Properties

Label 3381.2.a.bc.1.6
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7997584.1
Defining polynomial: \(x^{6} - x^{5} - 7 x^{4} + 5 x^{3} + 12 x^{2} - 4 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.04340\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.04340 q^{2} -1.00000 q^{3} +2.17548 q^{4} +1.94349 q^{5} -2.04340 q^{6} +0.358585 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.04340 q^{2} -1.00000 q^{3} +2.17548 q^{4} +1.94349 q^{5} -2.04340 q^{6} +0.358585 q^{8} +1.00000 q^{9} +3.97133 q^{10} -5.66529 q^{11} -2.17548 q^{12} -6.17812 q^{13} -1.94349 q^{15} -3.61824 q^{16} +7.23463 q^{17} +2.04340 q^{18} +0.790220 q^{19} +4.22803 q^{20} -11.5765 q^{22} -1.00000 q^{23} -0.358585 q^{24} -1.22285 q^{25} -12.6244 q^{26} -1.00000 q^{27} -5.34085 q^{29} -3.97133 q^{30} +0.911316 q^{31} -8.11067 q^{32} +5.66529 q^{33} +14.7832 q^{34} +2.17548 q^{36} -0.973320 q^{37} +1.61473 q^{38} +6.17812 q^{39} +0.696907 q^{40} -1.51843 q^{41} -2.70342 q^{43} -12.3247 q^{44} +1.94349 q^{45} -2.04340 q^{46} -1.52839 q^{47} +3.61824 q^{48} -2.49877 q^{50} -7.23463 q^{51} -13.4404 q^{52} -8.95803 q^{53} -2.04340 q^{54} -11.0104 q^{55} -0.790220 q^{57} -10.9135 q^{58} -10.8931 q^{59} -4.22803 q^{60} -6.31775 q^{61} +1.86218 q^{62} -9.33688 q^{64} -12.0071 q^{65} +11.5765 q^{66} +7.31833 q^{67} +15.7388 q^{68} +1.00000 q^{69} -8.75692 q^{71} +0.358585 q^{72} -0.875251 q^{73} -1.98888 q^{74} +1.22285 q^{75} +1.71911 q^{76} +12.6244 q^{78} +5.96655 q^{79} -7.03200 q^{80} +1.00000 q^{81} -3.10277 q^{82} -0.273554 q^{83} +14.0604 q^{85} -5.52417 q^{86} +5.34085 q^{87} -2.03149 q^{88} +17.6270 q^{89} +3.97133 q^{90} -2.17548 q^{92} -0.911316 q^{93} -3.12312 q^{94} +1.53578 q^{95} +8.11067 q^{96} +3.09620 q^{97} -5.66529 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} - 3q^{10} - 14q^{11} - 3q^{12} - 3q^{15} - 7q^{16} + 15q^{17} - q^{18} - q^{19} + 17q^{20} - 6q^{22} - 6q^{23} + 3q^{24} + 9q^{25} - 15q^{26} - 6q^{27} - 6q^{29} + 3q^{30} - 11q^{31} + 3q^{32} + 14q^{33} + 15q^{34} + 3q^{36} - 5q^{37} + 14q^{38} - 17q^{40} + 18q^{41} - 37q^{43} - 10q^{44} + 3q^{45} + q^{46} + 3q^{47} + 7q^{48} - 30q^{50} - 15q^{51} - 7q^{52} - 15q^{53} + q^{54} - 2q^{55} + q^{57} + 4q^{58} - 2q^{59} - 17q^{60} - 12q^{61} + 36q^{62} - 23q^{64} - 17q^{65} + 6q^{66} - 10q^{67} - q^{68} + 6q^{69} - 21q^{71} - 3q^{72} - 8q^{73} - 16q^{74} - 9q^{75} + 18q^{76} + 15q^{78} - 17q^{79} + 3q^{80} + 6q^{81} - 48q^{82} + 12q^{83} - 13q^{85} + 22q^{86} + 6q^{87} - 2q^{88} + 18q^{89} - 3q^{90} - 3q^{92} + 11q^{93} + 3q^{94} - 16q^{95} - 3q^{96} - 2q^{97} - 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.04340 1.44490 0.722451 0.691422i \(-0.243015\pi\)
0.722451 + 0.691422i \(0.243015\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.17548 1.08774
\(5\) 1.94349 0.869155 0.434577 0.900634i \(-0.356898\pi\)
0.434577 + 0.900634i \(0.356898\pi\)
\(6\) −2.04340 −0.834215
\(7\) 0 0
\(8\) 0.358585 0.126779
\(9\) 1.00000 0.333333
\(10\) 3.97133 1.25584
\(11\) −5.66529 −1.70815 −0.854074 0.520151i \(-0.825876\pi\)
−0.854074 + 0.520151i \(0.825876\pi\)
\(12\) −2.17548 −0.628008
\(13\) −6.17812 −1.71350 −0.856751 0.515730i \(-0.827521\pi\)
−0.856751 + 0.515730i \(0.827521\pi\)
\(14\) 0 0
\(15\) −1.94349 −0.501807
\(16\) −3.61824 −0.904559
\(17\) 7.23463 1.75466 0.877328 0.479892i \(-0.159324\pi\)
0.877328 + 0.479892i \(0.159324\pi\)
\(18\) 2.04340 0.481634
\(19\) 0.790220 0.181289 0.0906444 0.995883i \(-0.471107\pi\)
0.0906444 + 0.995883i \(0.471107\pi\)
\(20\) 4.22803 0.945417
\(21\) 0 0
\(22\) −11.5765 −2.46811
\(23\) −1.00000 −0.208514
\(24\) −0.358585 −0.0731960
\(25\) −1.22285 −0.244570
\(26\) −12.6244 −2.47584
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.34085 −0.991770 −0.495885 0.868388i \(-0.665156\pi\)
−0.495885 + 0.868388i \(0.665156\pi\)
\(30\) −3.97133 −0.725062
\(31\) 0.911316 0.163677 0.0818386 0.996646i \(-0.473921\pi\)
0.0818386 + 0.996646i \(0.473921\pi\)
\(32\) −8.11067 −1.43378
\(33\) 5.66529 0.986200
\(34\) 14.7832 2.53531
\(35\) 0 0
\(36\) 2.17548 0.362581
\(37\) −0.973320 −0.160013 −0.0800064 0.996794i \(-0.525494\pi\)
−0.0800064 + 0.996794i \(0.525494\pi\)
\(38\) 1.61473 0.261945
\(39\) 6.17812 0.989291
\(40\) 0.696907 0.110191
\(41\) −1.51843 −0.237140 −0.118570 0.992946i \(-0.537831\pi\)
−0.118570 + 0.992946i \(0.537831\pi\)
\(42\) 0 0
\(43\) −2.70342 −0.412268 −0.206134 0.978524i \(-0.566088\pi\)
−0.206134 + 0.978524i \(0.566088\pi\)
\(44\) −12.3247 −1.85803
\(45\) 1.94349 0.289718
\(46\) −2.04340 −0.301283
\(47\) −1.52839 −0.222939 −0.111470 0.993768i \(-0.535556\pi\)
−0.111470 + 0.993768i \(0.535556\pi\)
\(48\) 3.61824 0.522247
\(49\) 0 0
\(50\) −2.49877 −0.353379
\(51\) −7.23463 −1.01305
\(52\) −13.4404 −1.86385
\(53\) −8.95803 −1.23048 −0.615240 0.788340i \(-0.710941\pi\)
−0.615240 + 0.788340i \(0.710941\pi\)
\(54\) −2.04340 −0.278072
\(55\) −11.0104 −1.48465
\(56\) 0 0
