Properties

Label 6039.2.a.k.1.10
Level $6039$
Weight $2$
Character 6039.1
Self dual yes
Analytic conductor $48.222$
Analytic rank $0$
Dimension $19$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 5 x^{18} - 18 x^{17} + 122 x^{16} + 78 x^{15} - 1177 x^{14} + 387 x^{13} + 5755 x^{12} - 4673 x^{11} - 15053 x^{10} + 16875 x^{9} + 20141 x^{8} - 28019 x^{7} - 11589 x^{6} + \cdots - 43 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 671)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(0.488177\) of defining polynomial
Character \(\chi\) \(=\) 6039.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.488177 q^{2} -1.76168 q^{4} -1.92726 q^{5} +0.259128 q^{7} +1.83637 q^{8} +O(q^{10})\) \(q-0.488177 q^{2} -1.76168 q^{4} -1.92726 q^{5} +0.259128 q^{7} +1.83637 q^{8} +0.940847 q^{10} -1.00000 q^{11} -2.14915 q^{13} -0.126500 q^{14} +2.62689 q^{16} -4.30678 q^{17} +7.44254 q^{19} +3.39523 q^{20} +0.488177 q^{22} -3.82807 q^{23} -1.28565 q^{25} +1.04917 q^{26} -0.456501 q^{28} +3.58210 q^{29} -8.02405 q^{31} -4.95513 q^{32} +2.10248 q^{34} -0.499408 q^{35} -6.82822 q^{37} -3.63328 q^{38} -3.53917 q^{40} -5.88081 q^{41} +7.55874 q^{43} +1.76168 q^{44} +1.86878 q^{46} +13.1755 q^{47} -6.93285 q^{49} +0.627626 q^{50} +3.78612 q^{52} +10.1230 q^{53} +1.92726 q^{55} +0.475854 q^{56} -1.74870 q^{58} -5.99815 q^{59} -1.00000 q^{61} +3.91716 q^{62} -2.83480 q^{64} +4.14198 q^{65} -1.73926 q^{67} +7.58719 q^{68} +0.243800 q^{70} -11.3363 q^{71} -1.24305 q^{73} +3.33338 q^{74} -13.1114 q^{76} -0.259128 q^{77} -9.01217 q^{79} -5.06272 q^{80} +2.87088 q^{82} +13.5489 q^{83} +8.30031 q^{85} -3.69001 q^{86} -1.83637 q^{88} -5.65666 q^{89} -0.556904 q^{91} +6.74385 q^{92} -6.43198 q^{94} -14.3437 q^{95} -17.0543 q^{97} +3.38446 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 5 q^{2} + 23 q^{4} + 9 q^{7} - 9 q^{8} + 7 q^{10} - 19 q^{11} + 8 q^{13} + 11 q^{14} + 31 q^{16} - 9 q^{17} + 17 q^{19} + 6 q^{20} + 5 q^{22} + 10 q^{23} + 45 q^{25} - 5 q^{26} + 36 q^{28} - 27 q^{29} + 7 q^{31} - 8 q^{32} - 5 q^{34} - 17 q^{35} + 20 q^{37} + 37 q^{38} + 10 q^{40} - 19 q^{41} + 20 q^{43} - 23 q^{44} + 41 q^{46} + 19 q^{47} + 42 q^{49} - 36 q^{50} - 28 q^{52} - 3 q^{53} + 44 q^{56} + 23 q^{58} + 28 q^{59} - 19 q^{61} + 11 q^{62} + 47 q^{64} - 25 q^{65} + 3 q^{67} - 38 q^{68} + 3 q^{70} + 19 q^{71} + 20 q^{73} + 22 q^{74} - 25 q^{76} - 9 q^{77} + 69 q^{79} + 36 q^{80} - 61 q^{82} - q^{83} + 24 q^{85} + 27 q^{86} + 9 q^{88} + 24 q^{91} + 67 q^{92} + 64 q^{94} + 3 q^{95} + 21 q^{97} + 87 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.488177 −0.345194 −0.172597 0.984993i \(-0.555216\pi\)
−0.172597 + 0.984993i \(0.555216\pi\)
\(3\) 0 0
\(4\) −1.76168 −0.880841
\(5\) −1.92726 −0.861899 −0.430949 0.902376i \(-0.641821\pi\)
−0.430949 + 0.902376i \(0.641821\pi\)
\(6\) 0 0
\(7\) 0.259128 0.0979411 0.0489706 0.998800i \(-0.484406\pi\)
0.0489706 + 0.998800i \(0.484406\pi\)
\(8\) 1.83637 0.649254
\(9\) 0 0
\(10\) 0.940847 0.297522
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −2.14915 −0.596067 −0.298033 0.954555i \(-0.596331\pi\)
−0.298033 + 0.954555i \(0.596331\pi\)
\(14\) −0.126500 −0.0338086
\(15\) 0 0
\(16\) 2.62689 0.656723
\(17\) −4.30678 −1.04455 −0.522274 0.852778i \(-0.674916\pi\)
−0.522274 + 0.852778i \(0.674916\pi\)
\(18\) 0 0
\(19\) 7.44254 1.70744 0.853718 0.520736i \(-0.174342\pi\)
0.853718 + 0.520736i \(0.174342\pi\)
\(20\) 3.39523 0.759196
\(21\) 0 0
\(22\) 0.488177 0.104080
\(23\) −3.82807 −0.798208 −0.399104 0.916906i \(-0.630679\pi\)
−0.399104 + 0.916906i \(0.630679\pi\)
\(24\) 0 0
\(25\) −1.28565 −0.257130
\(26\) 1.04917 0.205758
\(27\) 0 0
\(28\) −0.456501 −0.0862706
\(29\) 3.58210 0.665179 0.332589 0.943072i \(-0.392078\pi\)
0.332589 + 0.943072i \(0.392078\pi\)
\(30\) 0 0
\(31\) −8.02405 −1.44116 −0.720581 0.693371i \(-0.756125\pi\)
−0.720581 + 0.693371i \(0.756125\pi\)
\(32\) −4.95513 −0.875951
\(33\) 0 0
\(34\) 2.10248 0.360571
\(35\) −0.499408 −0.0844153
\(36\) 0 0
\(37\) −6.82822 −1.12255 −0.561276 0.827628i \(-0.689690\pi\)
−0.561276 + 0.827628i \(0.689690\pi\)
\(38\) −3.63328 −0.589396
\(39\) 0 0
\(40\) −3.53917 −0.559592
\(41\) −5.88081 −0.918429 −0.459214 0.888326i \(-0.651869\pi\)
−0.459214 + 0.888326i \(0.651869\pi\)
\(42\) 0 0
\(43\) 7.55874 1.15270 0.576348 0.817204i \(-0.304477\pi\)
0.576348 + 0.817204i \(0.304477\pi\)
\(44\) 1.76168 0.265584
\(45\) 0 0
\(46\) 1.86878 0.275536
\(47\) 13.1755 1.92184 0.960922 0.276820i \(-0.0892807\pi\)
0.960922 + 0.276820i \(0.0892807\pi\)
\(48\) 0 0
\(49\) −6.93285 −0.990408
\(50\) 0.627626 0.0887597
\(51\) 0 0
