Properties

Label 1521.2.a.v.1.3
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,2,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1521.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1997632.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 8x^{4} + 19x^{2} - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.06082\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.14367 q^{2} -0.692021 q^{4} -3.48695 q^{5} -3.35690 q^{7} +3.07878 q^{8} +O(q^{10})\) \(q-1.14367 q^{2} -0.692021 q^{4} -3.48695 q^{5} -3.35690 q^{7} +3.07878 q^{8} +3.98792 q^{10} +2.97797 q^{11} +3.83918 q^{14} -2.13706 q^{16} +6.91796 q^{17} +0.295897 q^{19} +2.41305 q^{20} -3.40581 q^{22} +2.46899 q^{23} +7.15883 q^{25} +2.32304 q^{28} +0.609790 q^{29} -9.93900 q^{31} -3.71347 q^{32} -7.91185 q^{34} +11.7053 q^{35} +8.50365 q^{37} -0.338408 q^{38} -10.7356 q^{40} +7.30123 q^{41} -6.60388 q^{43} -2.06082 q^{44} -2.82371 q^{46} -4.63062 q^{47} +4.26875 q^{49} -8.18734 q^{50} -9.38695 q^{53} -10.3840 q^{55} -10.3351 q^{56} -0.697398 q^{58} +0.816341 q^{59} +5.08815 q^{61} +11.3669 q^{62} +8.52111 q^{64} -10.2349 q^{67} -4.78738 q^{68} -13.3870 q^{70} -10.1224 q^{71} -8.74094 q^{73} -9.72536 q^{74} -0.204767 q^{76} -9.99674 q^{77} +3.52111 q^{79} +7.45184 q^{80} -8.35019 q^{82} +11.7751 q^{83} -24.1226 q^{85} +7.55265 q^{86} +9.16852 q^{88} +7.77916 q^{89} -1.70859 q^{92} +5.29590 q^{94} -1.03178 q^{95} -2.56033 q^{97} -4.88204 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{4} - 12 q^{7} - 14 q^{10} - 2 q^{16} - 26 q^{19} + 6 q^{22} + 26 q^{25} - 26 q^{28} - 40 q^{31} - 40 q^{34} - 12 q^{37} - 42 q^{40} - 22 q^{43} - 2 q^{46} + 10 q^{49} - 42 q^{55} + 12 q^{58} + 38 q^{61} + 20 q^{64} - 14 q^{67} + 28 q^{70} - 24 q^{73} - 54 q^{76} - 10 q^{79} + 2 q^{82} - 14 q^{85} - 6 q^{88} + 4 q^{94} - 10 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.14367 −0.808696 −0.404348 0.914605i \(-0.632502\pi\)
−0.404348 + 0.914605i \(0.632502\pi\)
\(3\) 0 0
\(4\) −0.692021 −0.346011
\(5\) −3.48695 −1.55941 −0.779706 0.626146i \(-0.784632\pi\)
−0.779706 + 0.626146i \(0.784632\pi\)
\(6\) 0 0
\(7\) −3.35690 −1.26879 −0.634394 0.773010i \(-0.718750\pi\)
−0.634394 + 0.773010i \(0.718750\pi\)
\(8\) 3.07878 1.08851
\(9\) 0 0
\(10\) 3.98792 1.26109
\(11\) 2.97797 0.897892 0.448946 0.893559i \(-0.351800\pi\)
0.448946 + 0.893559i \(0.351800\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 3.83918 1.02606
\(15\) 0 0
\(16\) −2.13706 −0.534266
\(17\) 6.91796 1.67785 0.838926 0.544246i \(-0.183184\pi\)
0.838926 + 0.544246i \(0.183184\pi\)
\(18\) 0 0
\(19\) 0.295897 0.0678834 0.0339417 0.999424i \(-0.489194\pi\)
0.0339417 + 0.999424i \(0.489194\pi\)
\(20\) 2.41305 0.539573
\(21\) 0 0
\(22\) −3.40581 −0.726122
\(23\) 2.46899 0.514820 0.257410 0.966302i \(-0.417131\pi\)
0.257410 + 0.966302i \(0.417131\pi\)
\(24\) 0 0
\(25\) 7.15883 1.43177
\(26\) 0 0
\(27\) 0 0
\(28\) 2.32304 0.439014
\(29\) 0.609790 0.113235 0.0566176 0.998396i \(-0.481968\pi\)
0.0566176 + 0.998396i \(0.481968\pi\)
\(30\) 0 0
\(31\) −9.93900 −1.78510 −0.892549 0.450951i \(-0.851085\pi\)
−0.892549 + 0.450951i \(0.851085\pi\)
\(32\) −3.71347 −0.656455
\(33\) 0 0
\(34\) −7.91185 −1.35687
\(35\) 11.7053 1.97856
\(36\) 0 0
\(37\) 8.50365 1.39799 0.698996 0.715126i \(-0.253631\pi\)
0.698996 + 0.715126i \(0.253631\pi\)
\(38\) −0.338408 −0.0548970
\(39\) 0 0
\(40\) −10.7356 −1.69744
\(41\) 7.30123 1.14026 0.570130 0.821554i \(-0.306893\pi\)
0.570130 + 0.821554i \(0.306893\pi\)
\(42\) 0 0
\(43\) −6.60388 −1.00708 −0.503541 0.863971i \(-0.667970\pi\)
−0.503541 + 0.863971i \(0.667970\pi\)
\(44\) −2.06082 −0.310680
\(45\) 0 0
\(46\) −2.82371 −0.416333
\(47\) −4.63062 −0.675445 −0.337723 0.941246i \(-0.609657\pi\)
−0.337723 + 0.941246i \(0.609657\pi\)
\(48\) 0 0
\(49\) 4.26875 0.609821
\(50\) −8.18734 −1.15786
\(51\) 0 0
\(52\) 0 0
\(53\) −9.38695 −1.28940 −0.644698 0.764437i \(-0.723017\pi\)
−0.644698 + 0.764437i \(0.723017\pi\)
\(54\) 0 0
\(55\) −10.3840 −1.40018
\(56\) −10.3351 −1.38109
\(57\) 0 0
\(58\) −0.697398 −0.0915729
\(59\) 0.816341 0.106279 0.0531393 0.998587i \(-0.483077\pi\)
0.0531393 + 0.998587i \(0.483077\pi\)
\(60\) 0 0
\(61\) 5.08815 0.651470 0.325735 0.945461i \(-0.394388\pi\)
0.325735 + 0.945461i \(0.394388\pi\)
\(62\) 11.3669 1.44360
\(63\) 0 0
\(64\) 8.52111 1.06514
\(65\) 0 0
\(66\) 0 0
\(67\) −10.2349 −1.25039 −0.625196 0.780468i \(-0.714981\pi\)
−0.625196 + 0.780468i \(0.714981\pi\)