\(57\) −0.790220 −0.104667
\(58\) −10.9135 −1.43301
\(59\) −10.8931 −1.41816 −0.709081 0.705127i \(-0.750890\pi\)
−0.709081 + 0.705127i \(0.750890\pi\)
\(60\) −4.22803 −0.545837
\(61\) −6.31775 −0.808905 −0.404452 0.914559i \(-0.632538\pi\)
−0.404452 + 0.914559i \(0.632538\pi\)
\(62\) 1.86218 0.236497
\(63\) 0 0
\(64\) −9.33688 −1.16711
\(65\) −12.0071 −1.48930
\(66\) 11.5765 1.42496
\(67\) 7.31833 0.894077 0.447038 0.894515i \(-0.352479\pi\)
0.447038 + 0.894515i \(0.352479\pi\)
\(68\) 15.7388 1.90861
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −8.75692 −1.03926 −0.519628 0.854393i \(-0.673929\pi\)
−0.519628 + 0.854393i \(0.673929\pi\)
\(72\) 0.358585 0.0422597
\(73\) −0.875251 −0.102440 −0.0512202 0.998687i \(-0.516311\pi\)
−0.0512202 + 0.998687i \(0.516311\pi\)
\(74\) −1.98888 −0.231203
\(75\) 1.22285 0.141202
\(76\) 1.71911 0.197196
\(77\) 0 0
\(78\) 12.6244 1.42943
\(79\) 5.96655 0.671289 0.335644 0.941989i \(-0.391046\pi\)
0.335644 + 0.941989i \(0.391046\pi\)
\(80\) −7.03200 −0.786202
\(81\) 1.00000 0.111111
\(82\) −3.10277 −0.342644
\(83\) −0.273554 −0.0300265 −0.0150132 0.999887i \(-0.504779\pi\)
−0.0150132 + 0.999887i \(0.504779\pi\)
\(84\) 0 0
\(85\) 14.0604 1.52507
\(86\) −5.52417 −0.595686
\(87\) 5.34085 0.572599
\(88\) −2.03149 −0.216558
\(89\) 17.6270 1.86846 0.934229 0.356672i \(-0.116089\pi\)
0.934229 + 0.356672i \(0.116089\pi\)
\(90\) 3.97133 0.418615
\(91\) 0 0
\(92\) −2.17548 −0.226810
\(93\) −0.911316 −0.0944991
\(94\) −3.12312 −0.322125
\(95\) 1.53578 0.157568
\(96\) 8.11067 0.827792
\(97\) 3.09620 0.314372 0.157186 0.987569i \(-0.449758\pi\)
0.157186 + 0.987569i \(0.449758\pi\)
\(98\) 0 0
\(99\) −5.66529 −0.569383
\(100\) −2.66029 −0.266029
\(101\) 13.4276 1.33610 0.668050 0.744116i \(-0.267129\pi\)
0.668050 + 0.744116i \(0.267129\pi\)
\(102\) −14.7832 −1.46376
\(103\) 6.33653 0.624357 0.312178 0.950023i \(-0.398941\pi\)
0.312178 + 0.950023i \(0.398941\pi\)
\(104\) −2.21538 −0.217236
\(105\) 0 0
\(106\) −18.3049 −1.77792
\(107\) 15.9609 1.54299 0.771497 0.636232i \(-0.219508\pi\)
0.771497 + 0.636232i \(0.219508\pi\)
\(108\) −2.17548 −0.209336
\(109\) −0.0470715 −0.00450863 −0.00225431 0.999997i \(-0.500718\pi\)
−0.00225431 + 0.999997i \(0.500718\pi\)
\(110\) −22.4987 −2.14517
\(111\) 0.973320 0.0923835
\(112\) 0 0
\(113\) −6.27141 −0.589965 −0.294982 0.955503i \(-0.595314\pi\)
−0.294982 + 0.955503i \(0.595314\pi\)
\(114\) −1.61473 −0.151234
\(115\) −1.94349 −0.181231
\(116\) −11.6189 −1.07879
\(117\) −6.17812 −0.571167
\(118\) −22.2590 −2.04911
\(119\) 0 0
\(120\) −0.696907 −0.0636186
\(121\) 21.0955 1.91777
\(122\) −12.9097 −1.16879
\(123\) 1.51843 0.136913
\(124\) 1.98255 0.178039
\(125\) −12.0940 −1.08172
\(126\) 0 0
\(127\) −12.4325 −1.10320 −0.551601 0.834108i \(-0.685983\pi\)
−0.551601 + 0.834108i \(0.685983\pi\)
\(128\) −2.85764 −0.252582
\(129\) 2.70342 0.238023
\(130\) −24.5353 −2.15189
\(131\) −3.31098 −0.289281 −0.144641 0.989484i \(-0.546203\pi\)
−0.144641 + 0.989484i \(0.546203\pi\)
\(132\) 12.3247 1.07273
\(133\) 0 0
\(134\) 14.9543 1.29185
\(135\) −1.94349 −0.167269
\(136\) 2.59423 0.222454
\(137\) −13.0442 −1.11444 −0.557220 0.830365i \(-0.688132\pi\)
−0.557220 + 0.830365i \(0.688132\pi\)
\(138\) 2.04340 0.173946
\(139\) −11.3674 −0.964169 −0.482084 0.876125i \(-0.660120\pi\)
−0.482084 + 0.876125i \(0.660120\pi\)
\(140\) 0 0
\(141\) 1.52839 0.128714
\(142\) −17.8939 −1.50162
\(143\) 35.0008 2.92692
\(144\) −3.61824 −0.301520
\(145\) −10.3799 −0.862002
\(146\) −1.78849 −0.148016
\(147\) 0 0
\(148\) −2.11744 −0.174053
\(149\) 16.2060 1.32765 0.663824 0.747889i \(-0.268932\pi\)
0.663824 + 0.747889i \(0.268932\pi\)
\(150\) 2.49877 0.204024
\(151\) −12.9337 −1.05253 −0.526266 0.850320i \(-0.676408\pi\)
−0.526266 + 0.850320i \(0.676408\pi\)
\(152\) 0.283361 0.0229836
\(153\) 7.23463 0.584885
\(154\) 0 0
\(155\) 1.77113 0.142261
\(156\) 13.4404 1.07609
\(157\) −9.73442 −0.776892 −0.388446 0.921472i \(-0.626988\pi\)
−0.388446 + 0.921472i \(0.626988\pi\)
\(158\) 12.1920 0.969947
\(159\) 8.95803 0.710418
\(160\) −15.7630 −1.24618
\(161\) 0 0
\(162\) 2.04340 0.160545
\(163\) −24.0922 −1.88705 −0.943525 0.331300i \(-0.892513\pi\)
−0.943525 + 0.331300i \(0.892513\pi\)
\(164\) −3.30333 −0.257947
\(165\) 11.0104 0.857161
\(166\) −0.558981 −0.0433853
\(167\) −12.2582 −0.948566 −0.474283 0.880373i \(-0.657293\pi\)
−0.474283 + 0.880373i \(0.657293\pi\)
\(168\) 0 0
\(169\) 25.1692 1.93609
\(170\) 28.7311 2.20357
\(171\) 0.790220 0.0604296
\(172\) −5.88125 −0.448441
\(173\) −5.88796 −0.447653 −0.223826 0.974629i \(-0.571855\pi\)
−0.223826 + 0.974629i \(0.571855\pi\)
\(174\) 10.9135 0.827349
\(175\) 0 0
\(176\) 20.4984 1.54512
\(177\) 10.8931 0.818777
\(178\) 36.0190 2.69974
\(179\) −16.9732 −1.26864 −0.634319 0.773071i \(-0.718719\pi\)
−0.634319 + 0.773071i \(0.718719\pi\)
\(180\) 4.22803 0.315139
\(181\) 19.1241 1.42149 0.710743 0.703452i \(-0.248359\pi\)
0.710743 + 0.703452i \(0.248359\pi\)