\(52\) 3.78612 0.525040
\(53\) 10.1230 1.39050 0.695249 0.718769i \(-0.255294\pi\)
0.695249 + 0.718769i \(0.255294\pi\)
\(54\) 0 0
\(55\) 1.92726 0.259872
\(56\) 0.475854 0.0635887
\(57\) 0 0
\(58\) −1.74870 −0.229616
\(59\) −5.99815 −0.780892 −0.390446 0.920626i \(-0.627679\pi\)
−0.390446 + 0.920626i \(0.627679\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037
\(62\) 3.91716 0.497480
\(63\) 0 0
\(64\) −2.83480 −0.354350
\(65\) 4.14198 0.513749
\(66\) 0 0
\(67\) −1.73926 −0.212484 −0.106242 0.994340i \(-0.533882\pi\)
−0.106242 + 0.994340i \(0.533882\pi\)
\(68\) 7.58719 0.920082
\(69\) 0 0
\(70\) 0.243800 0.0291396
\(71\) −11.3363 −1.34537 −0.672687 0.739927i \(-0.734860\pi\)
−0.672687 + 0.739927i \(0.734860\pi\)
\(72\) 0 0
\(73\) −1.24305 −0.145488 −0.0727440 0.997351i \(-0.523176\pi\)
−0.0727440 + 0.997351i \(0.523176\pi\)
\(74\) 3.33338 0.387498
\(75\) 0 0
\(76\) −13.1114 −1.50398
\(77\) −0.259128 −0.0295304
\(78\) 0 0
\(79\) −9.01217 −1.01395 −0.506974 0.861961i \(-0.669236\pi\)
−0.506974 + 0.861961i \(0.669236\pi\)
\(80\) −5.06272 −0.566029
\(81\) 0 0
\(82\) 2.87088 0.317036
\(83\) 13.5489 1.48719 0.743595 0.668631i \(-0.233119\pi\)
0.743595 + 0.668631i \(0.233119\pi\)
\(84\) 0 0
\(85\) 8.30031 0.900295
\(86\) −3.69001 −0.397903
\(87\) 0 0
\(88\) −1.83637 −0.195758
\(89\) −5.65666 −0.599605 −0.299803 0.954001i \(-0.596921\pi\)
−0.299803 + 0.954001i \(0.596921\pi\)
\(90\) 0 0
\(91\) −0.556904 −0.0583794
\(92\) 6.74385 0.703095
\(93\) 0 0
\(94\) −6.43198 −0.663408
\(95\) −14.3437 −1.47164
\(96\) 0 0
\(97\) −17.0543 −1.73161 −0.865803 0.500385i \(-0.833192\pi\)
−0.865803 + 0.500385i \(0.833192\pi\)
\(98\) 3.38446 0.341882
\(99\) 0 0
\(100\) 2.26491 0.226491
\(101\) −17.3123 −1.72264 −0.861321 0.508062i \(-0.830362\pi\)
−0.861321 + 0.508062i \(0.830362\pi\)
\(102\) 0 0
\(103\) 1.07195 0.105622 0.0528112 0.998605i \(-0.483182\pi\)
0.0528112 + 0.998605i \(0.483182\pi\)
\(104\) −3.94663 −0.386999
\(105\) 0 0
\(106\) −4.94181 −0.479991
\(107\) 7.23733 0.699660 0.349830 0.936813i \(-0.386239\pi\)
0.349830 + 0.936813i \(0.386239\pi\)
\(108\) 0 0
\(109\) 12.0213 1.15143 0.575714 0.817651i \(-0.304724\pi\)
0.575714 + 0.817651i \(0.304724\pi\)
\(110\) −0.940847 −0.0897063
\(111\) 0 0
\(112\) 0.680701 0.0643202
\(113\) −18.4118 −1.73204 −0.866018 0.500012i \(-0.833329\pi\)
−0.866018 + 0.500012i \(0.833329\pi\)
\(114\) 0 0
\(115\) 7.37770 0.687975
\(116\) −6.31052 −0.585917
\(117\) 0 0
\(118\) 2.92816 0.269559
\(119\) −1.11601 −0.102304
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.488177 0.0441975
\(123\) 0 0
\(124\) 14.1358 1.26943
\(125\) 12.1141 1.08352
\(126\) 0 0
\(127\) −7.91532 −0.702371 −0.351185 0.936306i \(-0.614221\pi\)
−0.351185 + 0.936306i \(0.614221\pi\)
\(128\) 11.2941 0.998270
\(129\) 0 0
\(130\) −2.02202 −0.177343
\(131\) 13.2420 1.15695 0.578477 0.815698i \(-0.303647\pi\)
0.578477 + 0.815698i \(0.303647\pi\)
\(132\) 0 0
\(133\) 1.92857 0.167228
\(134\) 0.849067 0.0733482
\(135\) 0 0
\(136\) −7.90884 −0.678178
\(137\) −7.41315 −0.633349 −0.316674 0.948534i \(-0.602566\pi\)
−0.316674 + 0.948534i \(0.602566\pi\)
\(138\) 0 0
\(139\) −3.45579 −0.293116 −0.146558 0.989202i \(-0.546819\pi\)
−0.146558 + 0.989202i \(0.546819\pi\)
\(140\) 0.879798 0.0743565
\(141\) 0 0
\(142\) 5.53414 0.464414
\(143\) 2.14915 0.179721
\(144\) 0 0
\(145\) −6.90365 −0.573317
\(146\) 0.606829 0.0502215
\(147\) 0 0
\(148\) 12.0292 0.988791
\(149\) 8.16171 0.668634 0.334317 0.942461i \(-0.391494\pi\)
0.334317 + 0.942461i \(0.391494\pi\)
\(150\) 0 0
\(151\) 5.09846 0.414906 0.207453 0.978245i \(-0.433483\pi\)
0.207453 + 0.978245i \(0.433483\pi\)
\(152\) 13.6672 1.10856
\(153\) 0 0
\(154\) 0.126500 0.0101937
\(155\) 15.4645 1.24214
\(156\) 0 0
\(157\) 7.46388 0.595682 0.297841 0.954615i \(-0.403733\pi\)
0.297841 + 0.954615i \(0.403733\pi\)
\(158\) 4.39954 0.350009
\(159\) 0 0
\(160\) 9.54984 0.754981
\(161\) −0.991960 −0.0781774
\(162\) 0 0
\(163\) 11.6957 0.916077 0.458038 0.888932i \(-0.348552\pi\)
0.458038 + 0.888932i \(0.348552\pi\)
\(164\) 10.3601 0.808990
\(165\) 0 0
\(166\) −6.61429 −0.513368
\(167\) −10.6559 −0.824581 −0.412291 0.911052i \(-0.635271\pi\)
−0.412291 + 0.911052i \(0.635271\pi\)
\(168\) 0 0
\(169\) −8.38116 −0.644705
\(170\) −4.05203 −0.310776
\(171\) 0 0
\(172\) −13.3161 −1.01534
\(173\) 2.15144 0.163571 0.0817855 0.996650i \(-0.473938\pi\)
0.0817855 + 0.996650i \(0.473938\pi\)
\(174\) 0 0
\(175\) −0.333148 −0.0251836
\(176\) −2.62689 −0.198009
\(177\) 0 0
\(178\) 2.76146 0.206980
\(179\) 10.1067 0.755414 0.377707 0.925925i \(-0.376713\pi\)