\(68\) −4.78738 −0.580555
\(69\) 0 0
\(70\) −13.3870 −1.60006
\(71\) −10.1224 −1.20131 −0.600657 0.799507i \(-0.705094\pi\)
−0.600657 + 0.799507i \(0.705094\pi\)
\(72\) 0 0
\(73\) −8.74094 −1.02305 −0.511525 0.859269i \(-0.670919\pi\)
−0.511525 + 0.859269i \(0.670919\pi\)
\(74\) −9.72536 −1.13055
\(75\) 0 0
\(76\) −0.204767 −0.0234884
\(77\) −9.99674 −1.13923
\(78\) 0 0
\(79\) 3.52111 0.396155 0.198078 0.980186i \(-0.436530\pi\)
0.198078 + 0.980186i \(0.436530\pi\)
\(80\) 7.45184 0.833141
\(81\) 0 0
\(82\) −8.35019 −0.922124
\(83\) 11.7751 1.29248 0.646242 0.763132i \(-0.276339\pi\)
0.646242 + 0.763132i \(0.276339\pi\)
\(84\) 0 0
\(85\) −24.1226 −2.61646
\(86\) 7.55265 0.814423
\(87\) 0 0
\(88\) 9.16852 0.977368
\(89\) 7.77916 0.824590 0.412295 0.911050i \(-0.364727\pi\)
0.412295 + 0.911050i \(0.364727\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1.70859 −0.178133
\(93\) 0 0
\(94\) 5.29590 0.546230
\(95\) −1.03178 −0.105858
\(96\) 0 0
\(97\) −2.56033 −0.259963 −0.129981 0.991516i \(-0.541492\pi\)
−0.129981 + 0.991516i \(0.541492\pi\)
\(98\) −4.88204 −0.493160
\(99\) 0 0
\(100\) −4.95407 −0.495407
\(101\) −14.7841 −1.47107 −0.735537 0.677484i \(-0.763070\pi\)
−0.735537 + 0.677484i \(0.763070\pi\)
\(102\) 0 0
\(103\) 2.09783 0.206706 0.103353 0.994645i \(-0.467043\pi\)
0.103353 + 0.994645i \(0.467043\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.7356 1.04273
\(107\) 6.64658 0.642549 0.321274 0.946986i \(-0.395889\pi\)
0.321274 + 0.946986i \(0.395889\pi\)
\(108\) 0 0
\(109\) −3.06100 −0.293191 −0.146595 0.989197i \(-0.546831\pi\)
−0.146595 + 0.989197i \(0.546831\pi\)
\(110\) 11.8759 1.13232
\(111\) 0 0
\(112\) 7.17390 0.677870
\(113\) −5.81915 −0.547420 −0.273710 0.961812i \(-0.588251\pi\)
−0.273710 + 0.961812i \(0.588251\pi\)
\(114\) 0 0
\(115\) −8.60925 −0.802817
\(116\) −0.421988 −0.0391806
\(117\) 0 0
\(118\) −0.933624 −0.0859471
\(119\) −23.2229 −2.12884
\(120\) 0 0
\(121\) −2.13169 −0.193790
\(122\) −5.81915 −0.526841
\(123\) 0 0
\(124\) 6.87800 0.617663
\(125\) −7.52775 −0.673302
\(126\) 0 0
\(127\) 15.9879 1.41870 0.709349 0.704857i \(-0.248989\pi\)
0.709349 + 0.704857i \(0.248989\pi\)
\(128\) −2.31839 −0.204918
\(129\) 0 0
\(130\) 0 0
\(131\) 2.80740 0.245284 0.122642 0.992451i \(-0.460863\pi\)
0.122642 + 0.992451i \(0.460863\pi\)
\(132\) 0 0
\(133\) −0.993295 −0.0861296
\(134\) 11.7053 1.01119
\(135\) 0 0
\(136\) 21.2989 1.82636
\(137\) −1.80326 −0.154062 −0.0770312 0.997029i \(-0.524544\pi\)
−0.0770312 + 0.997029i \(0.524544\pi\)
\(138\) 0 0
\(139\) −4.26875 −0.362071 −0.181035 0.983477i \(-0.557945\pi\)
−0.181035 + 0.983477i \(0.557945\pi\)
\(140\) −8.10034 −0.684604
\(141\) 0 0
\(142\) 11.5767 0.971497
\(143\) 0 0
\(144\) 0 0
\(145\) −2.12631 −0.176580
\(146\) 9.99674 0.827336
\(147\) 0 0
\(148\) −5.88471 −0.483720
\(149\) 21.9784 1.80054 0.900270 0.435332i \(-0.143369\pi\)
0.900270 + 0.435332i \(0.143369\pi\)
\(150\) 0 0
\(151\) −7.08277 −0.576388 −0.288194 0.957572i \(-0.593055\pi\)
−0.288194 + 0.957572i \(0.593055\pi\)
\(152\) 0.911002 0.0738920
\(153\) 0 0
\(154\) 11.4330 0.921294
\(155\) 34.6568 2.78370
\(156\) 0 0
\(157\) −2.95646 −0.235951 −0.117976 0.993016i \(-0.537640\pi\)
−0.117976 + 0.993016i \(0.537640\pi\)
\(158\) −4.02698 −0.320369
\(159\) 0 0
\(160\) 12.9487 1.02368
\(161\) −8.28815 −0.653197
\(162\) 0 0
\(163\) 6.18060 0.484102 0.242051 0.970263i \(-0.422180\pi\)
0.242051 + 0.970263i \(0.422180\pi\)
\(164\) −5.05261 −0.394542
\(165\) 0 0
\(166\) −13.4668 −1.04523
\(167\) 3.78938 0.293231 0.146616 0.989194i \(-0.453162\pi\)
0.146616 + 0.989194i \(0.453162\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 27.5883 2.11592
\(171\) 0 0
\(172\) 4.57002 0.348461
\(173\) −10.4858 −0.797217 −0.398609 0.917121i \(-0.630507\pi\)
−0.398609 + 0.917121i \(0.630507\pi\)
\(174\) 0 0
\(175\) −24.0315 −1.81661
\(176\) −6.36411 −0.479713
\(177\) 0 0
\(178\) −8.89679 −0.666842
\(179\) −10.1473 −0.758448 −0.379224 0.925305i \(-0.623809\pi\)
−0.379224 + 0.925305i \(0.623809\pi\)
\(180\) 0 0
\(181\) −21.4426 −1.59382 −0.796910 0.604098i \(-0.793533\pi\)
−0.796910 + 0.604098i \(0.793533\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 7.60148 0.560389
\(185\) −29.6518 −2.18004
\(186\) 0 0
\(187\) 20.6015 1.50653
\(188\) 3.20449 0.233711
\(189\) 0 0
\(190\) 1.18001 0.0856071
\(191\) −21.3637 −1.54582 −0.772910 0.634515i \(-0.781200\pi\)
−0.772910 + 0.634515i \(0.781200\pi\)
\(192\) 0 0
\(193\) 2.23191 0.160657 0.0803283 0.996768i \(-0.474403\pi\)
