Properties

Label 144.5.q.a.113.1
Level $144$
Weight $5$
Character 144.113
Analytic conductor $14.885$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [144,5,Mod(65,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(14.8852746841\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.39400128.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 11x^{4} + 14x^{3} + 98x^{2} + 20x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 113.1
Root \(-0.102534 + 0.177594i\) of defining polynomial
Character \(\chi\) \(=\) 144.113
Dual form 144.5.q.a.65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-8.32172 + 3.42768i) q^{3} +(-30.0804 - 17.3669i) q^{5} +(-15.6054 - 27.0294i) q^{7} +(57.5020 - 57.0484i) q^{9} +(-49.9968 + 28.8657i) q^{11} +(36.6478 - 63.4758i) q^{13} +(309.848 + 41.4165i) q^{15} +386.985i q^{17} -115.791 q^{19} +(222.512 + 171.441i) q^{21} +(474.852 + 274.156i) q^{23} +(290.719 + 503.539i) q^{25} +(-282.971 + 671.839i) q^{27} +(-680.082 + 392.645i) q^{29} +(272.367 - 471.753i) q^{31} +(317.117 - 411.585i) q^{33} +1084.07i q^{35} +898.827 q^{37} +(-87.3975 + 653.845i) q^{39} +(2242.01 + 1294.43i) q^{41} +(1000.05 + 1732.14i) q^{43} +(-2720.43 + 717.406i) q^{45} +(702.646 - 405.673i) q^{47} +(713.441 - 1235.72i) q^{49} +(-1326.46 - 3220.38i) q^{51} -2221.00i q^{53} +2005.23 q^{55} +(963.579 - 396.895i) q^{57} +(-1309.65 - 756.128i) q^{59} +(951.281 + 1647.67i) q^{61} +(-2439.33 - 663.979i) q^{63} +(-2204.76 + 1272.92i) q^{65} +(2253.55 - 3903.26i) q^{67} +(-4891.30 - 653.806i) q^{69} +3993.54i q^{71} -3436.70 q^{73} +(-4145.25 - 3193.82i) q^{75} +(1560.44 + 900.923i) q^{77} +(601.388 + 1041.63i) q^{79} +(51.9555 - 6560.79i) q^{81} +(-8016.22 + 4628.17i) q^{83} +(6720.72 - 11640.6i) q^{85} +(4313.58 - 5598.59i) q^{87} +8929.99i q^{89} -2287.62 q^{91} +(-649.539 + 4859.38i) q^{93} +(3483.03 + 2010.93i) q^{95} +(-3335.14 - 5776.64i) q^{97} +(-1228.18 + 4512.07i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 12 q^{5} - 12 q^{7} + 99 q^{9} - 483 q^{11} - 6 q^{13} + 1026 q^{15} + 258 q^{19} + 480 q^{21} + 282 q^{23} - 273 q^{25} - 54 q^{27} - 1056 q^{29} - 1290 q^{31} + 279 q^{33} + 12 q^{37}+ \cdots + 9126 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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 0
\(3\) −8.32172 + 3.42768i −0.924635 + 0.380854i
\(4\) 0 0
\(5\) −30.0804 17.3669i −1.20321 0.694676i −0.241946 0.970290i \(-0.577786\pi\)
−0.961268 + 0.275614i \(0.911119\pi\)
\(6\) 0 0