\(182\) 0 0
\(183\) 6.31775 0.467021
\(184\) −0.358585 −0.0264353
\(185\) −1.89164 −0.139076
\(186\) −1.86218 −0.136542
\(187\) −40.9863 −2.99721
\(188\) −3.32500 −0.242500
\(189\) 0 0
\(190\) 3.13822 0.227670
\(191\) −20.6871 −1.49686 −0.748432 0.663212i \(-0.769193\pi\)
−0.748432 + 0.663212i \(0.769193\pi\)
\(192\) 9.33688 0.673832
\(193\) 2.22901 0.160447 0.0802237 0.996777i \(-0.474437\pi\)
0.0802237 + 0.996777i \(0.474437\pi\)
\(194\) 6.32678 0.454237
\(195\) 12.0071 0.859847
\(196\) 0 0
\(197\) 19.3710 1.38013 0.690064 0.723748i \(-0.257582\pi\)
0.690064 + 0.723748i \(0.257582\pi\)
\(198\) −11.5765 −0.822703
\(199\) 24.6840 1.74980 0.874902 0.484300i \(-0.160926\pi\)
0.874902 + 0.484300i \(0.160926\pi\)
\(200\) −0.438496 −0.0310063
\(201\) −7.31833 −0.516195
\(202\) 27.4381 1.93054
\(203\) 0 0
\(204\) −15.7388 −1.10194
\(205\) −2.95106 −0.206111
\(206\) 12.9481 0.902135
\(207\) −1.00000 −0.0695048
\(208\) 22.3539 1.54996
\(209\) −4.47682 −0.309668
\(210\) 0 0
\(211\) 9.90378 0.681804 0.340902 0.940099i \(-0.389267\pi\)
0.340902 + 0.940099i \(0.389267\pi\)
\(212\) −19.4881 −1.33845
\(213\) 8.75692 0.600014
\(214\) 32.6144 2.22948
\(215\) −5.25407 −0.358324
\(216\) −0.358585 −0.0243987
\(217\) 0 0
\(218\) −0.0961859 −0.00651453
\(219\) 0.875251 0.0591440
\(220\) −23.9530 −1.61491
\(221\) −44.6964 −3.00661
\(222\) 1.98888 0.133485
\(223\) 2.21461 0.148301 0.0741507 0.997247i \(-0.476375\pi\)
0.0741507 + 0.997247i \(0.476375\pi\)
\(224\) 0 0
\(225\) −1.22285 −0.0815232
\(226\) −12.8150 −0.852441
\(227\) −5.58033 −0.370380 −0.185190 0.982703i \(-0.559290\pi\)
−0.185190 + 0.982703i \(0.559290\pi\)
\(228\) −1.71911 −0.113851
\(229\) 6.97859 0.461158 0.230579 0.973054i \(-0.425938\pi\)
0.230579 + 0.973054i \(0.425938\pi\)
\(230\) −3.97133 −0.261862
\(231\) 0 0
\(232\) −1.91515 −0.125736
\(233\) 24.4042 1.59878 0.799388 0.600816i \(-0.205157\pi\)
0.799388 + 0.600816i \(0.205157\pi\)
\(234\) −12.6244 −0.825281
\(235\) −2.97042 −0.193769
\(236\) −23.6978 −1.54260
\(237\) −5.96655 −0.387569
\(238\) 0 0
\(239\) 0.424159 0.0274365 0.0137183 0.999906i \(-0.495633\pi\)
0.0137183 + 0.999906i \(0.495633\pi\)
\(240\) 7.03200 0.453914
\(241\) −16.3125 −1.05078 −0.525389 0.850862i \(-0.676080\pi\)
−0.525389 + 0.850862i \(0.676080\pi\)
\(242\) 43.1065 2.77099
\(243\) −1.00000 −0.0641500
\(244\) −13.7442 −0.879880
\(245\) 0 0
\(246\) 3.10277 0.197825
\(247\) −4.88207 −0.310639
\(248\) 0.326785 0.0207508
\(249\) 0.273554 0.0173358
\(250\) −24.7130 −1.56299
\(251\) −15.9242 −1.00513 −0.502564 0.864540i \(-0.667610\pi\)
−0.502564 + 0.864540i \(0.667610\pi\)
\(252\) 0 0
\(253\) 5.66529 0.356174
\(254\) −25.4045 −1.59402
\(255\) −14.0604 −0.880498
\(256\) 12.8345 0.802154
\(257\) −30.1569 −1.88114 −0.940569 0.339602i \(-0.889708\pi\)
−0.940569 + 0.339602i \(0.889708\pi\)
\(258\) 5.52417 0.343920
\(259\) 0 0
\(260\) −26.1213 −1.61997
\(261\) −5.34085 −0.330590
\(262\) −6.76565 −0.417983
\(263\) 15.1656 0.935153 0.467576 0.883953i \(-0.345127\pi\)
0.467576 + 0.883953i \(0.345127\pi\)
\(264\) 2.03149 0.125030
\(265\) −17.4098 −1.06948
\(266\) 0 0
\(267\) −17.6270 −1.07876
\(268\) 15.9209 0.972525
\(269\) 0.470833 0.0287072 0.0143536 0.999897i \(-0.495431\pi\)
0.0143536 + 0.999897i \(0.495431\pi\)
\(270\) −3.97133 −0.241687
\(271\) −7.28736 −0.442676 −0.221338 0.975197i \(-0.571042\pi\)
−0.221338 + 0.975197i \(0.571042\pi\)
\(272\) −26.1766 −1.58719
\(273\) 0 0
\(274\) −26.6545 −1.61026
\(275\) 6.92779 0.417761
\(276\) 2.17548 0.130949
\(277\) 2.96021 0.177862 0.0889310 0.996038i \(-0.471655\pi\)
0.0889310 + 0.996038i \(0.471655\pi\)
\(278\) −23.2281 −1.39313
\(279\) 0.911316 0.0545591
\(280\) 0 0
\(281\) 7.16141 0.427214 0.213607 0.976920i \(-0.431479\pi\)
0.213607 + 0.976920i \(0.431479\pi\)
\(282\) 3.12312 0.185979
\(283\) −20.0106 −1.18951 −0.594753 0.803908i \(-0.702750\pi\)
−0.594753 + 0.803908i \(0.702750\pi\)
\(284\) −19.0505 −1.13044
\(285\) −1.53578 −0.0909720
\(286\) 71.5207 4.22911
\(287\) 0 0
\(288\) −8.11067 −0.477926
\(289\) 35.3399 2.07882
\(290\) −21.2103 −1.24551
\(291\) −3.09620 −0.181503
\(292\) −1.90409 −0.111429
\(293\) 6.27910 0.366829 0.183414 0.983036i \(-0.441285\pi\)
0.183414 + 0.983036i \(0.441285\pi\)
\(294\) 0 0
\(295\) −21.1707 −1.23260
\(296\) −0.349019 −0.0202863
\(297\) 5.66529 0.328733
\(298\) 33.1154 1.91832
\(299\) 6.17812 0.357290
\(300\) 2.66029 0.153592
\(301\) 0 0
\(302\) −26.4288 −1.52081
\(303\) −13.4276 −0.771398
\(304\) −2.85920 −0.163986
\(305\) −12.2785 −0.703064
\(306\) 14.7832 0.845102
\(307\) 25.1923 1.43780 0.718900 0.695114i \(-0.244646\pi\)
0.718900 + 0.695114i \(0.244646\pi\)
\(308\) 0 0
\(309\) −6.33653 −0.360473
\(310\) 3.61913 0.205553
\(311\) 11.9221 0.676038 0.338019 0.941139i \(-0.390243\pi\)
0.338019 + 0.941139i \(0.390243\pi\)
\(312\) 2.21538 0.125421
\(313\) 10.0423 0.567625 0.283812 0.958880i \(-0.408401\pi\)
0.283812 + 0.958880i \(0.408401\pi\)