0.0803283 + 0.996768i \(0.474403\pi\)
\(194\) 2.92818 0.210231
\(195\) 0 0
\(196\) −2.95407 −0.211005
\(197\) −2.51386 −0.179105 −0.0895524 0.995982i \(-0.528544\pi\)
−0.0895524 + 0.995982i \(0.528544\pi\)
\(198\) 0 0
\(199\) −11.5646 −0.819796 −0.409898 0.912131i \(-0.634436\pi\)
−0.409898 + 0.912131i \(0.634436\pi\)
\(200\) 22.0405 1.55850
\(201\) 0 0
\(202\) 16.9081 1.18965
\(203\) −2.04700 −0.143671
\(204\) 0 0
\(205\) −25.4590 −1.77814
\(206\) −2.39923 −0.167162
\(207\) 0 0
\(208\) 0 0
\(209\) 0.881173 0.0609520
\(210\) 0 0
\(211\) −19.9041 −1.37025 −0.685127 0.728424i \(-0.740253\pi\)
−0.685127 + 0.728424i \(0.740253\pi\)
\(212\) 6.49597 0.446145
\(213\) 0 0
\(214\) −7.60148 −0.519627
\(215\) 23.0274 1.57046
\(216\) 0 0
\(217\) 33.3642 2.26491
\(218\) 3.50077 0.237102
\(219\) 0 0
\(220\) 7.18598 0.484479
\(221\) 0 0
\(222\) 0 0
\(223\) −18.1468 −1.21520 −0.607598 0.794245i \(-0.707867\pi\)
−0.607598 + 0.794245i \(0.707867\pi\)
\(224\) 12.4657 0.832902
\(225\) 0 0
\(226\) 6.65519 0.442696
\(227\) −11.5796 −0.768567 −0.384283 0.923215i \(-0.625551\pi\)
−0.384283 + 0.923215i \(0.625551\pi\)
\(228\) 0 0
\(229\) 11.8605 0.783767 0.391883 0.920015i \(-0.371824\pi\)
0.391883 + 0.920015i \(0.371824\pi\)
\(230\) 9.84613 0.649235
\(231\) 0 0
\(232\) 1.87741 0.123258
\(233\) −19.2331 −1.26000 −0.630001 0.776595i \(-0.716945\pi\)
−0.630001 + 0.776595i \(0.716945\pi\)
\(234\) 0 0
\(235\) 16.1468 1.05330
\(236\) −0.564926 −0.0367735
\(237\) 0 0
\(238\) 26.5593 1.72158
\(239\) −1.35637 −0.0877362 −0.0438681 0.999037i \(-0.513968\pi\)
−0.0438681 + 0.999037i \(0.513968\pi\)
\(240\) 0 0
\(241\) −9.64310 −0.621167 −0.310583 0.950546i \(-0.600524\pi\)
−0.310583 + 0.950546i \(0.600524\pi\)
\(242\) 2.43794 0.156717
\(243\) 0 0
\(244\) −3.52111 −0.225416
\(245\) −14.8849 −0.950963
\(246\) 0 0
\(247\) 0 0
\(248\) −30.6000 −1.94310
\(249\) 0 0
\(250\) 8.60925 0.544497
\(251\) −26.9114 −1.69863 −0.849317 0.527882i \(-0.822986\pi\)
−0.849317 + 0.527882i \(0.822986\pi\)
\(252\) 0 0
\(253\) 7.35258 0.462253
\(254\) −18.2849 −1.14730
\(255\) 0 0
\(256\) −14.3907 −0.899422
\(257\) −9.72536 −0.606651 −0.303326 0.952887i \(-0.598097\pi\)
−0.303326 + 0.952887i \(0.598097\pi\)
\(258\) 0 0
\(259\) −28.5459 −1.77375
\(260\) 0 0
\(261\) 0 0
\(262\) −3.21073 −0.198360
\(263\) 15.2061 0.937649 0.468824 0.883291i \(-0.344678\pi\)
0.468824 + 0.883291i \(0.344678\pi\)
\(264\) 0 0
\(265\) 32.7318 2.01070
\(266\) 1.13600 0.0696527
\(267\) 0 0
\(268\) 7.08277 0.432649
\(269\) −1.43721 −0.0876284 −0.0438142 0.999040i \(-0.513951\pi\)
−0.0438142 + 0.999040i \(0.513951\pi\)
\(270\) 0 0
\(271\) −7.32006 −0.444662 −0.222331 0.974971i \(-0.571367\pi\)
−0.222331 + 0.974971i \(0.571367\pi\)
\(272\) −14.7841 −0.896419
\(273\) 0 0
\(274\) 2.06233 0.124590
\(275\) 21.3188 1.28557
\(276\) 0 0
\(277\) −0.610580 −0.0366862 −0.0183431 0.999832i \(-0.505839\pi\)
−0.0183431 + 0.999832i \(0.505839\pi\)
\(278\) 4.88204 0.292805
\(279\) 0 0
\(280\) 36.0382 2.15369
\(281\) 11.2772 0.672741 0.336371 0.941730i \(-0.390801\pi\)
0.336371 + 0.941730i \(0.390801\pi\)
\(282\) 0 0
\(283\) −10.5894 −0.629475 −0.314737 0.949179i \(-0.601916\pi\)
−0.314737 + 0.949179i \(0.601916\pi\)
\(284\) 7.00495 0.415667
\(285\) 0 0
\(286\) 0 0
\(287\) −24.5095 −1.44675
\(288\) 0 0
\(289\) 30.8582 1.81519
\(290\) 2.43179 0.142800
\(291\) 0 0
\(292\) 6.04892 0.353986
\(293\) −13.4028 −0.783003 −0.391501 0.920178i \(-0.628044\pi\)
−0.391501 + 0.920178i \(0.628044\pi\)
\(294\) 0 0
\(295\) −2.84654 −0.165732
\(296\) 26.1809 1.52173
\(297\) 0 0
\(298\) −25.1360 −1.45609
\(299\) 0 0
\(300\) 0 0
\(301\) 22.1685 1.27777
\(302\) 8.10034 0.466122
\(303\) 0 0
\(304\) −0.632351 −0.0362678
\(305\) −17.7421 −1.01591
\(306\) 0 0
\(307\) 18.0476 1.03003 0.515015 0.857181i \(-0.327786\pi\)
0.515015 + 0.857181i \(0.327786\pi\)
\(308\) 6.91796 0.394187
\(309\) 0 0
\(310\) −39.6359 −2.25117
\(311\) −28.4993 −1.61604 −0.808022 0.589152i \(-0.799462\pi\)
−0.808022 + 0.589152i \(0.799462\pi\)
\(312\) 0 0
\(313\) 20.2911 1.14692 0.573461 0.819233i \(-0.305601\pi\)
0.573461 + 0.819233i \(0.305601\pi\)
\(314\) 3.38121 0.190813
\(315\) 0 0
\(316\) −2.43668 −0.137074
\(317\) −2.54983 −0.143213 −0.0716065 0.997433i \(-0.522813\pi\)
−0.0716065 + 0.997433i \(0.522813\pi\)
\(318\) 0 0
\(319\) 1.81594 0.101673
\(320\) −29.7127 −1.66099
\(321\) 0 0
\(322\) 9.47889 0.528238
\(323\) 2.04700 0.113898
\(324\) 0 0
\(325\) 0 0