\(7\) −15.6054 27.0294i −0.318478 0.551620i 0.661693 0.749775i \(-0.269838\pi\)
−0.980171 + 0.198155i \(0.936505\pi\)
\(8\) 0 0
\(9\) 57.5020 57.0484i 0.709901 0.704301i
\(10\) 0 0
\(11\) −49.9968 + 28.8657i −0.413197 + 0.238559i −0.692162 0.721742i \(-0.743342\pi\)
0.278965 + 0.960301i \(0.410009\pi\)
\(12\) 0 0
\(13\) 36.6478 63.4758i 0.216851 0.375597i −0.736993 0.675901i \(-0.763755\pi\)
0.953844 + 0.300304i \(0.0970882\pi\)
\(14\) 0 0
\(15\) 309.848 + 41.4165i 1.37710 + 0.184074i
\(16\) 0 0
\(17\) 386.985i 1.33905i 0.742791 + 0.669524i \(0.233502\pi\)
−0.742791 + 0.669524i \(0.766498\pi\)
\(18\) 0 0
\(19\) −115.791 −0.320750 −0.160375 0.987056i \(-0.551270\pi\)
−0.160375 + 0.987056i \(0.551270\pi\)
\(20\) 0 0
\(21\) 222.512 + 171.441i 0.504563 + 0.388754i
\(22\) 0 0
\(23\) 474.852 + 274.156i 0.897641 + 0.518253i 0.876434 0.481522i \(-0.159916\pi\)
0.0212066 + 0.999775i \(0.493249\pi\)
\(24\) 0 0
\(25\) 290.719 + 503.539i 0.465150 + 0.805663i
\(26\) 0 0
\(27\) −282.971 + 671.839i −0.388164 + 0.921590i
\(28\) 0 0
\(29\) −680.082 + 392.645i −0.808658 + 0.466879i −0.846490 0.532405i \(-0.821288\pi\)
0.0378313 + 0.999284i \(0.487955\pi\)
\(30\) 0 0
\(31\) 272.367 471.753i 0.283420 0.490898i −0.688805 0.724947i \(-0.741864\pi\)
0.972225 + 0.234049i \(0.0751976\pi\)
\(32\) 0 0
\(33\) 317.117 411.585i 0.291200 0.377948i
\(34\) 0 0
\(35\) 1084.07i 0.884957i
\(36\) 0 0
\(37\) 898.827 0.656557 0.328279 0.944581i \(-0.393532\pi\)
0.328279 + 0.944581i \(0.393532\pi\)
\(38\) 0 0
\(39\) −87.3975 + 653.845i −0.0574606 + 0.429878i
\(40\) 0 0
\(41\) 2242.01 + 1294.43i 1.33374 + 0.770034i 0.985870 0.167510i \(-0.0535727\pi\)
0.347867 + 0.937544i \(0.386906\pi\)
\(42\) 0 0
\(43\) 1000.05 + 1732.14i 0.540862 + 0.936801i 0.998855 + 0.0478447i \(0.0152352\pi\)
−0.457993 + 0.888956i \(0.651431\pi\)
\(44\) 0 0
\(45\) −2720.43 + 717.406i −1.34342 + 0.354274i
\(46\) 0 0
\(47\) 702.646 405.673i 0.318083 0.183645i −0.332455 0.943119i \(-0.607877\pi\)
0.650538 + 0.759474i \(0.274544\pi\)
\(48\) 0 0
\(49\) 713.441 1235.72i 0.297143 0.514667i
\(50\) 0 0
\(51\) −1326.46 3220.38i −0.509981 1.23813i
\(52\) 0 0
\(53\) 2221.00i 0.790672i −0.918537 0.395336i \(-0.870628\pi\)
0.918537 0.395336i \(-0.129372\pi\)
\(54\) 0 0
\(55\) 2005.23 0.662886
\(56\) 0 0
\(57\) 963.579 396.895i 0.296577 0.122159i
\(58\) 0 0
\(59\) −1309.65 756.128i −0.376229 0.217216i 0.299947 0.953956i \(-0.403031\pi\)
−0.676176 + 0.736740i \(0.736364\pi\)
\(60\) 0 0
\(61\) 951.281 + 1647.67i 0.255652 + 0.442802i 0.965072 0.261983i \(-0.0843765\pi\)