\(314\) −19.8913 −1.12253
\(315\) 0 0
\(316\) 12.9801 0.730189
\(317\) −8.86210 −0.497745 −0.248872 0.968536i \(-0.580060\pi\)
−0.248872 + 0.968536i \(0.580060\pi\)
\(318\) 18.3049 1.02649
\(319\) 30.2574 1.69409
\(320\) −18.1461 −1.01440
\(321\) −15.9609 −0.890849
\(322\) 0 0
\(323\) 5.71695 0.318099
\(324\) 2.17548 0.120860
\(325\) 7.55490 0.419071
\(326\) −49.2301 −2.72660
\(327\) 0.0470715 0.00260306
\(328\) −0.544489 −0.0300644
\(329\) 0 0
\(330\) 22.4987 1.23851
\(331\) −22.8992 −1.25865 −0.629327 0.777141i \(-0.716669\pi\)
−0.629327 + 0.777141i \(0.716669\pi\)
\(332\) −0.595113 −0.0326611
\(333\) −0.973320 −0.0533376
\(334\) −25.0483 −1.37058
\(335\) 14.2231 0.777091
\(336\) 0 0
\(337\) 12.2517 0.667392 0.333696 0.942681i \(-0.391704\pi\)
0.333696 + 0.942681i \(0.391704\pi\)
\(338\) 51.4307 2.79746
\(339\) 6.27141 0.340616
\(340\) 30.5882 1.65888
\(341\) −5.16287 −0.279585
\(342\) 1.61473 0.0873149
\(343\) 0 0
\(344\) −0.969407 −0.0522669
\(345\) 1.94349 0.104634
\(346\) −12.0314 −0.646815
\(347\) −6.45833 −0.346701 −0.173351 0.984860i \(-0.555459\pi\)
−0.173351 + 0.984860i \(0.555459\pi\)
\(348\) 11.6189 0.622840
\(349\) 20.5047 1.09759 0.548795 0.835957i \(-0.315087\pi\)
0.548795 + 0.835957i \(0.315087\pi\)
\(350\) 0 0
\(351\) 6.17812 0.329764
\(352\) 45.9493 2.44911
\(353\) 14.5113 0.772357 0.386178 0.922424i \(-0.373795\pi\)
0.386178 + 0.922424i \(0.373795\pi\)
\(354\) 22.2590 1.18305
\(355\) −17.0190 −0.903274
\(356\) 38.3473 2.03240
\(357\) 0 0
\(358\) −34.6831 −1.83306
\(359\) 4.67780 0.246885 0.123442 0.992352i \(-0.460607\pi\)
0.123442 + 0.992352i \(0.460607\pi\)
\(360\) 0.696907 0.0367302
\(361\) −18.3756 −0.967134
\(362\) 39.0782 2.05391
\(363\) −21.0955 −1.10723
\(364\) 0 0
\(365\) −1.70104 −0.0890366
\(366\) 12.9097 0.674800
\(367\) −28.0660 −1.46503 −0.732515 0.680750i \(-0.761654\pi\)
−0.732515 + 0.680750i \(0.761654\pi\)
\(368\) 3.61824 0.188614
\(369\) −1.51843 −0.0790465
\(370\) −3.86537 −0.200951
\(371\) 0 0
\(372\) −1.98255 −0.102791
\(373\) 14.6303 0.757527 0.378763 0.925494i \(-0.376349\pi\)
0.378763 + 0.925494i \(0.376349\pi\)
\(374\) −83.7514 −4.33068
\(375\) 12.0940 0.624534
\(376\) −0.548060 −0.0282640
\(377\) 32.9964 1.69940
\(378\) 0 0
\(379\) −2.44369 −0.125524 −0.0627620 0.998029i \(-0.519991\pi\)
−0.0627620 + 0.998029i \(0.519991\pi\)
\(380\) 3.34107 0.171393
\(381\) 12.4325 0.636934
\(382\) −42.2720 −2.16282
\(383\) 30.7792 1.57274 0.786371 0.617755i \(-0.211958\pi\)
0.786371 + 0.617755i \(0.211958\pi\)
\(384\) 2.85764 0.145828
\(385\) 0 0
\(386\) 4.55475 0.231831
\(387\) −2.70342 −0.137423
\(388\) 6.73575 0.341956
\(389\) 12.1855 0.617830 0.308915 0.951090i \(-0.400034\pi\)
0.308915 + 0.951090i \(0.400034\pi\)
\(390\) 24.5353 1.24239
\(391\) −7.23463 −0.365871
\(392\) 0 0
\(393\) 3.31098 0.167017
\(394\) 39.5827 1.99415
\(395\) 11.5959 0.583454
\(396\) −12.3247 −0.619342
\(397\) 21.3044 1.06924 0.534618 0.845094i \(-0.320455\pi\)
0.534618 + 0.845094i \(0.320455\pi\)
\(398\) 50.4394 2.52830
\(399\) 0 0
\(400\) 4.42455 0.221228
\(401\) 7.40280 0.369678 0.184839 0.982769i \(-0.440824\pi\)
0.184839 + 0.982769i \(0.440824\pi\)
\(402\) −14.9543 −0.745852
\(403\) −5.63022 −0.280461
\(404\) 29.2116 1.45333
\(405\) 1.94349 0.0965728
\(406\) 0 0
\(407\) 5.51414 0.273326
\(408\) −2.59423 −0.128434
\(409\) 26.4674 1.30873 0.654364 0.756180i \(-0.272936\pi\)
0.654364 + 0.756180i \(0.272936\pi\)
\(410\) −6.03020 −0.297810
\(411\) 13.0442 0.643423
\(412\) 13.7850 0.679140
\(413\) 0 0
\(414\) −2.04340 −0.100428
\(415\) −0.531650 −0.0260977
\(416\) 50.1087 2.45678
\(417\) 11.3674 0.556663
\(418\) −9.14794 −0.447440
\(419\) −30.7294 −1.50123 −0.750614 0.660741i \(-0.770242\pi\)
−0.750614 + 0.660741i \(0.770242\pi\)
\(420\) 0 0
\(421\) 2.65571 0.129432 0.0647158 0.997904i \(-0.479386\pi\)
0.0647158 + 0.997904i \(0.479386\pi\)
\(422\) 20.2374 0.985141
\(423\) −1.52839 −0.0743130
\(424\) −3.21222 −0.155999
\(425\) −8.84685 −0.429135
\(426\) 17.8939 0.866962
\(427\) 0 0
\(428\) 34.7226 1.67838
\(429\) −35.0008 −1.68986
\(430\) −10.7362 −0.517744
\(431\) 21.2541 1.02377 0.511887 0.859053i \(-0.328947\pi\)
0.511887 + 0.859053i \(0.328947\pi\)
\(432\) 3.61824 0.174082
\(433\) −16.8730 −0.810864 −0.405432 0.914125i \(-0.632879\pi\)
−0.405432 + 0.914125i \(0.632879\pi\)
\(434\) 0 0
\(435\) 10.3799 0.497677
\(436\) −0.102403 −0.00490423
\(437\) −0.790220 −0.0378013
\(438\) 1.78849 0.0854573
\(439\) −11.0905 −0.529320 −0.264660 0.964342i \(-0.585260\pi\)
−0.264660 + 0.964342i \(0.585260\pi\)
\(440\) −3.94818 −0.188222
\(441\) 0 0
\(442\) −91.3327 −4.34425
\(443\) −12.7742 −0.606920 −0.303460 0.952844i \(-0.598142\pi\)
−0.303460 + 0.952844i \(0.598142\pi\)
\(444\) 2.11744 0.100489
\(445\) 34.2579 1.62398
\(446\) 4.52534 0.214281
\(447\) −16.2060 −0.766518
\(448\) 0 0
\(449\) 38.7094 1.82681 0.913405 0.407052i \(-0.133443\pi\)
0.913405 + 0.407052i \(0.133443\pi\)
\(450\) −2.49877 −0.117793