\(326\) −7.06856 −0.391492
\(327\) 0 0
\(328\) 22.4789 1.24119
\(329\) 15.5445 0.856997
\(330\) 0 0
\(331\) 0.993295 0.0545964 0.0272982 0.999627i \(-0.491310\pi\)
0.0272982 + 0.999627i \(0.491310\pi\)
\(332\) −8.14862 −0.447214
\(333\) 0 0
\(334\) −4.33380 −0.237135
\(335\) 35.6886 1.94988
\(336\) 0 0
\(337\) 20.4959 1.11648 0.558241 0.829679i \(-0.311477\pi\)
0.558241 + 0.829679i \(0.311477\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 16.6933 0.905324
\(341\) −29.5981 −1.60282
\(342\) 0 0
\(343\) 9.16852 0.495054
\(344\) −20.3319 −1.09622
\(345\) 0 0
\(346\) 11.9922 0.644706
\(347\) 15.3939 0.826388 0.413194 0.910643i \(-0.364413\pi\)
0.413194 + 0.910643i \(0.364413\pi\)
\(348\) 0 0
\(349\) −3.72587 −0.199441 −0.0997207 0.995015i \(-0.531795\pi\)
−0.0997207 + 0.995015i \(0.531795\pi\)
\(350\) 27.4840 1.46908
\(351\) 0 0
\(352\) −11.0586 −0.589426
\(353\) 13.0307 0.693552 0.346776 0.937948i \(-0.387276\pi\)
0.346776 + 0.937948i \(0.387276\pi\)
\(354\) 0 0
\(355\) 35.2965 1.87334
\(356\) −5.38335 −0.285317
\(357\) 0 0
\(358\) 11.6052 0.613354
\(359\) 25.2339 1.33179 0.665897 0.746044i \(-0.268049\pi\)
0.665897 + 0.746044i \(0.268049\pi\)
\(360\) 0 0
\(361\) −18.9124 −0.995392
\(362\) 24.5233 1.28892
\(363\) 0 0
\(364\) 0 0
\(365\) 30.4792 1.59536
\(366\) 0 0
\(367\) 5.87369 0.306604 0.153302 0.988179i \(-0.451009\pi\)
0.153302 + 0.988179i \(0.451009\pi\)
\(368\) −5.27639 −0.275051
\(369\) 0 0
\(370\) 33.9119 1.76299
\(371\) 31.5110 1.63597
\(372\) 0 0
\(373\) 11.4330 0.591976 0.295988 0.955192i \(-0.404351\pi\)
0.295988 + 0.955192i \(0.404351\pi\)
\(374\) −23.5613 −1.21832
\(375\) 0 0
\(376\) −14.2567 −0.735232
\(377\) 0 0
\(378\) 0 0
\(379\) −26.5773 −1.36519 −0.682593 0.730799i \(-0.739148\pi\)
−0.682593 + 0.730799i \(0.739148\pi\)
\(380\) 0.714013 0.0366281
\(381\) 0 0
\(382\) 24.4330 1.25010
\(383\) −3.74725 −0.191476 −0.0957379 0.995407i \(-0.530521\pi\)
−0.0957379 + 0.995407i \(0.530521\pi\)
\(384\) 0 0
\(385\) 34.8582 1.77654
\(386\) −2.55257 −0.129922
\(387\) 0 0
\(388\) 1.77181 0.0899499
\(389\) 6.57955 0.333596 0.166798 0.985991i \(-0.446657\pi\)
0.166798 + 0.985991i \(0.446657\pi\)
\(390\) 0 0
\(391\) 17.0804 0.863792
\(392\) 13.1425 0.663799
\(393\) 0 0
\(394\) 2.87502 0.144841
\(395\) −12.2779 −0.617770
\(396\) 0 0
\(397\) 11.2101 0.562621 0.281310 0.959617i \(-0.409231\pi\)
0.281310 + 0.959617i \(0.409231\pi\)
\(398\) 13.2261 0.662966
\(399\) 0 0
\(400\) −15.2989 −0.764944
\(401\) −20.8547 −1.04143 −0.520717 0.853730i \(-0.674335\pi\)
−0.520717 + 0.853730i \(0.674335\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 10.2309 0.509008
\(405\) 0 0
\(406\) 2.34109 0.116187
\(407\) 25.3236 1.25525
\(408\) 0 0
\(409\) −36.5163 −1.80562 −0.902808 0.430044i \(-0.858498\pi\)
−0.902808 + 0.430044i \(0.858498\pi\)
\(410\) 29.1167 1.43797
\(411\) 0 0
\(412\) −1.45175 −0.0715224
\(413\) −2.74037 −0.134845
\(414\) 0 0
\(415\) −41.0592 −2.01552
\(416\) 0 0
\(417\) 0 0
\(418\) −1.00777 −0.0492916
\(419\) 9.99674 0.488373 0.244186 0.969728i \(-0.421479\pi\)
0.244186 + 0.969728i \(0.421479\pi\)
\(420\) 0 0
\(421\) 0.770479 0.0375508 0.0187754 0.999824i \(-0.494023\pi\)
0.0187754 + 0.999824i \(0.494023\pi\)
\(422\) 22.7637 1.10812
\(423\) 0 0
\(424\) −28.9004 −1.40353
\(425\) 49.5245 2.40229
\(426\) 0 0
\(427\) −17.0804 −0.826577
\(428\) −4.59957 −0.222329
\(429\) 0 0
\(430\) −26.3357 −1.27002
\(431\) −30.2838 −1.45872 −0.729359 0.684131i \(-0.760181\pi\)
−0.729359 + 0.684131i \(0.760181\pi\)
\(432\) 0 0
\(433\) −28.0713 −1.34902 −0.674510 0.738266i \(-0.735645\pi\)
−0.674510 + 0.738266i \(0.735645\pi\)
\(434\) −38.1576 −1.83162
\(435\) 0 0
\(436\) 2.11828 0.101447
\(437\) 0.730567 0.0349478
\(438\) 0 0
\(439\) −20.1347 −0.960975 −0.480488 0.877001i \(-0.659540\pi\)
−0.480488 + 0.877001i \(0.659540\pi\)
\(440\) −31.9702 −1.52412
\(441\) 0 0
\(442\) 0 0
\(443\) 31.8494 1.51321 0.756606 0.653871i \(-0.226856\pi\)
0.756606 + 0.653871i \(0.226856\pi\)
\(444\) 0 0
\(445\) −27.1256 −1.28588
\(446\) 20.7539 0.982724
\(447\) 0 0
\(448\) −28.6045 −1.35143
\(449\) −3.47313 −0.163907 −0.0819537 0.996636i \(-0.526116\pi\)
−0.0819537 + 0.996636i \(0.526116\pi\)
\(450\) 0 0
\(451\) 21.7429 1.02383
\(452\) 4.02698 0.189413
\(453\) 0 0
\(454\) 13.2433 0.621537
\(455\) 0 0
\(456\) 0 0
\(457\) −21.1021 −0.987117 −0.493558 0.869713i \(-0.664304\pi\)
−0.493558 + 0.869713i \(0.664304\pi\)
\(458\) −13.5645 −0.633829
\(459\) 0 0
\(460\) 5.95779 0.277783