0.377707 + 0.925925i \(0.376713\pi\)
\(180\) 0 0
\(181\) 0.277428 0.0206211 0.0103105 0.999947i \(-0.496718\pi\)
0.0103105 + 0.999947i \(0.496718\pi\)
\(182\) 0.271868 0.0201522
\(183\) 0 0
\(184\) −7.02975 −0.518240
\(185\) 13.1598 0.967527
\(186\) 0 0
\(187\) 4.30678 0.314943
\(188\) −23.2110 −1.69284
\(189\) 0 0
\(190\) 7.00229 0.508000
\(191\) 6.09681 0.441150 0.220575 0.975370i \(-0.429207\pi\)
0.220575 + 0.975370i \(0.429207\pi\)
\(192\) 0 0
\(193\) 9.45136 0.680323 0.340162 0.940367i \(-0.389518\pi\)
0.340162 + 0.940367i \(0.389518\pi\)
\(194\) 8.32555 0.597739
\(195\) 0 0
\(196\) 12.2135 0.872392
\(197\) 18.3986 1.31085 0.655425 0.755261i \(-0.272490\pi\)
0.655425 + 0.755261i \(0.272490\pi\)
\(198\) 0 0
\(199\) 7.30293 0.517691 0.258846 0.965919i \(-0.416658\pi\)
0.258846 + 0.965919i \(0.416658\pi\)
\(200\) −2.36093 −0.166943
\(201\) 0 0
\(202\) 8.45149 0.594645
\(203\) 0.928221 0.0651484
\(204\) 0 0
\(205\) 11.3339 0.791593
\(206\) −0.523302 −0.0364601
\(207\) 0 0
\(208\) −5.64558 −0.391451
\(209\) −7.44254 −0.514811
\(210\) 0 0
\(211\) 18.0652 1.24366 0.621828 0.783154i \(-0.286390\pi\)
0.621828 + 0.783154i \(0.286390\pi\)
\(212\) −17.8335 −1.22481
\(213\) 0 0
\(214\) −3.53310 −0.241518
\(215\) −14.5677 −0.993508
\(216\) 0 0
\(217\) −2.07925 −0.141149
\(218\) −5.86851 −0.397466
\(219\) 0 0
\(220\) −3.39523 −0.228906
\(221\) 9.25592 0.622621
\(222\) 0 0
\(223\) 9.04349 0.605597 0.302799 0.953055i \(-0.402079\pi\)
0.302799 + 0.953055i \(0.402079\pi\)
\(224\) −1.28401 −0.0857916
\(225\) 0 0
\(226\) 8.98823 0.597888
\(227\) 13.3724 0.887560 0.443780 0.896136i \(-0.353637\pi\)
0.443780 + 0.896136i \(0.353637\pi\)
\(228\) 0 0
\(229\) 16.6227 1.09846 0.549228 0.835672i \(-0.314922\pi\)
0.549228 + 0.835672i \(0.314922\pi\)
\(230\) −3.60163 −0.237484
\(231\) 0 0
\(232\) 6.57805 0.431870
\(233\) 0.993972 0.0651173 0.0325586 0.999470i \(-0.489634\pi\)
0.0325586 + 0.999470i \(0.489634\pi\)
\(234\) 0 0
\(235\) −25.3927 −1.65643
\(236\) 10.5668 0.687842
\(237\) 0 0
\(238\) 0.544810 0.0353148
\(239\) 7.06734 0.457148 0.228574 0.973527i \(-0.426594\pi\)
0.228574 + 0.973527i \(0.426594\pi\)
\(240\) 0 0
\(241\) −19.5749 −1.26093 −0.630467 0.776216i \(-0.717136\pi\)
−0.630467 + 0.776216i \(0.717136\pi\)
\(242\) −0.488177 −0.0313812
\(243\) 0 0
\(244\) 1.76168 0.112780
\(245\) 13.3614 0.853631
\(246\) 0 0
\(247\) −15.9951 −1.01775
\(248\) −14.7351 −0.935680
\(249\) 0 0
\(250\) −5.91384 −0.374024
\(251\) 1.82154 0.114974 0.0574872 0.998346i \(-0.481691\pi\)
0.0574872 + 0.998346i \(0.481691\pi\)
\(252\) 0 0
\(253\) 3.82807 0.240669
\(254\) 3.86408 0.242454
\(255\) 0 0
\(256\) 0.156059 0.00975369
\(257\) 1.98706 0.123949 0.0619747 0.998078i \(-0.480260\pi\)
0.0619747 + 0.998078i \(0.480260\pi\)
\(258\) 0 0
\(259\) −1.76938 −0.109944
\(260\) −7.29685 −0.452532
\(261\) 0 0
\(262\) −6.46442 −0.399373
\(263\) −8.50019 −0.524144 −0.262072 0.965048i \(-0.584406\pi\)
−0.262072 + 0.965048i \(0.584406\pi\)
\(264\) 0 0
\(265\) −19.5097 −1.19847
\(266\) −0.941484 −0.0577261
\(267\) 0 0
\(268\) 3.06402 0.187165
\(269\) −19.4329 −1.18485 −0.592423 0.805627i \(-0.701829\pi\)
−0.592423 + 0.805627i \(0.701829\pi\)
\(270\) 0 0
\(271\) 19.6791 1.19542 0.597710 0.801712i \(-0.296077\pi\)
0.597710 + 0.801712i \(0.296077\pi\)
\(272\) −11.3135 −0.685979
\(273\) 0 0
\(274\) 3.61893 0.218628
\(275\) 1.28565 0.0775277
\(276\) 0 0
\(277\) −18.0557 −1.08486 −0.542430 0.840101i \(-0.682496\pi\)
−0.542430 + 0.840101i \(0.682496\pi\)
\(278\) 1.68704 0.101182
\(279\) 0 0
\(280\) −0.917097 −0.0548070
\(281\) 20.0260 1.19465 0.597326 0.801999i \(-0.296230\pi\)
0.597326 + 0.801999i \(0.296230\pi\)
\(282\) 0 0
\(283\) 30.7189 1.82605 0.913026 0.407902i \(-0.133739\pi\)
0.913026 + 0.407902i \(0.133739\pi\)
\(284\) 19.9710 1.18506
\(285\) 0 0
\(286\) −1.04917 −0.0620385
\(287\) −1.52388 −0.0899519
\(288\) 0 0
\(289\) 1.54839 0.0910818
\(290\) 3.37021 0.197905
\(291\) 0 0
\(292\) 2.18986 0.128152
\(293\) −30.1972 −1.76414 −0.882069 0.471121i \(-0.843850\pi\)
−0.882069 + 0.471121i \(0.843850\pi\)
\(294\) 0 0
\(295\) 11.5600 0.673050
\(296\) −12.5391 −0.728822
\(297\) 0 0
\(298\) −3.98436 −0.230808
\(299\) 8.22709 0.475785
\(300\) 0 0
\(301\) 1.95868 0.112896
\(302\) −2.48895 −0.143223
\(303\) 0 0
\(304\) 19.5507 1.12131
\(305\) 1.92726 0.110355
\(306\) 0 0
\(307\) 6.02731 0.343997 0.171998 0.985097i \(-0.444978\pi\)
0.171998 + 0.985097i \(0.444978\pi\)
\(308\) 0.456501 0.0260116
\(309\) 0 0
\(310\) −7.54940 −0.428777
\(311\) 15.9585 0.904922 0.452461 0.891784i \(-0.350546\pi\)
0.452461 + 0.891784i \(0.350546\pi\)