\(451\) 8.60237 0.405070
\(452\) −13.6434 −0.641729
\(453\) 12.9337 0.607680
\(454\) −11.4029 −0.535162
\(455\) 0 0
\(456\) −0.283361 −0.0132696
\(457\) −36.2907 −1.69761 −0.848803 0.528709i \(-0.822676\pi\)
−0.848803 + 0.528709i \(0.822676\pi\)
\(458\) 14.2601 0.666329
\(459\) −7.23463 −0.337684
\(460\) −4.22803 −0.197133
\(461\) −3.70588 −0.172600 −0.0862999 0.996269i \(-0.527504\pi\)
−0.0862999 + 0.996269i \(0.527504\pi\)
\(462\) 0 0
\(463\) −29.9926 −1.39387 −0.696937 0.717132i \(-0.745454\pi\)
−0.696937 + 0.717132i \(0.745454\pi\)
\(464\) 19.3244 0.897115
\(465\) −1.77113 −0.0821343
\(466\) 49.8676 2.31007
\(467\) 28.0845 1.29959 0.649797 0.760108i \(-0.274854\pi\)
0.649797 + 0.760108i \(0.274854\pi\)
\(468\) −13.4404 −0.621283
\(469\) 0 0
\(470\) −6.06975 −0.279977
\(471\) 9.73442 0.448539
\(472\) −3.90611 −0.179793
\(473\) 15.3156 0.704214
\(474\) −12.1920 −0.559999
\(475\) −0.966319 −0.0443377
\(476\) 0 0
\(477\) −8.95803 −0.410160
\(478\) 0.866726 0.0396431
\(479\) −12.8184 −0.585688 −0.292844 0.956160i \(-0.594602\pi\)
−0.292844 + 0.956160i \(0.594602\pi\)
\(480\) 15.7630 0.719480
\(481\) 6.01329 0.274182
\(482\) −33.3329 −1.51827
\(483\) 0 0
\(484\) 45.8929 2.08604
\(485\) 6.01744 0.273238
\(486\) −2.04340 −0.0926905
\(487\) 9.33744 0.423120 0.211560 0.977365i \(-0.432146\pi\)
0.211560 + 0.977365i \(0.432146\pi\)
\(488\) −2.26545 −0.102552
\(489\) 24.0922 1.08949
\(490\) 0 0
\(491\) −1.98455 −0.0895616 −0.0447808 0.998997i \(-0.514259\pi\)
−0.0447808 + 0.998997i \(0.514259\pi\)
\(492\) 3.30333 0.148926
\(493\) −38.6391 −1.74022
\(494\) −9.97603 −0.448843
\(495\) −11.0104 −0.494882
\(496\) −3.29736 −0.148056
\(497\) 0 0
\(498\) 0.558981 0.0250485
\(499\) −40.8742 −1.82978 −0.914890 0.403702i \(-0.867723\pi\)
−0.914890 + 0.403702i \(0.867723\pi\)
\(500\) −26.3104 −1.17664
\(501\) 12.2582 0.547655
\(502\) −32.5396 −1.45231
\(503\) −2.08575 −0.0929989 −0.0464994 0.998918i \(-0.514807\pi\)
−0.0464994 + 0.998918i \(0.514807\pi\)
\(504\) 0 0
\(505\) 26.0965 1.16128
\(506\) 11.5765 0.514636
\(507\) −25.1692 −1.11780
\(508\) −27.0466 −1.20000
\(509\) −17.4199 −0.772123 −0.386062 0.922473i \(-0.626165\pi\)
−0.386062 + 0.922473i \(0.626165\pi\)
\(510\) −28.7311 −1.27223
\(511\) 0 0
\(512\) 31.9412 1.41162
\(513\) −0.790220 −0.0348890
\(514\) −61.6227 −2.71806
\(515\) 12.3150 0.542663
\(516\) 5.88125 0.258907
\(517\) 8.65879 0.380813
\(518\) 0 0
\(519\) 5.88796 0.258453
\(520\) −4.30558 −0.188812
\(521\) −13.2487 −0.580438 −0.290219 0.956960i \(-0.593728\pi\)
−0.290219 + 0.956960i \(0.593728\pi\)
\(522\) −10.9135 −0.477670
\(523\) 6.49239 0.283892 0.141946 0.989874i \(-0.454664\pi\)
0.141946 + 0.989874i \(0.454664\pi\)
\(524\) −7.20298 −0.314664
\(525\) 0 0
\(526\) 30.9894 1.35120
\(527\) 6.59303 0.287197
\(528\) −20.4984 −0.892076
\(529\) 1.00000 0.0434783
\(530\) −35.5753 −1.54529
\(531\) −10.8931 −0.472721
\(532\) 0 0
\(533\) 9.38107 0.406339
\(534\) −36.0190 −1.55870
\(535\) 31.0198 1.34110
\(536\) 2.62425 0.113350
\(537\) 16.9732 0.732449
\(538\) 0.962101 0.0414791
\(539\) 0 0
\(540\) −4.22803 −0.181946
\(541\) 17.0384 0.732536 0.366268 0.930509i \(-0.380635\pi\)
0.366268 + 0.930509i \(0.380635\pi\)
\(542\) −14.8910 −0.639623
\(543\) −19.1241 −0.820695
\(544\) −58.6777 −2.51579
\(545\) −0.0914829 −0.00391870
\(546\) 0 0
\(547\) −29.4820 −1.26056 −0.630280 0.776368i \(-0.717060\pi\)
−0.630280 + 0.776368i \(0.717060\pi\)
\(548\) −28.3774 −1.21222
\(549\) −6.31775 −0.269635
\(550\) 14.1562 0.603624
\(551\) −4.22044 −0.179797
\(552\) 0.358585 0.0152624
\(553\) 0 0
\(554\) 6.04890 0.256993
\(555\) 1.89164 0.0802956
\(556\) −24.7296 −1.04877
\(557\) −10.2316 −0.433526 −0.216763 0.976224i \(-0.569550\pi\)
−0.216763 + 0.976224i \(0.569550\pi\)
\(558\) 1.86218 0.0788325
\(559\) 16.7020 0.706421
\(560\) 0 0
\(561\) 40.9863 1.73044
\(562\) 14.6336 0.617282
\(563\) 23.2599 0.980288 0.490144 0.871641i \(-0.336944\pi\)
0.490144 + 0.871641i \(0.336944\pi\)
\(564\) 3.32500 0.140008
\(565\) −12.1884 −0.512771
\(566\) −40.8897 −1.71872
\(567\) 0 0
\(568\) −3.14010 −0.131756
\(569\) 19.6402 0.823362 0.411681 0.911328i \(-0.364942\pi\)
0.411681 + 0.911328i \(0.364942\pi\)
\(570\) −3.13822 −0.131446
\(571\) −41.5761 −1.73991 −0.869953 0.493134i \(-0.835851\pi\)
−0.869953 + 0.493134i \(0.835851\pi\)
\(572\) 76.1438 3.18373
\(573\) 20.6871 0.864215
\(574\) 0 0
\(575\) 1.22285 0.0509963
\(576\) −9.33688 −0.389037
\(577\) −1.41768 −0.0590187 −0.0295093 0.999565i \(-0.509394\pi\)
−0.0295093 + 0.999565i \(0.509394\pi\)
\(578\) 72.2135 3.00369
\(579\) −2.22901 −0.0926344
\(580\) −22.5813 −0.937636
\(581\) 0 0
\(582\) −6.32678 −0.262254
\(583\) 50.7499 2.10184
\(584\) −0.313852 −0.0129873
\(585\) −12.0071 −0.496433
\(586\) 12.8307 0.530032
\(587\) 28.5127 1.17685 0.588423 0.808553i \(-0.299749\pi\)
0.588423 + 0.808553i \(0.299749\pi\)
\(588\) 0 0
\(589\) 0.720140 0.0296728
\(590\) −43.2601 −1.78099
\(591\) −19.3710 −0.796817