−0.709420 + 0.704786i \(0.751043\pi\)
\(62\) 0 0
\(63\) −2439.33 663.979i −0.614595 0.167291i
\(64\) 0 0
\(65\) −2204.76 + 1272.92i −0.521836 + 0.301282i
\(66\) 0 0
\(67\) 2253.55 3903.26i 0.502015 0.869516i −0.497982 0.867187i \(-0.665925\pi\)
0.999997 0.00232883i \(-0.000741290\pi\)
\(68\) 0 0
\(69\) −4891.30 653.806i −1.02737 0.137325i
\(70\) 0 0
\(71\) 3993.54i 0.792213i 0.918205 + 0.396106i \(0.129639\pi\)
−0.918205 + 0.396106i \(0.870361\pi\)
\(72\) 0 0
\(73\) −3436.70 −0.644905 −0.322452 0.946586i \(-0.604507\pi\)
−0.322452 + 0.946586i \(0.604507\pi\)
\(74\) 0 0
\(75\) −4145.25 3193.82i −0.736934 0.567790i
\(76\) 0 0
\(77\) 1560.44 + 900.923i 0.263188 + 0.151952i
\(78\) 0 0
\(79\) 601.388 + 1041.63i 0.0963608 + 0.166902i 0.910176 0.414222i \(-0.135946\pi\)
−0.813815 + 0.581124i \(0.802613\pi\)
\(80\) 0 0
\(81\) 51.9555 6560.79i 0.00791883 0.999969i
\(82\) 0 0
\(83\) −8016.22 + 4628.17i −1.16363 + 0.671820i −0.952170 0.305567i \(-0.901154\pi\)
−0.211456 + 0.977387i \(0.567821\pi\)
\(84\) 0 0
\(85\) 6720.72 11640.6i 0.930204 1.61116i
\(86\) 0 0
\(87\) 4313.58 5598.59i 0.569902 0.739673i
\(88\) 0 0
\(89\) 8929.99i 1.12738i 0.825986 + 0.563691i \(0.190619\pi\)
−0.825986 + 0.563691i \(0.809381\pi\)
\(90\) 0 0
\(91\) −2287.62 −0.276249
\(92\) 0 0
\(93\) −649.539 + 4859.38i −0.0751000 + 0.561843i
\(94\) 0 0
\(95\) 3483.03 + 2010.93i 0.385932 + 0.222818i
\(96\) 0 0
\(97\) −3335.14 5776.64i −0.354463 0.613948i 0.632563 0.774509i \(-0.282003\pi\)
−0.987026 + 0.160561i \(0.948670\pi\)
\(98\) 0 0
\(99\) −1228.18 + 4512.07i −0.125311 + 0.460369i
\(100\) 0 0
\(101\) −7924.19 + 4575.03i −0.776805 + 0.448489i −0.835297 0.549799i \(-0.814704\pi\)
0.0584918 + 0.998288i \(0.481371\pi\)
\(102\) 0 0
\(103\) 7656.20 13260.9i 0.721670 1.24997i −0.238660 0.971103i \(-0.576708\pi\)
0.960330 0.278866i \(-0.0899586\pi\)
\(104\) 0 0
\(105\) −3715.85 9021.34i −0.337039 0.818262i
\(106\) 0 0
\(107\) 6099.28i 0.532735i 0.963872 + 0.266367i \(0.0858234\pi\)
−0.963872 + 0.266367i \(0.914177\pi\)
\(108\) 0 0
\(109\) 15169.5 1.27679 0.638393 0.769710i \(-0.279599\pi\)
0.638393 + 0.769710i \(0.279599\pi\)
\(110\) 0 0
\(111\) −7479.78 + 3080.89i −0.607076 + 0.250052i
\(112\) 0 0
\(113\) −1189.11 686.532i −0.0931246 0.0537655i 0.452714 0.891656i \(-0.350456\pi\)
−0.545839 + 0.837890i \(0.683789\pi\)
\(114\) 0 0
\(115\) −9522.48 16493.4i −0.720036 1.24714i
\(116\) 0 0
\(117\) −1513.88 5740.68i −0.110591 0.419365i
\(118\) 0 0
\(119\) 10460.0 6039.06i 0.738646 0.426457i
\(120\) 0 0