\(312\) 0 0
\(313\) 3.72668 0.210644 0.105322 0.994438i \(-0.466413\pi\)
0.105322 + 0.994438i \(0.466413\pi\)
\(314\) −3.64370 −0.205626
\(315\) 0 0
\(316\) 15.8766 0.893128
\(317\) 12.4598 0.699811 0.349905 0.936785i \(-0.386214\pi\)
0.349905 + 0.936785i \(0.386214\pi\)
\(318\) 0 0
\(319\) −3.58210 −0.200559
\(320\) 5.46341 0.305414
\(321\) 0 0
\(322\) 0.484252 0.0269863
\(323\) −32.0534 −1.78350
\(324\) 0 0
\(325\) 2.76306 0.153267
\(326\) −5.70957 −0.316224
\(327\) 0 0
\(328\) −10.7993 −0.596294
\(329\) 3.41414 0.188227
\(330\) 0 0
\(331\) 25.4673 1.39981 0.699905 0.714236i \(-0.253226\pi\)
0.699905 + 0.714236i \(0.253226\pi\)
\(332\) −23.8689 −1.30998
\(333\) 0 0
\(334\) 5.20199 0.284640
\(335\) 3.35201 0.183140
\(336\) 0 0
\(337\) 31.8147 1.73306 0.866529 0.499126i \(-0.166346\pi\)
0.866529 + 0.499126i \(0.166346\pi\)
\(338\) 4.09149 0.222548
\(339\) 0 0
\(340\) −14.6225 −0.793017
\(341\) 8.02405 0.434526
\(342\) 0 0
\(343\) −3.61039 −0.194943
\(344\) 13.8806 0.748393
\(345\) 0 0
\(346\) −1.05028 −0.0564636
\(347\) 17.9415 0.963149 0.481575 0.876405i \(-0.340065\pi\)
0.481575 + 0.876405i \(0.340065\pi\)
\(348\) 0 0
\(349\) 34.8428 1.86509 0.932547 0.361049i \(-0.117581\pi\)
0.932547 + 0.361049i \(0.117581\pi\)
\(350\) 0.162635 0.00869323
\(351\) 0 0
\(352\) 4.95513 0.264109
\(353\) −21.0412 −1.11991 −0.559955 0.828523i \(-0.689182\pi\)
−0.559955 + 0.828523i \(0.689182\pi\)
\(354\) 0 0
\(355\) 21.8481 1.15958
\(356\) 9.96525 0.528157
\(357\) 0 0
\(358\) −4.93389 −0.260764
\(359\) 7.15114 0.377423 0.188711 0.982033i \(-0.439569\pi\)
0.188711 + 0.982033i \(0.439569\pi\)
\(360\) 0 0
\(361\) 36.3914 1.91534
\(362\) −0.135434 −0.00711826
\(363\) 0 0
\(364\) 0.981089 0.0514230
\(365\) 2.39569 0.125396
\(366\) 0 0
\(367\) 10.4282 0.544348 0.272174 0.962248i \(-0.412257\pi\)
0.272174 + 0.962248i \(0.412257\pi\)
\(368\) −10.0559 −0.524201
\(369\) 0 0
\(370\) −6.42431 −0.333984
\(371\) 2.62315 0.136187
\(372\) 0 0
\(373\) 8.18290 0.423694 0.211847 0.977303i \(-0.432052\pi\)
0.211847 + 0.977303i \(0.432052\pi\)
\(374\) −2.10248 −0.108716
\(375\) 0 0
\(376\) 24.1951 1.24777
\(377\) −7.69846 −0.396491
\(378\) 0 0
\(379\) −21.8456 −1.12213 −0.561067 0.827770i \(-0.689609\pi\)
−0.561067 + 0.827770i \(0.689609\pi\)
\(380\) 25.2691 1.29628
\(381\) 0 0
\(382\) −2.97633 −0.152282
\(383\) 11.7434 0.600058 0.300029 0.953930i \(-0.403004\pi\)
0.300029 + 0.953930i \(0.403004\pi\)
\(384\) 0 0
\(385\) 0.499408 0.0254522
\(386\) −4.61394 −0.234843
\(387\) 0 0
\(388\) 30.0443 1.52527
\(389\) −1.07865 −0.0546896 −0.0273448 0.999626i \(-0.508705\pi\)
−0.0273448 + 0.999626i \(0.508705\pi\)
\(390\) 0 0
\(391\) 16.4867 0.833767
\(392\) −12.7313 −0.643026
\(393\) 0 0
\(394\) −8.98180 −0.452497
\(395\) 17.3688 0.873921
\(396\) 0 0
\(397\) −0.649240 −0.0325844 −0.0162922 0.999867i \(-0.505186\pi\)
−0.0162922 + 0.999867i \(0.505186\pi\)
\(398\) −3.56513 −0.178704
\(399\) 0 0
\(400\) −3.37727 −0.168863
\(401\) −8.09547 −0.404269 −0.202134 0.979358i \(-0.564788\pi\)
−0.202134 + 0.979358i \(0.564788\pi\)
\(402\) 0 0
\(403\) 17.2449 0.859028
\(404\) 30.4988 1.51737
\(405\) 0 0
\(406\) −0.453137 −0.0224888
\(407\) 6.82822 0.338462
\(408\) 0 0
\(409\) 29.5573 1.46152 0.730758 0.682636i \(-0.239167\pi\)
0.730758 + 0.682636i \(0.239167\pi\)
\(410\) −5.53295 −0.273253
\(411\) 0 0
\(412\) −1.88843 −0.0930365
\(413\) −1.55429 −0.0764814
\(414\) 0 0
\(415\) −26.1124 −1.28181
\(416\) 10.6493 0.522125
\(417\) 0 0
\(418\) 3.63328 0.177710
\(419\) 17.0334 0.832134 0.416067 0.909334i \(-0.363408\pi\)
0.416067 + 0.909334i \(0.363408\pi\)
\(420\) 0 0
\(421\) −17.2184 −0.839172 −0.419586 0.907716i \(-0.637825\pi\)
−0.419586 + 0.907716i \(0.637825\pi\)
\(422\) −8.81900 −0.429302
\(423\) 0 0
\(424\) 18.5895 0.902787
\(425\) 5.53702 0.268585
\(426\) 0 0
\(427\) −0.259128 −0.0125401
\(428\) −12.7499 −0.616289
\(429\) 0 0
\(430\) 7.11162 0.342953
\(431\) 6.62757 0.319239 0.159619 0.987179i \(-0.448973\pi\)
0.159619 + 0.987179i \(0.448973\pi\)
\(432\) 0 0
\(433\) 31.5499 1.51619 0.758095 0.652144i \(-0.226130\pi\)
0.758095 + 0.652144i \(0.226130\pi\)
\(434\) 1.01504 0.0487237
\(435\) 0 0
\(436\) −21.1776 −1.01423
\(437\) −28.4906 −1.36289
\(438\) 0 0
\(439\) 1.99975 0.0954430 0.0477215 0.998861i \(-0.484804\pi\)
0.0477215 + 0.998861i \(0.484804\pi\)
\(440\) 3.53917 0.168723
\(441\) 0 0
\(442\) −4.51853 −0.214925
\(443\) −34.0277 −1.61670 −0.808352 0.588700i \(-0.799640\pi\)
−0.808352 + 0.588700i \(0.799640\pi\)
\(444\) 0 0
\(445\) 10.9019 0.516799
\(446\) −4.41483 −0.209048
\(447\) 0 0
\(448\) −0.734576 −0.0347055