\(461\) −7.19427 −0.335071 −0.167535 0.985866i \(-0.553581\pi\)
−0.167535 + 0.985866i \(0.553581\pi\)
\(462\) 0 0
\(463\) −33.1594 −1.54105 −0.770525 0.637410i \(-0.780006\pi\)
−0.770525 + 0.637410i \(0.780006\pi\)
\(464\) −1.30316 −0.0604977
\(465\) 0 0
\(466\) 21.9963 1.01896
\(467\) 21.6350 1.00115 0.500575 0.865693i \(-0.333122\pi\)
0.500575 + 0.865693i \(0.333122\pi\)
\(468\) 0 0
\(469\) 34.3575 1.58648
\(470\) −18.4665 −0.851798
\(471\) 0 0
\(472\) 2.51334 0.115686
\(473\) −19.6662 −0.904251
\(474\) 0 0
\(475\) 2.11828 0.0971932
\(476\) 16.0707 0.736600
\(477\) 0 0
\(478\) 1.55124 0.0709519
\(479\) 31.0020 1.41652 0.708259 0.705952i \(-0.249481\pi\)
0.708259 + 0.705952i \(0.249481\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 11.0285 0.502335
\(483\) 0 0
\(484\) 1.47517 0.0670533
\(485\) 8.92776 0.405389
\(486\) 0 0
\(487\) 27.3478 1.23925 0.619624 0.784899i \(-0.287285\pi\)
0.619624 + 0.784899i \(0.287285\pi\)
\(488\) 15.6653 0.709134
\(489\) 0 0
\(490\) 17.0234 0.769040
\(491\) 15.0017 0.677019 0.338510 0.940963i \(-0.390077\pi\)
0.338510 + 0.940963i \(0.390077\pi\)
\(492\) 0 0
\(493\) 4.21850 0.189992
\(494\) 0 0
\(495\) 0 0
\(496\) 21.2403 0.953716
\(497\) 33.9800 1.52421
\(498\) 0 0
\(499\) −5.85862 −0.262268 −0.131134 0.991365i \(-0.541862\pi\)
−0.131134 + 0.991365i \(0.541862\pi\)
\(500\) 5.20936 0.232970
\(501\) 0 0
\(502\) 30.7778 1.37368
\(503\) −25.8797 −1.15392 −0.576958 0.816773i \(-0.695761\pi\)
−0.576958 + 0.816773i \(0.695761\pi\)
\(504\) 0 0
\(505\) 51.5515 2.29401
\(506\) −8.40892 −0.373822
\(507\) 0 0
\(508\) −11.0640 −0.490885
\(509\) 10.5306 0.466761 0.233381 0.972385i \(-0.425021\pi\)
0.233381 + 0.972385i \(0.425021\pi\)
\(510\) 0 0
\(511\) 29.3424 1.29803
\(512\) 21.0950 0.932277
\(513\) 0 0
\(514\) 11.1226 0.490596
\(515\) −7.31505 −0.322340
\(516\) 0 0
\(517\) −13.7899 −0.606477
\(518\) 32.6470 1.43443
\(519\) 0 0
\(520\) 0 0
\(521\) −10.5395 −0.461744 −0.230872 0.972984i \(-0.574158\pi\)
−0.230872 + 0.972984i \(0.574158\pi\)
\(522\) 0 0
\(523\) −4.97152 −0.217390 −0.108695 0.994075i \(-0.534667\pi\)
−0.108695 + 0.994075i \(0.534667\pi\)
\(524\) −1.94278 −0.0848708
\(525\) 0 0
\(526\) −17.3907 −0.758273
\(527\) −68.7576 −2.99513
\(528\) 0 0
\(529\) −16.9041 −0.734960
\(530\) −37.4344 −1.62605
\(531\) 0 0
\(532\) 0.687382 0.0298018
\(533\) 0 0
\(534\) 0 0
\(535\) −23.1763 −1.00200
\(536\) −31.5110 −1.36107
\(537\) 0 0
\(538\) 1.64370 0.0708647
\(539\) 12.7122 0.547554
\(540\) 0 0
\(541\) 1.99462 0.0857555 0.0428778 0.999080i \(-0.486347\pi\)
0.0428778 + 0.999080i \(0.486347\pi\)
\(542\) 8.37173 0.359596
\(543\) 0 0
\(544\) −25.6896 −1.10143
\(545\) 10.6736 0.457205
\(546\) 0 0
\(547\) −37.6872 −1.61139 −0.805695 0.592331i \(-0.798208\pi\)
−0.805695 + 0.592331i \(0.798208\pi\)
\(548\) 1.24789 0.0533073
\(549\) 0 0
\(550\) −24.3817 −1.03964
\(551\) 0.180435 0.00768679
\(552\) 0 0
\(553\) −11.8200 −0.502637
\(554\) 0.698302 0.0296680
\(555\) 0 0
\(556\) 2.95407 0.125280
\(557\) 26.4695 1.12155 0.560774 0.827969i \(-0.310504\pi\)
0.560774 + 0.827969i \(0.310504\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −25.0150 −1.05708
\(561\) 0 0
\(562\) −12.8974 −0.544043
\(563\) 6.79718 0.286467 0.143234 0.989689i \(-0.454250\pi\)
0.143234 + 0.989689i \(0.454250\pi\)
\(564\) 0 0
\(565\) 20.2911 0.853653
\(566\) 12.1108 0.509054
\(567\) 0 0
\(568\) −31.1648 −1.30765
\(569\) 38.2784 1.60471 0.802356 0.596845i \(-0.203579\pi\)
0.802356 + 0.596845i \(0.203579\pi\)
\(570\) 0 0
\(571\) 29.7375 1.24447 0.622237 0.782829i \(-0.286224\pi\)
0.622237 + 0.782829i \(0.286224\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 28.0307 1.16998
\(575\) 17.6751 0.737102
\(576\) 0 0
\(577\) −11.7549 −0.489365 −0.244682 0.969603i \(-0.578684\pi\)
−0.244682 + 0.969603i \(0.578684\pi\)
\(578\) −35.2915 −1.46793
\(579\) 0 0
\(580\) 1.47145 0.0610987
\(581\) −39.5278 −1.63989
\(582\) 0 0
\(583\) −27.9541 −1.15774
\(584\) −26.9114 −1.11360
\(585\) 0 0
\(586\) 15.3284 0.633211
\(587\) 18.2090 0.751565 0.375782 0.926708i \(-0.377374\pi\)
0.375782 + 0.926708i \(0.377374\pi\)
\(588\) 0 0
\(589\) −2.94092 −0.121178
\(590\) 3.25550 0.134027
\(591\) 0 0
\(592\) −18.1728 −0.746899
\(593\) −31.5421 −1.29528 −0.647639 0.761948i \(-0.724243\pi\)
−0.647639 + 0.761948i \(0.724243\pi\)
\(594\) 0 0
\(595\) 80.9770 3.31973
\(596\) −15.2095 −0.623006
\(597\) 0 0
\(598\) 0 0
\(599\) −32.4592 −1.32625 −0.663124 0.748510i \(-0.730770\pi\)