\(121\) −5654.04 + 9793.09i −0.386179 + 0.668881i
\(122\) 0 0
\(123\) −23094.3 3086.94i −1.52649 0.204042i
\(124\) 0 0
\(125\) 1513.10i 0.0968386i
\(126\) 0 0
\(127\) 19152.4 1.18745 0.593726 0.804667i \(-0.297656\pi\)
0.593726 + 0.804667i \(0.297656\pi\)
\(128\) 0 0
\(129\) −14259.4 10986.5i −0.856884 0.660210i
\(130\) 0 0
\(131\) 2615.39 + 1510.00i 0.152403 + 0.0879902i 0.574262 0.818671i \(-0.305289\pi\)
−0.421859 + 0.906661i \(0.638622\pi\)
\(132\) 0 0
\(133\) 1806.97 + 3129.76i 0.102152 + 0.176932i
\(134\) 0 0
\(135\) 20179.6 15294.8i 1.10725 0.839223i
\(136\) 0 0
\(137\) −8788.30 + 5073.93i −0.468235 + 0.270336i −0.715501 0.698612i \(-0.753801\pi\)
0.247266 + 0.968948i \(0.420468\pi\)
\(138\) 0 0
\(139\) −17563.2 + 30420.3i −0.909021 + 1.57447i −0.0935938 + 0.995610i \(0.529835\pi\)
−0.815427 + 0.578860i \(0.803498\pi\)
\(140\) 0 0
\(141\) −4456.70 + 5784.34i −0.224169 + 0.290948i
\(142\) 0 0
\(143\) 4231.45i 0.206927i
\(144\) 0 0
\(145\) 27276.1 1.29732
\(146\) 0 0
\(147\) −1701.41 + 12728.7i −0.0787363 + 0.589048i
\(148\) 0 0
\(149\) 32373.7 + 18691.0i 1.45821 + 0.841899i 0.998923 0.0463890i \(-0.0147714\pi\)
0.459288 + 0.888288i \(0.348105\pi\)
\(150\) 0 0
\(151\) 16635.0 + 28812.7i 0.729573 + 1.26366i 0.957064 + 0.289878i \(0.0936146\pi\)
−0.227490 + 0.973780i \(0.573052\pi\)
\(152\) 0 0
\(153\) 22076.9 + 22252.4i 0.943093 + 0.950591i
\(154\) 0 0
\(155\) −16385.8 + 9460.33i −0.682030 + 0.393770i
\(156\) 0 0
\(157\) −4040.16 + 6997.77i −0.163908 + 0.283897i −0.936267 0.351289i \(-0.885743\pi\)
0.772359 + 0.635186i \(0.219077\pi\)
\(158\) 0 0
\(159\) 7612.87 + 18482.5i 0.301130 + 0.731083i
\(160\) 0 0
\(161\) 17113.3i 0.660209i
\(162\) 0 0
\(163\) 25427.1 0.957022 0.478511 0.878081i \(-0.341177\pi\)
0.478511 + 0.878081i \(0.341177\pi\)
\(164\) 0 0
\(165\) −16687.0 + 6873.29i −0.612928 + 0.252463i
\(166\) 0 0
\(167\) 30356.6 + 17526.4i 1.08848 + 0.628434i 0.933171 0.359432i \(-0.117029\pi\)
0.155309 + 0.987866i \(0.450363\pi\)
\(168\) 0 0
\(169\) 11594.4 + 20082.1i 0.405951 + 0.703129i
\(170\) 0 0
\(171\) −6658.21 + 6605.69i −0.227701 + 0.225905i
\(172\) 0 0
\(173\) 12731.0 7350.23i 0.425372 0.245589i −0.272001 0.962297i \(-0.587685\pi\)
0.697373 + 0.716708i \(0.254352\pi\)
\(174\) 0 0
\(175\) 9073.58 15715.9i 0.296280 0.513172i
\(176\) 0 0
\(177\) 13490.3 + 1803.21i 0.430602 + 0.0575573i
\(178\) 0 0
\(179\) 37052.5i 1.15641i −0.815892 0.578205i \(-0.803753\pi\)
0.815892 0.578205i \(-0.196247\pi\)
\(180\) 0 0
\(181\) −39664.7 −1.21073 −0.605365 0.795948i \(-0.706973\pi\)