\(592\) 3.52170 0.144741
\(593\) −8.25387 −0.338946 −0.169473 0.985535i \(-0.554206\pi\)
−0.169473 + 0.985535i \(0.554206\pi\)
\(594\) 11.5765 0.474988
\(595\) 0 0
\(596\) 35.2559 1.44414
\(597\) −24.6840 −1.01025
\(598\) 12.6244 0.516249
\(599\) 25.2430 1.03140 0.515700 0.856769i \(-0.327532\pi\)
0.515700 + 0.856769i \(0.327532\pi\)
\(600\) 0.438496 0.0179015
\(601\) 19.9481 0.813702 0.406851 0.913495i \(-0.366627\pi\)
0.406851 + 0.913495i \(0.366627\pi\)
\(602\) 0 0
\(603\) 7.31833 0.298026
\(604\) −28.1371 −1.14488
\(605\) 40.9989 1.66684
\(606\) −27.4381 −1.11459
\(607\) 28.0616 1.13899 0.569493 0.821996i \(-0.307139\pi\)
0.569493 + 0.821996i \(0.307139\pi\)
\(608\) −6.40921 −0.259928
\(609\) 0 0
\(610\) −25.0898 −1.01586
\(611\) 9.44260 0.382007
\(612\) 15.7388 0.636204
\(613\) −34.9865 −1.41309 −0.706545 0.707668i \(-0.749747\pi\)
−0.706545 + 0.707668i \(0.749747\pi\)
\(614\) 51.4779 2.07748
\(615\) 2.95106 0.118998
\(616\) 0 0
\(617\) −30.9952 −1.24782 −0.623911 0.781496i \(-0.714457\pi\)
−0.623911 + 0.781496i \(0.714457\pi\)
\(618\) −12.9481 −0.520848
\(619\) 31.2143 1.25461 0.627305 0.778774i \(-0.284158\pi\)
0.627305 + 0.778774i \(0.284158\pi\)
\(620\) 3.85307 0.154743
\(621\) 1.00000 0.0401286
\(622\) 24.3615 0.976809
\(623\) 0 0
\(624\) −22.3539 −0.894872
\(625\) −17.3904 −0.695616
\(626\) 20.5205 0.820162
\(627\) 4.47682 0.178787
\(628\) −21.1771 −0.845058
\(629\) −7.04161 −0.280767
\(630\) 0 0
\(631\) 19.1609 0.762782 0.381391 0.924414i \(-0.375445\pi\)
0.381391 + 0.924414i \(0.375445\pi\)
\(632\) 2.13952 0.0851054
\(633\) −9.90378 −0.393640
\(634\) −18.1088 −0.719193
\(635\) −24.1624 −0.958854
\(636\) 19.4881 0.772752
\(637\) 0 0
\(638\) 61.8281 2.44780
\(639\) −8.75692 −0.346418
\(640\) −5.55380 −0.219533
\(641\) 16.1599 0.638278 0.319139 0.947708i \(-0.396606\pi\)
0.319139 + 0.947708i \(0.396606\pi\)
\(642\) −32.6144 −1.28719
\(643\) −44.7116 −1.76325 −0.881627 0.471947i \(-0.843551\pi\)
−0.881627 + 0.471947i \(0.843551\pi\)
\(644\) 0 0
\(645\) 5.25407 0.206879
\(646\) 11.6820 0.459623
\(647\) 7.95077 0.312577 0.156288 0.987711i \(-0.450047\pi\)
0.156288 + 0.987711i \(0.450047\pi\)
\(648\) 0.358585 0.0140866
\(649\) 61.7126 2.42243
\(650\) 15.4377 0.605516
\(651\) 0 0
\(652\) −52.4123 −2.05262
\(653\) −42.7024 −1.67107 −0.835537 0.549434i \(-0.814843\pi\)
−0.835537 + 0.549434i \(0.814843\pi\)
\(654\) 0.0961859 0.00376117
\(655\) −6.43485 −0.251430
\(656\) 5.49405 0.214507
\(657\) −0.875251 −0.0341468
\(658\) 0 0
\(659\) −31.2099 −1.21577 −0.607883 0.794027i \(-0.707981\pi\)
−0.607883 + 0.794027i \(0.707981\pi\)
\(660\) 23.9530 0.932370
\(661\) 5.89998 0.229483 0.114741 0.993395i \(-0.463396\pi\)
0.114741 + 0.993395i \(0.463396\pi\)
\(662\) −46.7922 −1.81863
\(663\) 44.6964 1.73586
\(664\) −0.0980926 −0.00380673
\(665\) 0 0
\(666\) −1.98888 −0.0770677
\(667\) 5.34085 0.206798
\(668\) −26.6675 −1.03180
\(669\) −2.21461 −0.0856218
\(670\) 29.0635 1.12282
\(671\) 35.7919 1.38173
\(672\) 0 0
\(673\) −6.61159 −0.254858 −0.127429 0.991848i \(-0.540673\pi\)
−0.127429 + 0.991848i \(0.540673\pi\)
\(674\) 25.0351 0.964316
\(675\) 1.22285 0.0470675
\(676\) 54.7551 2.10597
\(677\) 3.71673 0.142845 0.0714227 0.997446i \(-0.477246\pi\)
0.0714227 + 0.997446i \(0.477246\pi\)
\(678\) 12.8150 0.492157
\(679\) 0 0
\(680\) 5.04187 0.193347
\(681\) 5.58033 0.213839
\(682\) −10.5498 −0.403973
\(683\) 29.5669 1.13135 0.565674 0.824629i \(-0.308616\pi\)
0.565674 + 0.824629i \(0.308616\pi\)
\(684\) 1.71911 0.0657318
\(685\) −25.3513 −0.968622
\(686\) 0 0
\(687\) −6.97859 −0.266250
\(688\) 9.78161 0.372920
\(689\) 55.3438 2.10843
\(690\) 3.97133 0.151186
\(691\) 0.603892 0.0229731 0.0114866 0.999934i \(-0.496344\pi\)
0.0114866 + 0.999934i \(0.496344\pi\)
\(692\) −12.8092 −0.486931
\(693\) 0 0
\(694\) −13.1969 −0.500949
\(695\) −22.0924 −0.838012
\(696\) 1.91515 0.0725936
\(697\) −10.9853 −0.416098
\(698\) 41.8992 1.58591
\(699\) −24.4042 −0.923053
\(700\) 0 0
\(701\) 8.15835 0.308136 0.154068 0.988060i \(-0.450762\pi\)
0.154068 + 0.988060i \(0.450762\pi\)
\(702\) 12.6244 0.476476
\(703\) −0.769137 −0.0290085
\(704\) 52.8961 1.99360
\(705\) 2.97042 0.111872
\(706\) 29.6523 1.11598
\(707\) 0 0
\(708\) 23.6978 0.890618
\(709\) −24.2270 −0.909863 −0.454932 0.890526i \(-0.650336\pi\)
−0.454932 + 0.890526i \(0.650336\pi\)
\(710\) −34.7766 −1.30514
\(711\) 5.96655 0.223763
\(712\) 6.32079 0.236882
\(713\) −0.911316 −0.0341290
\(714\) 0 0
\(715\) 68.0238 2.54394
\(716\) −36.9250 −1.37995
\(717\) −0.424159 −0.0158405
\(718\) 9.55861 0.356724
\(719\) 2.58504 0.0964058 0.0482029 0.998838i \(-0.484651\pi\)
0.0482029 + 0.998838i \(0.484651\pi\)
\(720\) −7.03200 −0.262067
\(721\) 0 0
\(722\) −37.5486 −1.39741
\(723\) 16.3125 0.606667
\(724\) 41.6042 1.54621
\(725\) 6.53105 0.242557
\(726\) −43.1065 −1.59983
\(727\) −1.39574 −0.0517650 −0.0258825 0.999665i \(-0.508240\pi\)