\(449\) −15.5859 −0.735544 −0.367772 0.929916i \(-0.619879\pi\)
−0.367772 + 0.929916i \(0.619879\pi\)
\(450\) 0 0
\(451\) 5.88081 0.276917
\(452\) 32.4358 1.52565
\(453\) 0 0
\(454\) −6.52812 −0.306380
\(455\) 1.07330 0.0503172
\(456\) 0 0
\(457\) −7.87791 −0.368513 −0.184257 0.982878i \(-0.558988\pi\)
−0.184257 + 0.982878i \(0.558988\pi\)
\(458\) −8.11481 −0.379180
\(459\) 0 0
\(460\) −12.9972 −0.605996
\(461\) −28.7163 −1.33745 −0.668725 0.743510i \(-0.733160\pi\)
−0.668725 + 0.743510i \(0.733160\pi\)
\(462\) 0 0
\(463\) 4.92149 0.228721 0.114360 0.993439i \(-0.463518\pi\)
0.114360 + 0.993439i \(0.463518\pi\)
\(464\) 9.40978 0.436838
\(465\) 0 0
\(466\) −0.485235 −0.0224781
\(467\) 18.6313 0.862152 0.431076 0.902316i \(-0.358134\pi\)
0.431076 + 0.902316i \(0.358134\pi\)
\(468\) 0 0
\(469\) −0.450690 −0.0208110
\(470\) 12.3961 0.571791
\(471\) 0 0
\(472\) −11.0148 −0.506998
\(473\) −7.55874 −0.347551
\(474\) 0 0
\(475\) −9.56851 −0.439033
\(476\) 1.96605 0.0901138
\(477\) 0 0
\(478\) −3.45011 −0.157805
\(479\) 18.3125 0.836720 0.418360 0.908281i \(-0.362605\pi\)
0.418360 + 0.908281i \(0.362605\pi\)
\(480\) 0 0
\(481\) 14.6749 0.669116
\(482\) 9.55605 0.435266
\(483\) 0 0
\(484\) −1.76168 −0.0800765
\(485\) 32.8682 1.49247
\(486\) 0 0
\(487\) −30.8841 −1.39949 −0.699745 0.714393i \(-0.746703\pi\)
−0.699745 + 0.714393i \(0.746703\pi\)
\(488\) −1.83637 −0.0831285
\(489\) 0 0
\(490\) −6.52275 −0.294668
\(491\) 7.76148 0.350271 0.175135 0.984544i \(-0.443964\pi\)
0.175135 + 0.984544i \(0.443964\pi\)
\(492\) 0 0
\(493\) −15.4273 −0.694812
\(494\) 7.80846 0.351319
\(495\) 0 0
\(496\) −21.0783 −0.946443
\(497\) −2.93756 −0.131767
\(498\) 0 0
\(499\) 12.8210 0.573947 0.286973 0.957939i \(-0.407351\pi\)
0.286973 + 0.957939i \(0.407351\pi\)
\(500\) −21.3412 −0.954409
\(501\) 0 0
\(502\) −0.889234 −0.0396884
\(503\) 14.4507 0.644325 0.322162 0.946684i \(-0.395590\pi\)
0.322162 + 0.946684i \(0.395590\pi\)
\(504\) 0 0
\(505\) 33.3654 1.48474
\(506\) −1.86878 −0.0830773
\(507\) 0 0
\(508\) 13.9443 0.618677
\(509\) −35.2748 −1.56353 −0.781764 0.623574i \(-0.785680\pi\)
−0.781764 + 0.623574i \(0.785680\pi\)
\(510\) 0 0
\(511\) −0.322109 −0.0142493
\(512\) −22.6645 −1.00164
\(513\) 0 0
\(514\) −0.970037 −0.0427865
\(515\) −2.06593 −0.0910357
\(516\) 0 0
\(517\) −13.1755 −0.579458
\(518\) 0.863772 0.0379520
\(519\) 0 0
\(520\) 7.60620 0.333554
\(521\) 4.52876 0.198408 0.0992042 0.995067i \(-0.468370\pi\)
0.0992042 + 0.995067i \(0.468370\pi\)
\(522\) 0 0
\(523\) −23.1945 −1.01422 −0.507112 0.861880i \(-0.669287\pi\)
−0.507112 + 0.861880i \(0.669287\pi\)
\(524\) −23.3281 −1.01909
\(525\) 0 0
\(526\) 4.14960 0.180931
\(527\) 34.5578 1.50536
\(528\) 0 0
\(529\) −8.34588 −0.362864
\(530\) 9.52418 0.413704
\(531\) 0 0
\(532\) −3.39753 −0.147301
\(533\) 12.6387 0.547445
\(534\) 0 0
\(535\) −13.9483 −0.603036
\(536\) −3.19392 −0.137956
\(537\) 0 0
\(538\) 9.48671 0.409001
\(539\) 6.93285 0.298619
\(540\) 0 0
\(541\) 4.64556 0.199728 0.0998641 0.995001i \(-0.468159\pi\)
0.0998641 + 0.995001i \(0.468159\pi\)
\(542\) −9.60690 −0.412652
\(543\) 0 0
\(544\) 21.3407 0.914973
\(545\) −23.1682 −0.992415
\(546\) 0 0
\(547\) 32.9951 1.41077 0.705384 0.708825i \(-0.250774\pi\)
0.705384 + 0.708825i \(0.250774\pi\)
\(548\) 13.0596 0.557880
\(549\) 0 0
\(550\) −0.627626 −0.0267621
\(551\) 26.6599 1.13575
\(552\) 0 0
\(553\) −2.33530 −0.0993073
\(554\) 8.81438 0.374487
\(555\) 0 0
\(556\) 6.08800 0.258189
\(557\) 39.8211 1.68727 0.843637 0.536914i \(-0.180410\pi\)
0.843637 + 0.536914i \(0.180410\pi\)
\(558\) 0 0
\(559\) −16.2449 −0.687084
\(560\) −1.31189 −0.0554375
\(561\) 0 0
\(562\) −9.77624 −0.412386
\(563\) −26.0693 −1.09869 −0.549345 0.835596i \(-0.685123\pi\)
−0.549345 + 0.835596i \(0.685123\pi\)
\(564\) 0 0
\(565\) 35.4844 1.49284
\(566\) −14.9963 −0.630341
\(567\) 0 0
\(568\) −20.8177 −0.873490
\(569\) 34.9916 1.46692 0.733462 0.679730i \(-0.237903\pi\)
0.733462 + 0.679730i \(0.237903\pi\)
\(570\) 0 0
\(571\) 3.45551 0.144609 0.0723043 0.997383i \(-0.476965\pi\)
0.0723043 + 0.997383i \(0.476965\pi\)
\(572\) −3.78612 −0.158306
\(573\) 0 0
\(574\) 0.743925 0.0310508
\(575\) 4.92156 0.205243
\(576\) 0 0
\(577\) −15.1331 −0.629998 −0.314999 0.949092i \(-0.602004\pi\)
−0.314999 + 0.949092i \(0.602004\pi\)
\(578\) −0.755889 −0.0314409
\(579\) 0 0
\(580\) 12.1620 0.505001
\(581\) 3.51091 0.145657
\(582\) 0 0
\(583\) −10.1230 −0.419251
\(584\) −2.28270 −0.0944587
\(585\) 0 0
\(586\) 14.7416 0.608969
\(587\) −18.7095 −0.772224 −0.386112 0.922452i \(-0.626182\pi\)
−0.386112 + 0.922452i \(0.626182\pi\)
\(588\) 0 0