−0.605365 + 0.795948i \(0.706973\pi\)
\(182\) 0 0
\(183\) −13564.0 10450.7i −0.405028 0.312065i
\(184\) 0 0
\(185\) −27037.0 15609.8i −0.789979 0.456094i
\(186\) 0 0
\(187\) −11170.6 19348.0i −0.319442 0.553290i
\(188\) 0 0
\(189\) 22575.3 2835.80i 0.631990 0.0793874i
\(190\) 0 0
\(191\) −49915.1 + 28818.5i −1.36825 + 0.789959i −0.990704 0.136032i \(-0.956565\pi\)
−0.377545 + 0.925991i \(0.623232\pi\)
\(192\) 0 0
\(193\) −2089.81 + 3619.67i −0.0561039 + 0.0971748i −0.892713 0.450625i \(-0.851201\pi\)
0.836609 + 0.547800i \(0.184534\pi\)
\(194\) 0 0
\(195\) 13984.2 18150.1i 0.367763 0.477319i
\(196\) 0 0
\(197\) 22191.5i 0.571812i 0.958258 + 0.285906i \(0.0922945\pi\)
−0.958258 + 0.285906i \(0.907705\pi\)
\(198\) 0 0
\(199\) −50608.7 −1.27797 −0.638983 0.769221i \(-0.720645\pi\)
−0.638983 + 0.769221i \(0.720645\pi\)
\(200\) 0 0
\(201\) −5374.25 + 40206.3i −0.133023 + 0.995180i
\(202\) 0 0
\(203\) 21225.9 + 12254.8i 0.515080 + 0.297382i
\(204\) 0 0
\(205\) −44960.4 77873.6i −1.06985 1.85303i
\(206\) 0 0
\(207\) 42945.1 11325.0i 1.00224 0.264301i
\(208\) 0 0
\(209\) 5789.18 3342.38i 0.132533 0.0765180i
\(210\) 0 0
\(211\) −4459.02 + 7723.26i −0.100156 + 0.173474i −0.911749 0.410749i \(-0.865267\pi\)
0.811593 + 0.584223i \(0.198601\pi\)
\(212\) 0 0
\(213\) −13688.6 33233.1i −0.301717 0.732508i
\(214\) 0 0
\(215\) 69471.4i 1.50290i
\(216\) 0 0
\(217\) −17001.6 −0.361052
\(218\) 0 0
\(219\) 28599.2 11779.9i 0.596302 0.245614i
\(220\) 0 0
\(221\) 24564.2 + 14182.1i 0.502941 + 0.290373i
\(222\) 0 0
\(223\) −4248.76 7359.07i −0.0854384 0.147984i 0.820140 0.572164i \(-0.193896\pi\)
−0.905578 + 0.424180i \(0.860562\pi\)
\(224\) 0 0
\(225\) 45443.0 + 12369.5i 0.897640 + 0.244335i
\(226\) 0 0
\(227\) −16543.5 + 9551.40i −0.321052 + 0.185360i −0.651862 0.758338i \(-0.726012\pi\)
0.330809 + 0.943698i \(0.392678\pi\)
\(228\) 0 0
\(229\) 5361.51 9286.40i 0.102239 0.177083i −0.810368 0.585921i \(-0.800733\pi\)
0.912607 + 0.408839i \(0.134066\pi\)
\(230\) 0 0
\(231\) −16073.7 2148.52i −0.301225 0.0402638i
\(232\) 0 0
\(233\) 102200.i 1.88251i −0.337694 0.941256i \(-0.609647\pi\)
0.337694 0.941256i \(-0.390353\pi\)
\(234\) 0 0
\(235\) −28181.1 −0.510296
\(236\) 0 0
\(237\) −8574.97 6606.82i −0.152664 0.117624i
\(238\) 0 0
\(239\) −13440.9 7760.09i −0.235305 0.135854i 0.377712 0.925923i \(-0.376711\pi\)
−0.613017 + 0.790070i \(0.710044\pi\)
\(240\) 0 0
\(241\) 35978.9 + 62317.3i 0.619461 + 1.07294i 0.989584 + 0.143955i \(0.0459820\pi\)
−0.370123 + 0.928983i \(0.620685\pi\)
