 2
5776.2.a.bi.1.2 3 4.3 odd 2
5776.2.a.br.1.2 3 76.75 even 2
9025.2.a.x.1.1 3 5.4 even 2
9025.2.a.bd.1.3 3 95.94 odd 2