\(589\) −59.7193 −2.46069
\(590\) −5.64334 −0.232333
\(591\) 0 0
\(592\) −17.9370 −0.737206
\(593\) 14.5930 0.599262 0.299631 0.954055i \(-0.403137\pi\)
0.299631 + 0.954055i \(0.403137\pi\)
\(594\) 0 0
\(595\) 2.15084 0.0881759
\(596\) −14.3783 −0.588960
\(597\) 0 0
\(598\) −4.01628 −0.164238
\(599\) 39.2459 1.60354 0.801771 0.597631i \(-0.203891\pi\)
0.801771 + 0.597631i \(0.203891\pi\)
\(600\) 0 0
\(601\) 3.52750 0.143890 0.0719449 0.997409i \(-0.477079\pi\)
0.0719449 + 0.997409i \(0.477079\pi\)
\(602\) −0.956183 −0.0389711
\(603\) 0 0
\(604\) −8.98186 −0.365467
\(605\) −1.92726 −0.0783544
\(606\) 0 0
\(607\) −0.496801 −0.0201645 −0.0100823 0.999949i \(-0.503209\pi\)
−0.0100823 + 0.999949i \(0.503209\pi\)
\(608\) −36.8787 −1.49563
\(609\) 0 0
\(610\) −0.940847 −0.0380938
\(611\) −28.3161 −1.14555
\(612\) 0 0
\(613\) −17.8350 −0.720349 −0.360174 0.932885i \(-0.617283\pi\)
−0.360174 + 0.932885i \(0.617283\pi\)
\(614\) −2.94240 −0.118746
\(615\) 0 0
\(616\) −0.475854 −0.0191727
\(617\) −30.2909 −1.21946 −0.609732 0.792608i \(-0.708723\pi\)
−0.609732 + 0.792608i \(0.708723\pi\)
\(618\) 0 0
\(619\) −15.8595 −0.637448 −0.318724 0.947847i \(-0.603254\pi\)
−0.318724 + 0.947847i \(0.603254\pi\)
\(620\) −27.2435 −1.09412
\(621\) 0 0
\(622\) −7.79057 −0.312373
\(623\) −1.46580 −0.0587260
\(624\) 0 0
\(625\) −16.9188 −0.676754
\(626\) −1.81928 −0.0727131
\(627\) 0 0
\(628\) −13.1490 −0.524701
\(629\) 29.4077 1.17256
\(630\) 0 0
\(631\) −24.1448 −0.961188 −0.480594 0.876943i \(-0.659579\pi\)
−0.480594 + 0.876943i \(0.659579\pi\)
\(632\) −16.5497 −0.658311
\(633\) 0 0
\(634\) −6.08258 −0.241570
\(635\) 15.2549 0.605373
\(636\) 0 0
\(637\) 14.8997 0.590349
\(638\) 1.74870 0.0692317
\(639\) 0 0
\(640\) −21.7668 −0.860408
\(641\) 11.9751 0.472987 0.236493 0.971633i \(-0.424002\pi\)
0.236493 + 0.971633i \(0.424002\pi\)
\(642\) 0 0
\(643\) −49.2315 −1.94150 −0.970750 0.240094i \(-0.922822\pi\)
−0.970750 + 0.240094i \(0.922822\pi\)
\(644\) 1.74752 0.0688619
\(645\) 0 0
\(646\) 15.6478 0.615653
\(647\) 19.8119 0.778886 0.389443 0.921050i \(-0.372667\pi\)
0.389443 + 0.921050i \(0.372667\pi\)
\(648\) 0 0
\(649\) 5.99815 0.235448
\(650\) −1.34886 −0.0529067
\(651\) 0 0
\(652\) −20.6041 −0.806918
\(653\) −3.15323 −0.123395 −0.0616976 0.998095i \(-0.519651\pi\)
−0.0616976 + 0.998095i \(0.519651\pi\)
\(654\) 0 0
\(655\) −25.5207 −0.997178
\(656\) −15.4483 −0.603153
\(657\) 0 0
\(658\) −1.66670 −0.0649749
\(659\) 5.25903 0.204863 0.102431 0.994740i \(-0.467338\pi\)
0.102431 + 0.994740i \(0.467338\pi\)
\(660\) 0 0
\(661\) 41.7574 1.62417 0.812087 0.583537i \(-0.198332\pi\)
0.812087 + 0.583537i \(0.198332\pi\)
\(662\) −12.4326 −0.483205
\(663\) 0 0
\(664\) 24.8808 0.965564
\(665\) −3.71686 −0.144134
\(666\) 0 0
\(667\) −13.7125 −0.530951
\(668\) 18.7724 0.726325
\(669\) 0 0
\(670\) −1.63638 −0.0632188
\(671\) 1.00000 0.0386046
\(672\) 0 0
\(673\) 43.4335 1.67424 0.837118 0.547022i \(-0.184238\pi\)
0.837118 + 0.547022i \(0.184238\pi\)
\(674\) −15.5312 −0.598241
\(675\) 0 0
\(676\) 14.7649 0.567882
\(677\) −35.0164 −1.34579 −0.672894 0.739739i \(-0.734949\pi\)
−0.672894 + 0.739739i \(0.734949\pi\)
\(678\) 0 0
\(679\) −4.41926 −0.169595
\(680\) 15.2424 0.584521
\(681\) 0 0
\(682\) −3.91716 −0.149996
\(683\) 41.6290 1.59289 0.796445 0.604711i \(-0.206711\pi\)
0.796445 + 0.604711i \(0.206711\pi\)
\(684\) 0 0
\(685\) 14.2871 0.545882
\(686\) 1.76251 0.0672930
\(687\) 0 0
\(688\) 19.8560 0.757002
\(689\) −21.7558 −0.828830
\(690\) 0 0
\(691\) 20.8432 0.792912 0.396456 0.918054i \(-0.370240\pi\)
0.396456 + 0.918054i \(0.370240\pi\)
\(692\) −3.79015 −0.144080
\(693\) 0 0
\(694\) −8.75863 −0.332473
\(695\) 6.66021 0.252636
\(696\) 0 0
\(697\) 25.3274 0.959343
\(698\) −17.0095 −0.643819
\(699\) 0 0
\(700\) 0.586901 0.0221828
\(701\) −33.5579 −1.26747 −0.633733 0.773552i \(-0.718478\pi\)
−0.633733 + 0.773552i \(0.718478\pi\)
\(702\) 0 0
\(703\) −50.8193 −1.91669
\(704\) 2.83480 0.106841
\(705\) 0 0
\(706\) 10.2718 0.386586
\(707\) −4.48611 −0.168717
\(708\) 0 0
\(709\) 36.6060 1.37477 0.687384 0.726294i \(-0.258759\pi\)
0.687384 + 0.726294i \(0.258759\pi\)
\(710\) −10.6657 −0.400278
\(711\) 0 0
\(712\) −10.3877 −0.389296
\(713\) 30.7166 1.15035
\(714\) 0 0
\(715\) −4.14198 −0.154901
\(716\) −17.8049 −0.665400
\(717\) 0 0
\(718\) −3.49102 −0.130284
\(719\) 18.2060 0.678970 0.339485 0.940611i \(-0.389747\pi\)
0.339485 + 0.940611i \(0.389747\pi\)
\(720\) 0 0
\(721\) 0.277772 0.0103448
\(722\) −17.7655 −0.661162
\(723\) 0 0
\(724\) −0.488740 −0.0181639
\(725\) −4.60533 −0.171038
\(726\) 0 0