−0.663124 + 0.748510i \(0.730770\pi\)
\(600\) 0 0
\(601\) 8.47112 0.345544 0.172772 0.984962i \(-0.444728\pi\)
0.172772 + 0.984962i \(0.444728\pi\)
\(602\) −25.3534 −1.03333
\(603\) 0 0
\(604\) 4.90143 0.199436
\(605\) 7.43309 0.302198
\(606\) 0 0
\(607\) −23.9342 −0.971460 −0.485730 0.874109i \(-0.661446\pi\)
−0.485730 + 0.874109i \(0.661446\pi\)
\(608\) −1.09880 −0.0445624
\(609\) 0 0
\(610\) 20.2911 0.821563
\(611\) 0 0
\(612\) 0 0
\(613\) 38.8864 1.57061 0.785303 0.619112i \(-0.212507\pi\)
0.785303 + 0.619112i \(0.212507\pi\)
\(614\) −20.6405 −0.832981
\(615\) 0 0
\(616\) −30.7778 −1.24007
\(617\) 34.1927 1.37655 0.688273 0.725452i \(-0.258369\pi\)
0.688273 + 0.725452i \(0.258369\pi\)
\(618\) 0 0
\(619\) −43.0398 −1.72992 −0.864958 0.501844i \(-0.832655\pi\)
−0.864958 + 0.501844i \(0.832655\pi\)
\(620\) −23.9833 −0.963191
\(621\) 0 0
\(622\) 32.5937 1.30689
\(623\) −26.1138 −1.04623
\(624\) 0 0
\(625\) −9.54527 −0.381811
\(626\) −23.2063 −0.927511
\(627\) 0 0
\(628\) 2.04593 0.0816416
\(629\) 58.8279 2.34562
\(630\) 0 0
\(631\) −4.92394 −0.196019 −0.0980094 0.995185i \(-0.531248\pi\)
−0.0980094 + 0.995185i \(0.531248\pi\)
\(632\) 10.8407 0.431221
\(633\) 0 0
\(634\) 2.91617 0.115816
\(635\) −55.7491 −2.21234
\(636\) 0 0
\(637\) 0 0
\(638\) −2.07683 −0.0822226
\(639\) 0 0
\(640\) 8.08410 0.319552
\(641\) −29.8396 −1.17859 −0.589297 0.807917i \(-0.700595\pi\)
−0.589297 + 0.807917i \(0.700595\pi\)
\(642\) 0 0
\(643\) −13.7885 −0.543766 −0.271883 0.962330i \(-0.587646\pi\)
−0.271883 + 0.962330i \(0.587646\pi\)
\(644\) 5.73557 0.226013
\(645\) 0 0
\(646\) −2.34109 −0.0921091
\(647\) −41.4540 −1.62972 −0.814862 0.579654i \(-0.803188\pi\)
−0.814862 + 0.579654i \(0.803188\pi\)
\(648\) 0 0
\(649\) 2.43104 0.0954267
\(650\) 0 0
\(651\) 0 0
\(652\) −4.27711 −0.167505
\(653\) 39.5948 1.54946 0.774732 0.632290i \(-0.217885\pi\)
0.774732 + 0.632290i \(0.217885\pi\)
\(654\) 0 0
\(655\) −9.78927 −0.382498
\(656\) −15.6032 −0.609202
\(657\) 0 0
\(658\) −17.7778 −0.693050
\(659\) 24.7437 0.963876 0.481938 0.876205i \(-0.339933\pi\)
0.481938 + 0.876205i \(0.339933\pi\)
\(660\) 0 0
\(661\) −21.2664 −0.827165 −0.413583 0.910467i \(-0.635723\pi\)
−0.413583 + 0.910467i \(0.635723\pi\)
\(662\) −1.13600 −0.0441519
\(663\) 0 0
\(664\) 36.2529 1.40689
\(665\) 3.46357 0.134312
\(666\) 0 0
\(667\) 1.50557 0.0582958
\(668\) −2.62233 −0.101461
\(669\) 0 0
\(670\) −40.8159 −1.57686
\(671\) 15.1524 0.584950
\(672\) 0 0
\(673\) 12.7821 0.492713 0.246357 0.969179i \(-0.420767\pi\)
0.246357 + 0.969179i \(0.420767\pi\)
\(674\) −23.4405 −0.902894
\(675\) 0 0
\(676\) 0 0
\(677\) −17.5915 −0.676097 −0.338048 0.941129i \(-0.609767\pi\)
−0.338048 + 0.941129i \(0.609767\pi\)
\(678\) 0 0
\(679\) 8.59478 0.329837
\(680\) −74.2682 −2.84805
\(681\) 0 0
\(682\) 33.8504 1.29620
\(683\) −27.2609 −1.04311 −0.521555 0.853218i \(-0.674648\pi\)
−0.521555 + 0.853218i \(0.674648\pi\)
\(684\) 0 0
\(685\) 6.28786 0.240247
\(686\) −10.4858 −0.400348
\(687\) 0 0
\(688\) 14.1129 0.538049
\(689\) 0 0
\(690\) 0 0
\(691\) −27.5851 −1.04939 −0.524693 0.851291i \(-0.675820\pi\)
−0.524693 + 0.851291i \(0.675820\pi\)
\(692\) 7.25637 0.275846
\(693\) 0 0
\(694\) −17.6055 −0.668297
\(695\) 14.8849 0.564617
\(696\) 0 0
\(697\) 50.5096 1.91319
\(698\) 4.26117 0.161287
\(699\) 0 0
\(700\) 16.6303 0.628566
\(701\) 20.1813 0.762236 0.381118 0.924526i \(-0.375539\pi\)
0.381118 + 0.924526i \(0.375539\pi\)
\(702\) 0 0
\(703\) 2.51620 0.0949004
\(704\) 25.3756 0.956379
\(705\) 0 0
\(706\) −14.9028 −0.560873
\(707\) 49.6287 1.86648
\(708\) 0 0
\(709\) −36.9764 −1.38868 −0.694339 0.719648i \(-0.744303\pi\)
−0.694339 + 0.719648i \(0.744303\pi\)
\(710\) −40.3675 −1.51497
\(711\) 0 0
\(712\) 23.9503 0.897577
\(713\) −24.5393 −0.919004
\(714\) 0 0
\(715\) 0 0
\(716\) 7.02218 0.262431
\(717\) 0 0
\(718\) −28.8592 −1.07702
\(719\) −2.87442 −0.107198 −0.0535990 0.998563i \(-0.517069\pi\)
−0.0535990 + 0.998563i \(0.517069\pi\)
\(720\) 0 0
\(721\) −7.04221 −0.262266
\(722\) 21.6296 0.804969
\(723\) 0 0
\(724\) 14.8388 0.551479
\(725\) 4.36539 0.162126
\(726\) 0 0
\(727\) 15.5459 0.576564 0.288282 0.957546i \(-0.406916\pi\)
0.288282 + 0.957546i \(0.406916\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −34.8582 −1.29016
\(731\) −45.6853 −1.68973
\(732\) 0 0
\(733\) −37.2723 −1.37668 −0.688342 0.725386i \(-0.741661\pi\)
−0.688342 + 0.725386i \(0.741661\pi\)
\(734\) −6.71756 −0.247950
\(735\) 0 0
\(736\) −9.16852 −0.337956