\(242\) 0 0
\(243\) 22056.0 + 54775.2i 0.373520 + 0.927622i
\(244\) 0 0
\(245\) −42921.1 + 24780.5i −0.715054 + 0.412837i
\(246\) 0 0
\(247\) −4243.48 + 7349.92i −0.0695550 + 0.120473i
\(248\) 0 0
\(249\) 50844.9 65991.4i 0.820065 1.06436i
\(250\) 0 0
\(251\) 41487.8i 0.658526i −0.944238 0.329263i \(-0.893200\pi\)
0.944238 0.329263i \(-0.106800\pi\)
\(252\) 0 0
\(253\) −31654.8 −0.494537
\(254\) 0 0
\(255\) −16027.6 + 119907.i −0.246483 + 1.84401i
\(256\) 0 0
\(257\) −62812.2 36264.6i −0.950994 0.549057i −0.0576045 0.998339i \(-0.518346\pi\)
−0.893390 + 0.449283i \(0.851680\pi\)
\(258\) 0 0
\(259\) −14026.6 24294.7i −0.209099 0.362170i
\(260\) 0 0
\(261\) −16706.2 + 61375.5i −0.245244 + 0.900977i
\(262\) 0 0
\(263\) 74383.1 42945.1i 1.07538 0.620872i 0.145735 0.989324i \(-0.453445\pi\)
0.929647 + 0.368451i \(0.120112\pi\)
\(264\) 0 0
\(265\) −38571.8 + 66808.4i −0.549261 + 0.951347i
\(266\) 0 0
\(267\) −30609.2 74312.8i −0.429367 1.04242i
\(268\) 0 0
\(269\) 88967.6i 1.22950i −0.788724 0.614748i \(-0.789258\pi\)
0.788724 0.614748i \(-0.210742\pi\)
\(270\) 0 0
\(271\) −96541.6 −1.31455 −0.657273 0.753652i \(-0.728290\pi\)
−0.657273 + 0.753652i \(0.728290\pi\)
\(272\) 0 0
\(273\) 19036.9 7841.23i 0.255429 0.105210i
\(274\) 0 0
\(275\) −29070.0 16783.6i −0.384397 0.221932i
\(276\) 0 0
\(277\) 11770.8 + 20387.6i 0.153407 + 0.265710i 0.932478 0.361227i \(-0.117642\pi\)
−0.779071 + 0.626936i \(0.784309\pi\)
\(278\) 0 0
\(279\) −11251.1 42664.8i −0.144540 0.548102i
\(280\) 0 0
\(281\) −50584.4 + 29204.9i −0.640626 + 0.369865i −0.784855 0.619679i \(-0.787263\pi\)
0.144230 + 0.989544i \(0.453930\pi\)
\(282\) 0 0
\(283\) −38058.6 + 65919.5i −0.475204 + 0.823078i −0.999597 0.0283989i \(-0.990959\pi\)
0.524393 + 0.851477i \(0.324292\pi\)
\(284\) 0 0
\(285\) −35877.6 4795.66i −0.441707 0.0590417i
\(286\) 0 0
\(287\) 80800.3i 0.980956i
\(288\) 0 0
\(289\) −66236.1 −0.793047
\(290\) 0 0
\(291\) 47554.6 + 36639.7i 0.561573 + 0.432680i
\(292\) 0 0
\(293\) 121166. + 69955.5i 1.41139 + 0.814866i 0.995519 0.0945573i \(-0.0301436\pi\)
0.415871 + 0.909424i \(0.363477\pi\)
\(294\) 0 0
\(295\) 26263.2 + 45489.2i 0.301789 + 0.522714i
\(296\) 0 0
\(297\) −5245.43 41758.0i −0.0594660 0.473399i
\(298\) 0 0
\(299\) 34804.5 20094.4i 0.389308 0.224767i
\(300\) 0 0
\(301\) 31212.5 54061.7i 0.344505 0.596701i
\(302\) 0 0
\(303\) 50261.1 65233.7i 0.547453 0.710537i
\(304\) 0 0
\(305\) 66083.2i 0.710381i
\(306\) 0 0
\(307\) 81796.8 0.867880 0.433940 0.900942i \(-0.357123\pi\)