\(727\) 16.2387 0.602259 0.301130 0.953583i \(-0.402636\pi\)
0.301130 + 0.953583i \(0.402636\pi\)
\(728\) −1.02268 −0.0379031
\(729\) 0 0
\(730\) −1.16952 −0.0432859
\(731\) −32.5538 −1.20405
\(732\) 0 0
\(733\) −29.3708 −1.08483 −0.542417 0.840109i \(-0.682491\pi\)
−0.542417 + 0.840109i \(0.682491\pi\)
\(734\) −5.09082 −0.187906
\(735\) 0 0
\(736\) 18.9686 0.699191
\(737\) 1.73926 0.0640664
\(738\) 0 0
\(739\) 21.4179 0.787872 0.393936 0.919138i \(-0.371113\pi\)
0.393936 + 0.919138i \(0.371113\pi\)
\(740\) −23.1834 −0.852238
\(741\) 0 0
\(742\) −1.28056 −0.0470109
\(743\) −32.8143 −1.20384 −0.601921 0.798556i \(-0.705598\pi\)
−0.601921 + 0.798556i \(0.705598\pi\)
\(744\) 0 0
\(745\) −15.7298 −0.576295
\(746\) −3.99471 −0.146257
\(747\) 0 0
\(748\) −7.58719 −0.277415
\(749\) 1.87539 0.0685254
\(750\) 0 0
\(751\) −42.9406 −1.56693 −0.783463 0.621439i \(-0.786548\pi\)
−0.783463 + 0.621439i \(0.786548\pi\)
\(752\) 34.6106 1.26212
\(753\) 0 0
\(754\) 3.75822 0.136866
\(755\) −9.82607 −0.357607
\(756\) 0 0
\(757\) −34.8797 −1.26772 −0.633861 0.773447i \(-0.718531\pi\)
−0.633861 + 0.773447i \(0.718531\pi\)
\(758\) 10.6645 0.387354
\(759\) 0 0
\(760\) −26.3404 −0.955467
\(761\) 42.2729 1.53239 0.766196 0.642607i \(-0.222147\pi\)
0.766196 + 0.642607i \(0.222147\pi\)
\(762\) 0 0
\(763\) 3.11504 0.112772
\(764\) −10.7407 −0.388583
\(765\) 0 0
\(766\) −5.73285 −0.207136
\(767\) 12.8909 0.465464
\(768\) 0 0
\(769\) 22.3299 0.805238 0.402619 0.915368i \(-0.368100\pi\)
0.402619 + 0.915368i \(0.368100\pi\)
\(770\) −0.243800 −0.00878593
\(771\) 0 0
\(772\) −16.6503 −0.599257
\(773\) −34.1247 −1.22738 −0.613689 0.789548i \(-0.710315\pi\)
−0.613689 + 0.789548i \(0.710315\pi\)
\(774\) 0 0
\(775\) 10.3161 0.370566
\(776\) −31.3181 −1.12425
\(777\) 0 0
\(778\) 0.526571 0.0188785
\(779\) −43.7682 −1.56816
\(780\) 0 0
\(781\) 11.3363 0.405645
\(782\) −8.04842 −0.287811
\(783\) 0 0
\(784\) −18.2119 −0.650423
\(785\) −14.3849 −0.513418
\(786\) 0 0
\(787\) 3.69963 0.131878 0.0659388 0.997824i \(-0.478996\pi\)
0.0659388 + 0.997824i \(0.478996\pi\)
\(788\) −32.4126 −1.15465
\(789\) 0 0
\(790\) −8.47908 −0.301672
\(791\) −4.77101 −0.169638
\(792\) 0 0
\(793\) 2.14915 0.0763185
\(794\) 0.316944 0.0112479
\(795\) 0 0
\(796\) −12.8655 −0.456004
\(797\) −15.0310 −0.532426 −0.266213 0.963914i \(-0.585773\pi\)
−0.266213 + 0.963914i \(0.585773\pi\)
\(798\) 0 0
\(799\) −56.7440 −2.00746
\(800\) 6.37057 0.225234
\(801\) 0 0
\(802\) 3.95203 0.139551
\(803\) 1.24305 0.0438663
\(804\) 0 0
\(805\) 1.91177 0.0673810
\(806\) −8.41856 −0.296531
\(807\) 0 0
\(808\) −31.7918 −1.11843
\(809\) 0.622132 0.0218730 0.0109365 0.999940i \(-0.496519\pi\)
0.0109365 + 0.999940i \(0.496519\pi\)
\(810\) 0 0
\(811\) 17.8338 0.626230 0.313115 0.949715i \(-0.398628\pi\)
0.313115 + 0.949715i \(0.398628\pi\)
\(812\) −1.63523 −0.0573854
\(813\) 0 0
\(814\) −3.33338 −0.116835
\(815\) −22.5407 −0.789566
\(816\) 0 0
\(817\) 56.2562 1.96816
\(818\) −14.4292 −0.504506
\(819\) 0 0
\(820\) −19.9667 −0.697268
\(821\) −3.48246 −0.121539 −0.0607694 0.998152i \(-0.519355\pi\)
−0.0607694 + 0.998152i \(0.519355\pi\)
\(822\) 0 0
\(823\) −10.6086 −0.369793 −0.184897 0.982758i \(-0.559195\pi\)
−0.184897 + 0.982758i \(0.559195\pi\)
\(824\) 1.96849 0.0685757
\(825\) 0 0
\(826\) 0.758768 0.0264009
\(827\) 18.3993 0.639805 0.319902 0.947451i \(-0.396350\pi\)
0.319902 + 0.947451i \(0.396350\pi\)
\(828\) 0 0
\(829\) −15.1655 −0.526720 −0.263360 0.964698i \(-0.584831\pi\)
−0.263360 + 0.964698i \(0.584831\pi\)
\(830\) 12.7475 0.442472
\(831\) 0 0
\(832\) 6.09241 0.211216
\(833\) 29.8583 1.03453
\(834\) 0 0
\(835\) 20.5368 0.710706
\(836\) 13.1114 0.453467
\(837\) 0 0
\(838\) −8.31530 −0.287247
\(839\) 17.5973 0.607528 0.303764 0.952747i \(-0.401757\pi\)
0.303764 + 0.952747i \(0.401757\pi\)
\(840\) 0 0
\(841\) −16.1686 −0.557537
\(842\) 8.40562 0.289677
\(843\) 0 0
\(844\) −31.8251 −1.09546
\(845\) 16.1527 0.555670
\(846\) 0 0
\(847\) 0.259128 0.00890374
\(848\) 26.5920 0.913172
\(849\) 0 0
\(850\) −2.70305 −0.0927138
\(851\) 26.1389 0.896030
\(852\) 0 0
\(853\) −57.7984 −1.97898 −0.989490 0.144602i \(-0.953810\pi\)
−0.989490 + 0.144602i \(0.953810\pi\)
\(854\) 0.126500 0.00432875
\(855\) 0 0
\(856\) 13.2904 0.454257
\(857\) 12.5542 0.428844 0.214422 0.976741i \(-0.431213\pi\)
0.214422 + 0.976741i \(0.431213\pi\)
\(858\) 0 0
\(859\) −36.2175 −1.23573 −0.617863 0.786286i \(-0.712001\pi\)
−0.617863 + 0.786286i \(0.712001\pi\)
\(860\) 25.6636 0.875123
\(861\) 0 0
\(862\) −3.23543 −0.110199
\(863\) −13.9997 −0.476554 −0.238277 0.971197i \(-0.576583\pi\)