\(737\) −30.4792 −1.12272
\(738\) 0 0
\(739\) 29.5652 1.08757 0.543787 0.839223i \(-0.316990\pi\)
0.543787 + 0.839223i \(0.316990\pi\)
\(740\) 20.5197 0.754319
\(741\) 0 0
\(742\) −36.0382 −1.32300
\(743\) 5.28528 0.193898 0.0969490 0.995289i \(-0.469092\pi\)
0.0969490 + 0.995289i \(0.469092\pi\)
\(744\) 0 0
\(745\) −76.6376 −2.80778
\(746\) −13.0755 −0.478729
\(747\) 0 0
\(748\) −14.2567 −0.521275
\(749\) −22.3119 −0.815258
\(750\) 0 0
\(751\) 11.7616 0.429188 0.214594 0.976703i \(-0.431157\pi\)
0.214594 + 0.976703i \(0.431157\pi\)
\(752\) 9.89593 0.360867
\(753\) 0 0
\(754\) 0 0
\(755\) 24.6973 0.898826
\(756\) 0 0
\(757\) −50.1226 −1.82174 −0.910868 0.412697i \(-0.864587\pi\)
−0.910868 + 0.412697i \(0.864587\pi\)
\(758\) 30.3957 1.10402
\(759\) 0 0
\(760\) −3.17662 −0.115228
\(761\) 20.5473 0.744840 0.372420 0.928064i \(-0.378528\pi\)
0.372420 + 0.928064i \(0.378528\pi\)
\(762\) 0 0
\(763\) 10.2755 0.371996
\(764\) 14.7841 0.534871
\(765\) 0 0
\(766\) 4.28562 0.154846
\(767\) 0 0
\(768\) 0 0
\(769\) 40.8745 1.47397 0.736987 0.675907i \(-0.236248\pi\)
0.736987 + 0.675907i \(0.236248\pi\)
\(770\) −39.8662 −1.43668
\(771\) 0 0
\(772\) −1.54453 −0.0555889
\(773\) 26.7857 0.963416 0.481708 0.876332i \(-0.340017\pi\)
0.481708 + 0.876332i \(0.340017\pi\)
\(774\) 0 0
\(775\) −71.1517 −2.55584
\(776\) −7.88271 −0.282973
\(777\) 0 0
\(778\) −7.52483 −0.269778
\(779\) 2.16041 0.0774048
\(780\) 0 0
\(781\) −30.1444 −1.07865
\(782\) −19.5343 −0.698545
\(783\) 0 0
\(784\) −9.12259 −0.325807
\(785\) 10.3090 0.367945
\(786\) 0 0
\(787\) −31.3631 −1.11797 −0.558987 0.829176i \(-0.688810\pi\)
−0.558987 + 0.829176i \(0.688810\pi\)
\(788\) 1.73964 0.0619722
\(789\) 0 0
\(790\) 14.0419 0.499588
\(791\) 19.5343 0.694560
\(792\) 0 0
\(793\) 0 0
\(794\) −12.8207 −0.454989
\(795\) 0 0
\(796\) 8.00298 0.283658
\(797\) 18.4727 0.654336 0.327168 0.944966i \(-0.393906\pi\)
0.327168 + 0.944966i \(0.393906\pi\)
\(798\) 0 0
\(799\) −32.0344 −1.13330
\(800\) −26.5841 −0.939890
\(801\) 0 0
\(802\) 23.8509 0.842203
\(803\) −26.0303 −0.918588
\(804\) 0 0
\(805\) 28.9004 1.01860
\(806\) 0 0
\(807\) 0 0
\(808\) −45.5171 −1.60128
\(809\) 37.1258 1.30527 0.652637 0.757671i \(-0.273663\pi\)
0.652637 + 0.757671i \(0.273663\pi\)
\(810\) 0 0
\(811\) −5.15319 −0.180953 −0.0904766 0.995899i \(-0.528839\pi\)
−0.0904766 + 0.995899i \(0.528839\pi\)
\(812\) 1.41657 0.0497119
\(813\) 0 0
\(814\) −28.9618 −1.01511
\(815\) −21.5515 −0.754915
\(816\) 0 0
\(817\) −1.95407 −0.0683641
\(818\) 41.7626 1.46019
\(819\) 0 0
\(820\) 17.6182 0.615254
\(821\) −14.4368 −0.503849 −0.251924 0.967747i \(-0.581063\pi\)
−0.251924 + 0.967747i \(0.581063\pi\)
\(822\) 0 0
\(823\) 29.8568 1.04074 0.520372 0.853940i \(-0.325793\pi\)
0.520372 + 0.853940i \(0.325793\pi\)
\(824\) 6.45877 0.225002
\(825\) 0 0
\(826\) 3.13408 0.109049
\(827\) −43.5486 −1.51433 −0.757167 0.653222i \(-0.773417\pi\)
−0.757167 + 0.653222i \(0.773417\pi\)
\(828\) 0 0
\(829\) −35.3327 −1.22716 −0.613578 0.789634i \(-0.710270\pi\)
−0.613578 + 0.789634i \(0.710270\pi\)
\(830\) 46.9581 1.62994
\(831\) 0 0
\(832\) 0 0
\(833\) 29.5310 1.02319
\(834\) 0 0
\(835\) −13.2134 −0.457268
\(836\) −0.609790 −0.0210900
\(837\) 0 0
\(838\) −11.4330 −0.394945
\(839\) −36.6107 −1.26394 −0.631971 0.774992i \(-0.717754\pi\)
−0.631971 + 0.774992i \(0.717754\pi\)
\(840\) 0 0
\(841\) −28.6282 −0.987178
\(842\) −0.881173 −0.0303672
\(843\) 0 0
\(844\) 13.7741 0.474122
\(845\) 0 0
\(846\) 0 0
\(847\) 7.15585 0.245878
\(848\) 20.0605 0.688881
\(849\) 0 0
\(850\) −56.6396 −1.94272
\(851\) 20.9954 0.719714
\(852\) 0 0
\(853\) −8.47757 −0.290266 −0.145133 0.989412i \(-0.546361\pi\)
−0.145133 + 0.989412i \(0.546361\pi\)
\(854\) 19.5343 0.668450
\(855\) 0 0
\(856\) 20.4634 0.699423
\(857\) 15.0555 0.514286 0.257143 0.966373i \(-0.417219\pi\)
0.257143 + 0.966373i \(0.417219\pi\)
\(858\) 0 0
\(859\) −53.1952 −1.81500 −0.907498 0.420056i \(-0.862011\pi\)
−0.907498 + 0.420056i \(0.862011\pi\)
\(860\) −15.9355 −0.543394
\(861\) 0 0
\(862\) 34.6346 1.17966
\(863\) −42.7433 −1.45500 −0.727500 0.686107i \(-0.759318\pi\)
−0.727500 + 0.686107i \(0.759318\pi\)
\(864\) 0 0
\(865\) 36.5633 1.24319
\(866\) 32.1042 1.09095
\(867\) 0 0
\(868\) −23.0887 −0.783683
\(869\) 10.4858 0.355705
\(870\) 0 0
\(871\) 0 0
\(872\) −9.42415 −0.319142
\(873\) 0 0
\(874\) −0.835527 −0.0282621
\(875\) 25.2699 0.854278
\(876\) 0 0
\(877\) 15.7767 0.532742 0.266371 0.963871i \(-0.414175\pi\)
0.266371 + 0.963871i \(0.414175\pi\)