−0.0258825 + 0.999665i \(0.508240\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −3.47591 −0.128649
\(731\) −19.5582 −0.723388
\(732\) 13.7442 0.507999
\(733\) 20.4900 0.756815 0.378408 0.925639i \(-0.376472\pi\)
0.378408 + 0.925639i \(0.376472\pi\)
\(734\) −57.3500 −2.11683
\(735\) 0 0
\(736\) 8.11067 0.298963
\(737\) −41.4605 −1.52722
\(738\) −3.10277 −0.114215
\(739\) −47.4816 −1.74664 −0.873320 0.487148i \(-0.838037\pi\)
−0.873320 + 0.487148i \(0.838037\pi\)
\(740\) −4.11523 −0.151279
\(741\) 4.88207 0.179347
\(742\) 0 0
\(743\) −41.4450 −1.52047 −0.760235 0.649648i \(-0.774916\pi\)
−0.760235 + 0.649648i \(0.774916\pi\)
\(744\) −0.326785 −0.0119805
\(745\) 31.4962 1.15393
\(746\) 29.8955 1.09455
\(747\) −0.273554 −0.0100088
\(748\) −89.1650 −3.26020
\(749\) 0 0
\(750\) 24.7130 0.902390
\(751\) 11.5886 0.422872 0.211436 0.977392i \(-0.432186\pi\)
0.211436 + 0.977392i \(0.432186\pi\)
\(752\) 5.53009 0.201662
\(753\) 15.9242 0.580311
\(754\) 67.4248 2.45547
\(755\) −25.1366 −0.914814
\(756\) 0 0
\(757\) −11.3101 −0.411073 −0.205536 0.978649i \(-0.565894\pi\)
−0.205536 + 0.978649i \(0.565894\pi\)
\(758\) −4.99344 −0.181370
\(759\) −5.66529 −0.205637
\(760\) 0.550710 0.0199763
\(761\) −21.8257 −0.791183 −0.395591 0.918427i \(-0.629460\pi\)
−0.395591 + 0.918427i \(0.629460\pi\)
\(762\) 25.4045 0.920308
\(763\) 0 0
\(764\) −45.0044 −1.62820
\(765\) 14.0604 0.508356
\(766\) 62.8941 2.27246
\(767\) 67.2990 2.43002
\(768\) −12.8345 −0.463124
\(769\) −24.1550 −0.871052 −0.435526 0.900176i \(-0.643438\pi\)
−0.435526 + 0.900176i \(0.643438\pi\)
\(770\) 0 0
\(771\) 30.1569 1.08608
\(772\) 4.84917 0.174525
\(773\) 14.7390 0.530124 0.265062 0.964231i \(-0.414608\pi\)
0.265062 + 0.964231i \(0.414608\pi\)
\(774\) −5.52417 −0.198562
\(775\) −1.11440 −0.0400305
\(776\) 1.11025 0.0398558
\(777\) 0 0
\(778\) 24.8999 0.892703
\(779\) −1.19990 −0.0429908
\(780\) 26.1213 0.935292
\(781\) 49.6105 1.77520
\(782\) −14.7832 −0.528648
\(783\) 5.34085 0.190866
\(784\) 0 0
\(785\) −18.9188 −0.675239
\(786\) 6.76565 0.241323
\(787\) 23.8103 0.848746 0.424373 0.905487i \(-0.360495\pi\)
0.424373 + 0.905487i \(0.360495\pi\)
\(788\) 42.1414 1.50122
\(789\) −15.1656 −0.539911
\(790\) 23.6951 0.843034
\(791\) 0 0
\(792\) −2.03149 −0.0721859
\(793\) 39.0318 1.38606
\(794\) 43.5334 1.54494
\(795\) 17.4098 0.617464
\(796\) 53.6997 1.90334
\(797\) 19.8938 0.704674 0.352337 0.935873i \(-0.385387\pi\)
0.352337 + 0.935873i \(0.385387\pi\)
\(798\) 0 0
\(799\) −11.0574 −0.391181
\(800\) 9.91812 0.350659
\(801\) 17.6270 0.622820
\(802\) 15.1269 0.534149
\(803\) 4.95855 0.174983
\(804\) −15.9209 −0.561488
\(805\) 0 0
\(806\) −11.5048 −0.405239
\(807\) −0.470833 −0.0165741
\(808\) 4.81496 0.169390
\(809\) −47.9127 −1.68452 −0.842260 0.539071i \(-0.818775\pi\)
−0.842260 + 0.539071i \(0.818775\pi\)
\(810\) 3.97133 0.139538
\(811\) −48.3843 −1.69900 −0.849500 0.527588i \(-0.823097\pi\)
−0.849500 + 0.527588i \(0.823097\pi\)
\(812\) 0 0
\(813\) 7.28736 0.255579
\(814\) 11.2676 0.394929
\(815\) −46.8230 −1.64014
\(816\) 26.1766 0.916364
\(817\) −2.13629 −0.0747395
\(818\) 54.0835 1.89098
\(819\) 0 0
\(820\) −6.41999 −0.224196
\(821\) −8.16367 −0.284914 −0.142457 0.989801i \(-0.545500\pi\)
−0.142457 + 0.989801i \(0.545500\pi\)
\(822\) 26.6545 0.929683
\(823\) −33.2648 −1.15954 −0.579769 0.814781i \(-0.696857\pi\)
−0.579769 + 0.814781i \(0.696857\pi\)
\(824\) 2.27219 0.0791554
\(825\) −6.92779 −0.241195
\(826\) 0 0
\(827\) 31.2711 1.08740 0.543701 0.839279i \(-0.317023\pi\)
0.543701 + 0.839279i \(0.317023\pi\)
\(828\) −2.17548 −0.0756033
\(829\) −17.8076 −0.618484 −0.309242 0.950983i \(-0.600075\pi\)
−0.309242 + 0.950983i \(0.600075\pi\)
\(830\) −1.08637 −0.0377086
\(831\) −2.96021 −0.102689
\(832\) 57.6844 1.99985
\(833\) 0 0
\(834\) 23.2281 0.804324
\(835\) −23.8236 −0.824451
\(836\) −9.73926 −0.336839
\(837\) −0.911316 −0.0314997
\(838\) −62.7924 −2.16913
\(839\) 29.7947 1.02863 0.514313 0.857603i \(-0.328047\pi\)
0.514313 + 0.857603i \(0.328047\pi\)
\(840\) 0 0
\(841\) −0.475352 −0.0163915
\(842\) 5.42669 0.187016
\(843\) −7.16141 −0.246652
\(844\) 21.5455 0.741628
\(845\) 48.9160 1.68276
\(846\) −3.12312 −0.107375
\(847\) 0 0
\(848\) 32.4123 1.11304
\(849\) 20.0106 0.686762
\(850\) −18.0777 −0.620059
\(851\) 0.973320 0.0333650
\(852\) 19.0505 0.652661
\(853\) −43.0902 −1.47538 −0.737690 0.675140i \(-0.764083\pi\)
−0.737690 + 0.675140i \(0.764083\pi\)
\(854\) 0 0
\(855\) 1.53578 0.0525227
\(856\) 5.72333 0.195620
\(857\) 48.0096 1.63998 0.819988 0.572381i \(-0.193980\pi\)
0.819988 + 0.572381i \(0.193980\pi\)
\(858\) −71.5207 −2.44168
\(859\) 11.7497 0.400896 0.200448 0.979704i \(-0.435760\pi\)
0.200448 + 0.979704i \(0.435760\pi\)
\(860\) −11.4301 −0.389765
\(861\) 0 0
\(862\) 43.4306 1.47925
\(863\) −5.60759 −0.190885 −0.0954424 0.995435i \(-0.530427\pi\)
−0.0954424 + 0.995435i \(0.530427\pi\)
\(864\) 8.11067 0.275931
\(865\) −11.4432 −0.389080
\(866\) −34.4783 −1.17162