−0.238277 + 0.971197i \(0.576583\pi\)
\(864\) 0 0
\(865\) −4.14639 −0.140982
\(866\) −15.4019 −0.523379
\(867\) 0 0
\(868\) 3.66298 0.124330
\(869\) 9.01217 0.305717
\(870\) 0 0
\(871\) 3.73793 0.126655
\(872\) 22.0755 0.747570
\(873\) 0 0
\(874\) 13.9085 0.470460
\(875\) 3.13910 0.106121
\(876\) 0 0
\(877\) 37.3773 1.26214 0.631071 0.775725i \(-0.282616\pi\)
0.631071 + 0.775725i \(0.282616\pi\)
\(878\) −0.976235 −0.0329463
\(879\) 0 0
\(880\) 5.06272 0.170664
\(881\) −20.9400 −0.705485 −0.352743 0.935720i \(-0.614751\pi\)
−0.352743 + 0.935720i \(0.614751\pi\)
\(882\) 0 0
\(883\) 34.8733 1.17358 0.586789 0.809740i \(-0.300392\pi\)
0.586789 + 0.809740i \(0.300392\pi\)
\(884\) −16.3060 −0.548430
\(885\) 0 0
\(886\) 16.6115 0.558076
\(887\) 1.28356 0.0430978 0.0215489 0.999768i \(-0.493140\pi\)
0.0215489 + 0.999768i \(0.493140\pi\)
\(888\) 0 0
\(889\) −2.05108 −0.0687910
\(890\) −5.32206 −0.178396
\(891\) 0 0
\(892\) −15.9318 −0.533435
\(893\) 98.0591 3.28142
\(894\) 0 0
\(895\) −19.4784 −0.651090
\(896\) 2.92663 0.0977717
\(897\) 0 0
\(898\) 7.60869 0.253905
\(899\) −28.7429 −0.958630
\(900\) 0 0
\(901\) −43.5975 −1.45244
\(902\) −2.87088 −0.0955899
\(903\) 0 0
\(904\) −33.8109 −1.12453
\(905\) −0.534677 −0.0177733
\(906\) 0 0
\(907\) 5.99757 0.199146 0.0995730 0.995030i \(-0.468252\pi\)
0.0995730 + 0.995030i \(0.468252\pi\)
\(908\) −23.5580 −0.781799
\(909\) 0 0
\(910\) −0.523962 −0.0173692
\(911\) 36.5132 1.20974 0.604869 0.796325i \(-0.293226\pi\)
0.604869 + 0.796325i \(0.293226\pi\)
\(912\) 0 0
\(913\) −13.5489 −0.448404
\(914\) 3.84582 0.127208
\(915\) 0 0
\(916\) −29.2839 −0.967566
\(917\) 3.43136 0.113313
\(918\) 0 0
\(919\) 33.2375 1.09640 0.548201 0.836346i \(-0.315313\pi\)
0.548201 + 0.836346i \(0.315313\pi\)
\(920\) 13.5482 0.446670
\(921\) 0 0
\(922\) 14.0186 0.461679
\(923\) 24.3634 0.801932
\(924\) 0 0
\(925\) 8.77871 0.288642
\(926\) −2.40256 −0.0789530
\(927\) 0 0
\(928\) −17.7498 −0.582664
\(929\) −41.6156 −1.36537 −0.682683 0.730715i \(-0.739187\pi\)
−0.682683 + 0.730715i \(0.739187\pi\)
\(930\) 0 0
\(931\) −51.5980 −1.69106
\(932\) −1.75106 −0.0573580
\(933\) 0 0
\(934\) −9.09536 −0.297609
\(935\) −8.30031 −0.271449
\(936\) 0 0
\(937\) 15.3565 0.501674 0.250837 0.968029i \(-0.419294\pi\)
0.250837 + 0.968029i \(0.419294\pi\)
\(938\) 0.220017 0.00718381
\(939\) 0 0
\(940\) 44.7338 1.45906
\(941\) 34.5345 1.12579 0.562897 0.826527i \(-0.309687\pi\)
0.562897 + 0.826527i \(0.309687\pi\)
\(942\) 0 0
\(943\) 22.5122 0.733097
\(944\) −15.7565 −0.512830
\(945\) 0 0
\(946\) 3.69001 0.119972
\(947\) 41.3303 1.34305 0.671527 0.740980i \(-0.265639\pi\)
0.671527 + 0.740980i \(0.265639\pi\)
\(948\) 0 0
\(949\) 2.67150 0.0867205
\(950\) 4.67113 0.151552
\(951\) 0 0
\(952\) −2.04940 −0.0664215
\(953\) −23.2336 −0.752610 −0.376305 0.926496i \(-0.622805\pi\)
−0.376305 + 0.926496i \(0.622805\pi\)
\(954\) 0 0
\(955\) −11.7502 −0.380227
\(956\) −12.4504 −0.402675
\(957\) 0 0
\(958\) −8.93975 −0.288830
\(959\) −1.92095 −0.0620309
\(960\) 0 0
\(961\) 33.3853 1.07695
\(962\) −7.16394 −0.230975
\(963\) 0 0
\(964\) 34.4848 1.11068
\(965\) −18.2153 −0.586370
\(966\) 0 0
\(967\) 0.466832 0.0150123 0.00750615 0.999972i \(-0.497611\pi\)
0.00750615 + 0.999972i \(0.497611\pi\)
\(968\) 1.83637 0.0590231
\(969\) 0 0
\(970\) −16.0455 −0.515191
\(971\) −4.43549 −0.142342 −0.0711709 0.997464i \(-0.522674\pi\)
−0.0711709 + 0.997464i \(0.522674\pi\)
\(972\) 0 0
\(973\) −0.895490 −0.0287081
\(974\) 15.0769 0.483095
\(975\) 0 0
\(976\) −2.62689 −0.0840848
\(977\) −15.5007 −0.495912 −0.247956 0.968771i \(-0.579759\pi\)
−0.247956 + 0.968771i \(0.579759\pi\)
\(978\) 0 0
\(979\) 5.65666 0.180788
\(980\) −23.5386 −0.751914
\(981\) 0 0
\(982\) −3.78898 −0.120911
\(983\) 36.2887 1.15743 0.578715 0.815530i \(-0.303554\pi\)
0.578715 + 0.815530i \(0.303554\pi\)
\(984\) 0 0
\(985\) −35.4591 −1.12982
\(986\) 7.53127 0.239845
\(987\) 0 0
\(988\) 28.1783 0.896472
\(989\) −28.9354 −0.920092
\(990\) 0 0
\(991\) 30.8702 0.980625 0.490313 0.871547i \(-0.336883\pi\)
0.490313 + 0.871547i \(0.336883\pi\)
\(992\) 39.7602 1.26239
\(993\) 0 0
\(994\) 1.43405 0.0454853
\(995\) −14.0747 −0.446198
\(996\) 0 0
\(997\) 2.22245 0.0703857 0.0351929 0.999381i \(-0.488795\pi\)
0.0351929 + 0.999381i \(0.488795\pi\)
\(998\) −6.25893 −0.198123
\(999\) 0 0
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 6039.2.a.k.1.10 19
3.2 odd 2 671.2.a.c.1.10 19
33.32 even 2 7381.2.a.i.1.10 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
671.2.a.c.1.10 19 3.2 odd 2
6039.2.a.k.1.10 19 1.1 even 1 trivial
7381.2.a.i.1.10 19 33.32 even 2