\(878\) 23.0274 0.777137
\(879\) 0 0
\(880\) 22.1914 0.748071
\(881\) −7.46072 −0.251358 −0.125679 0.992071i \(-0.540111\pi\)
−0.125679 + 0.992071i \(0.540111\pi\)
\(882\) 0 0
\(883\) −2.54586 −0.0856750 −0.0428375 0.999082i \(-0.513640\pi\)
−0.0428375 + 0.999082i \(0.513640\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −36.4252 −1.22373
\(887\) −3.48422 −0.116988 −0.0584942 0.998288i \(-0.518630\pi\)
−0.0584942 + 0.998288i \(0.518630\pi\)
\(888\) 0 0
\(889\) −53.6698 −1.80003
\(890\) 31.0227 1.03988
\(891\) 0 0
\(892\) 12.5579 0.420471
\(893\) −1.37019 −0.0458515
\(894\) 0 0
\(895\) 35.3833 1.18273
\(896\) 7.78258 0.259998
\(897\) 0 0
\(898\) 3.97212 0.132551
\(899\) −6.06071 −0.202136
\(900\) 0 0
\(901\) −64.9385 −2.16342
\(902\) −24.8666 −0.827968
\(903\) 0 0
\(904\) −17.9159 −0.595874
\(905\) 74.7695 2.48542
\(906\) 0 0
\(907\) 40.4209 1.34215 0.671077 0.741388i \(-0.265832\pi\)
0.671077 + 0.741388i \(0.265832\pi\)
\(908\) 8.01335 0.265932
\(909\) 0 0
\(910\) 0 0
\(911\) 48.5763 1.60941 0.804703 0.593678i \(-0.202325\pi\)
0.804703 + 0.593678i \(0.202325\pi\)
\(912\) 0 0
\(913\) 35.0659 1.16051
\(914\) 24.1339 0.798277
\(915\) 0 0
\(916\) −8.20775 −0.271192
\(917\) −9.42415 −0.311213
\(918\) 0 0
\(919\) −35.6937 −1.17743 −0.588713 0.808342i \(-0.700365\pi\)
−0.588713 + 0.808342i \(0.700365\pi\)
\(920\) −26.5060 −0.873877
\(921\) 0 0
\(922\) 8.22787 0.270970
\(923\) 0 0
\(924\) 0 0
\(925\) 60.8762 2.00160
\(926\) 37.9234 1.24624
\(927\) 0 0
\(928\) −2.26444 −0.0743338
\(929\) −22.5411 −0.739550 −0.369775 0.929121i \(-0.620565\pi\)
−0.369775 + 0.929121i \(0.620565\pi\)
\(930\) 0 0
\(931\) 1.26311 0.0413968
\(932\) 13.3097 0.435974
\(933\) 0 0
\(934\) −24.7433 −0.809627
\(935\) −71.8364 −2.34930
\(936\) 0 0
\(937\) −32.0344 −1.04652 −0.523260 0.852173i \(-0.675284\pi\)
−0.523260 + 0.852173i \(0.675284\pi\)
\(938\) −39.2936 −1.28298
\(939\) 0 0
\(940\) −11.1739 −0.364452
\(941\) −45.6156 −1.48703 −0.743513 0.668722i \(-0.766842\pi\)
−0.743513 + 0.668722i \(0.766842\pi\)
\(942\) 0 0
\(943\) 18.0267 0.587029
\(944\) −1.74457 −0.0567810
\(945\) 0 0
\(946\) 22.4916 0.731264
\(947\) −39.7494 −1.29168 −0.645840 0.763473i \(-0.723493\pi\)
−0.645840 + 0.763473i \(0.723493\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −2.42261 −0.0785998
\(951\) 0 0
\(952\) −71.4981 −2.31727
\(953\) 14.3621 0.465235 0.232617 0.972568i \(-0.425271\pi\)
0.232617 + 0.972568i \(0.425271\pi\)
\(954\) 0 0
\(955\) 74.4941 2.41057
\(956\) 0.938637 0.0303577
\(957\) 0 0
\(958\) −35.4561 −1.14553
\(959\) 6.05334 0.195473
\(960\) 0 0
\(961\) 67.7837 2.18657
\(962\) 0 0
\(963\) 0 0
\(964\) 6.67324 0.214930
\(965\) −7.78258 −0.250530
\(966\) 0 0
\(967\) 33.2261 1.06848 0.534239 0.845333i \(-0.320598\pi\)
0.534239 + 0.845333i \(0.320598\pi\)
\(968\) −6.56300 −0.210943
\(969\) 0 0
\(970\) −10.2104 −0.327836
\(971\) 25.0522 0.803965 0.401982 0.915647i \(-0.368321\pi\)
0.401982 + 0.915647i \(0.368321\pi\)
\(972\) 0 0
\(973\) 14.3297 0.459391
\(974\) −31.2768 −1.00217
\(975\) 0 0
\(976\) −10.8737 −0.348058
\(977\) −55.6599 −1.78072 −0.890359 0.455259i \(-0.849547\pi\)
−0.890359 + 0.455259i \(0.849547\pi\)
\(978\) 0 0
\(979\) 23.1661 0.740393
\(980\) 10.3007 0.329043
\(981\) 0 0
\(982\) −17.1570 −0.547503
\(983\) 28.2878 0.902240 0.451120 0.892463i \(-0.351025\pi\)
0.451120 + 0.892463i \(0.351025\pi\)
\(984\) 0 0
\(985\) 8.76569 0.279298
\(986\) −4.82457 −0.153646
\(987\) 0 0
\(988\) 0 0
\(989\) −16.3049 −0.518466
\(990\) 0 0
\(991\) 41.0863 1.30515 0.652575 0.757724i \(-0.273689\pi\)
0.652575 + 0.757724i \(0.273689\pi\)
\(992\) 36.9082 1.17184
\(993\) 0 0
\(994\) −38.8619 −1.23262
\(995\) 40.3254 1.27840
\(996\) 0 0
\(997\) 45.7663 1.44943 0.724716 0.689047i \(-0.241971\pi\)
0.724716 + 0.689047i \(0.241971\pi\)
\(998\) 6.70033 0.212095
\(999\) 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 1521.2.a.v.1.3 6
3.2 odd 2 inner 1521.2.a.v.1.4 yes 6
13.5 odd 4 1521.2.b.n.1351.7 12
13.8 odd 4 1521.2.b.n.1351.6 12
13.12 even 2 1521.2.a.w.1.4 yes 6
39.5 even 4 1521.2.b.n.1351.5 12
39.8 even 4 1521.2.b.n.1351.8 12
39.38 odd 2 1521.2.a.w.1.3 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1521.2.a.v.1.3 6 1.1 even 1 trivial
1521.2.a.v.1.4 yes 6 3.2 odd 2 inner
1521.2.a.w.1.3 yes 6 39.38 odd 2
1521.2.a.w.1.4 yes 6 13.12 even 2
1521.2.b.n.1351.5 12 39.5 even 4
1521.2.b.n.1351.6 12 13.8 odd 4
1521.2.b.n.1351.7 12 13.5 odd 4
1521.2.b.n.1351.8 12 39.8 even 4