\(867\) −35.3399 −1.20021
\(868\) 0 0
\(869\) −33.8022 −1.14666
\(870\) 21.2103 0.719095
\(871\) −45.2135 −1.53200
\(872\) −0.0168791 −0.000571600 0
\(873\) 3.09620 0.104791
\(874\) −1.61473 −0.0546192
\(875\) 0 0
\(876\) 1.90409 0.0643334
\(877\) 16.7638 0.566075 0.283037 0.959109i \(-0.408658\pi\)
0.283037 + 0.959109i \(0.408658\pi\)
\(878\) −22.6623 −0.764816
\(879\) −6.27910 −0.211789
\(880\) 39.8383 1.34295
\(881\) −39.4239 −1.32823 −0.664113 0.747632i \(-0.731191\pi\)
−0.664113 + 0.747632i \(0.731191\pi\)
\(882\) 0 0
\(883\) −17.4928 −0.588679 −0.294339 0.955701i \(-0.595100\pi\)
−0.294339 + 0.955701i \(0.595100\pi\)
\(884\) −97.2364 −3.27041
\(885\) 21.1707 0.711644
\(886\) −26.1028 −0.876940
\(887\) 2.75212 0.0924073 0.0462037 0.998932i \(-0.485288\pi\)
0.0462037 + 0.998932i \(0.485288\pi\)
\(888\) 0.349019 0.0117123
\(889\) 0 0
\(890\) 70.0026 2.34649
\(891\) −5.66529 −0.189794
\(892\) 4.81785 0.161314
\(893\) −1.20777 −0.0404164
\(894\) −33.1154 −1.10754
\(895\) −32.9873 −1.10264
\(896\) 0 0
\(897\) −6.17812 −0.206281
\(898\) 79.0988 2.63956
\(899\) −4.86720 −0.162330
\(900\) −2.66029 −0.0886763
\(901\) −64.8081 −2.15907
\(902\) 17.5781 0.585286
\(903\) 0 0
\(904\) −2.24884 −0.0747952
\(905\) 37.1675 1.23549
\(906\) 26.4288 0.878038
\(907\) −29.1794 −0.968886 −0.484443 0.874823i \(-0.660978\pi\)
−0.484443 + 0.874823i \(0.660978\pi\)
\(908\) −12.1399 −0.402878
\(909\) 13.4276 0.445367
\(910\) 0 0
\(911\) −6.24020 −0.206747 −0.103374 0.994643i \(-0.532964\pi\)
−0.103374 + 0.994643i \(0.532964\pi\)
\(912\) 2.85920 0.0946776
\(913\) 1.54976 0.0512897
\(914\) −74.1564 −2.45288
\(915\) 12.2785 0.405914
\(916\) 15.1818 0.501621
\(917\) 0 0
\(918\) −14.7832 −0.487920
\(919\) −36.4022 −1.20080 −0.600398 0.799701i \(-0.704991\pi\)
−0.600398 + 0.799701i \(0.704991\pi\)
\(920\) −0.696907 −0.0229763
\(921\) −25.1923 −0.830114
\(922\) −7.57259 −0.249390
\(923\) 54.1013 1.78077
\(924\) 0 0
\(925\) 1.19022 0.0391343
\(926\) −61.2869 −2.01401
\(927\) 6.33653 0.208119
\(928\) 43.3179 1.42198
\(929\) −0.0513202 −0.00168376 −0.000841881 1.00000i \(-0.500268\pi\)
−0.000841881 1.00000i \(0.500268\pi\)
\(930\) −3.61913 −0.118676
\(931\) 0 0
\(932\) 53.0911 1.73906
\(933\) −11.9221 −0.390311
\(934\) 57.3878 1.87779
\(935\) −79.6564 −2.60504
\(936\) −2.21538 −0.0724121
\(937\) −8.46139 −0.276421 −0.138211 0.990403i \(-0.544135\pi\)
−0.138211 + 0.990403i \(0.544135\pi\)
\(938\) 0 0
\(939\) −10.0423 −0.327718
\(940\) −6.46210 −0.210770
\(941\) −6.94143 −0.226284 −0.113142 0.993579i \(-0.536092\pi\)
−0.113142 + 0.993579i \(0.536092\pi\)
\(942\) 19.8913 0.648094
\(943\) 1.51843 0.0494470
\(944\) 39.4139 1.28281
\(945\) 0 0
\(946\) 31.2960 1.01752
\(947\) 6.05341 0.196709 0.0983547 0.995151i \(-0.468642\pi\)
0.0983547 + 0.995151i \(0.468642\pi\)
\(948\) −12.9801 −0.421575
\(949\) 5.40740 0.175532
\(950\) −1.97458 −0.0640637
\(951\) 8.86210 0.287373
\(952\) 0 0
\(953\) −52.7135 −1.70756 −0.853778 0.520637i \(-0.825695\pi\)
−0.853778 + 0.520637i \(0.825695\pi\)
\(954\) −18.3049 −0.592641
\(955\) −40.2051 −1.30101
\(956\) 0.922751 0.0298439
\(957\) −30.2574 −0.978084
\(958\) −26.1931 −0.846262
\(959\) 0 0
\(960\) 18.1461 0.585664
\(961\) −30.1695 −0.973210
\(962\) 12.2876 0.396167
\(963\) 15.9609 0.514332
\(964\) −35.4875 −1.14298
\(965\) 4.33205 0.139454
\(966\) 0 0
\(967\) −16.9757 −0.545900 −0.272950 0.962028i \(-0.587999\pi\)
−0.272950 + 0.962028i \(0.587999\pi\)
\(968\) 7.56454 0.243133
\(969\) −5.71695 −0.183655
\(970\) 12.2960 0.394802
\(971\) −29.0124 −0.931051 −0.465525 0.885034i \(-0.654135\pi\)
−0.465525 + 0.885034i \(0.654135\pi\)
\(972\) −2.17548 −0.0697787
\(973\) 0 0
\(974\) 19.0801 0.611367
\(975\) −7.55490 −0.241951
\(976\) 22.8591 0.731702
\(977\) −32.1866 −1.02974 −0.514871 0.857268i \(-0.672160\pi\)
−0.514871 + 0.857268i \(0.672160\pi\)
\(978\) 49.2301 1.57421
\(979\) −99.8621 −3.19161
\(980\) 0 0
\(981\) −0.0470715 −0.00150288
\(982\) −4.05523 −0.129408
\(983\) 6.03861 0.192602 0.0963009 0.995352i \(-0.469299\pi\)
0.0963009 + 0.995352i \(0.469299\pi\)
\(984\) 0.544489 0.0173577
\(985\) 37.6474 1.19955
\(986\) −78.9551 −2.51444
\(987\) 0 0
\(988\) −10.6209 −0.337895
\(989\) 2.70342 0.0859637
\(990\) −22.4987 −0.715056
\(991\) 20.8108 0.661075 0.330538 0.943793i \(-0.392770\pi\)
0.330538 + 0.943793i \(0.392770\pi\)
\(992\) −7.39139 −0.234677
\(993\) 22.8992 0.726684
\(994\) 0 0
\(995\) 47.9732 1.52085
\(996\) 0.595113 0.0188569
\(997\) −53.1594 −1.68358 −0.841788 0.539809i \(-0.818496\pi\)
−0.841788 + 0.539809i \(0.818496\pi\)
\(998\) −83.5224 −2.64385
\(999\) 0.973320 0.0307945
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 3381.2.a.bc.1.6 6
7.2 even 3 483.2.i.f.277.1 12
7.4 even 3 483.2.i.f.415.1 yes 12
7.6 odd 2 3381.2.a.bd.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.f.277.1 12 7.2 even 3
483.2.i.f.415.1 yes 12 7.4 even 3
3381.2.a.bc.1.6 6 1.1 even 1 trivial
3381.2.a.bd.1.6 6 7.6 odd 2