0.433940 + 0.900942i \(0.357123\pi\)
\(308\) 0 0
\(309\) −18258.5 + 136597.i −0.191226 + 1.43062i
\(310\) 0 0
\(311\) 1474.59 + 851.354i 0.0152458 + 0.00880216i 0.507604 0.861591i \(-0.330531\pi\)
−0.492358 + 0.870393i \(0.663865\pi\)
\(312\) 0 0
\(313\) 34980.1 + 60587.3i 0.357052 + 0.618433i 0.987467 0.157826i \(-0.0504484\pi\)
−0.630415 + 0.776259i \(0.717115\pi\)
\(314\) 0 0
\(315\) 61844.6 + 62336.3i 0.623276 + 0.628232i
\(316\) 0 0
\(317\) 101004. 58314.7i 1.00512 0.580309i 0.0953639 0.995442i \(-0.469599\pi\)
0.909760 + 0.415134i \(0.136265\pi\)
\(318\) 0 0
\(319\) 22668.0 39262.1i 0.222757 0.385826i
\(320\) 0 0
\(321\) −20906.4 50756.5i −0.202894 0.492585i
\(322\) 0 0
\(323\) 44809.3i 0.429500i
\(324\) 0 0
\(325\) 42616.8 0.403472
\(326\) 0 0
\(327\) −126236. + 51996.2i −1.18056 + 0.486269i
\(328\) 0 0
\(329\) −21930.2 12661.4i −0.202605 0.116974i
\(330\) 0 0
\(331\) 50418.1 + 87326.8i 0.460183 + 0.797061i 0.998970 0.0453817i \(-0.0144504\pi\)
−0.538787 + 0.842442i \(0.681117\pi\)
\(332\) 0 0
\(333\) 51684.3 51276.6i 0.466090 0.462414i
\(334\) 0 0
\(335\) −135575. + 78274.3i −1.20806 + 0.697476i
\(336\) 0 0
\(337\) −20047.2 + 34722.8i −0.176520 + 0.305742i −0.940686 0.339277i \(-0.889817\pi\)
0.764166 + 0.645020i \(0.223151\pi\)
\(338\) 0 0
\(339\) 12248.6 + 1637.24i 0.106583 + 0.0142466i
\(340\) 0 0
\(341\) 31448.2i 0.270450i
\(342\) 0 0
\(343\) −119471. −1.01549
\(344\) 0 0
\(345\) 135778. + 104613.i 1.14075 + 0.878920i
\(346\) 0 0
\(347\) 196035. + 113181.i 1.62808 + 0.939971i 0.984666 + 0.174451i \(0.0558150\pi\)
0.643412 + 0.765520i \(0.277518\pi\)
\(348\) 0 0
\(349\) −39799.1 68934.1i −0.326755 0.565957i 0.655111 0.755533i \(-0.272622\pi\)
−0.981866 + 0.189576i \(0.939289\pi\)
\(350\) 0 0
\(351\) 32275.3 + 42583.3i 0.261973 + 0.345641i
\(352\) 0 0
\(353\) −121173. + 69959.4i −0.972427 + 0.561431i −0.899975 0.435941i \(-0.856416\pi\)
−0.0724520 + 0.997372i \(0.523082\pi\)
\(354\) 0 0
\(355\) 69355.5 120127.i 0.550331 0.953202i
\(356\) 0 0
\(357\) −66344.9 + 86108.8i −0.520560 + 0.675633i
\(358\) 0 0
\(359\) 211616.i 1.64195i 0.570963 + 0.820976i \(0.306570\pi\)
−0.570963 + 0.820976i \(0.693430\pi\)
\(360\) 0 0
\(361\) −116913. −0.897119
\(362\) 0 0
\(363\) 13483.7 100876.i 0.102329 0.765549i
\(364\) 0 0
\(365\) 103377. + 59684.8i 0.775959 + 0.448000i
\(366\) 0 0
\(367\) −15996.8 27707.3i −0.118769 0.205713i 0.800511 0.599318i \(-0.204561\pi\)
−0.919280 + 0.393604i \(0.871228\pi\)
\(368\) 0 0
\(369\) 202765. 53471.2i 1.48916 0.392706i
\(370\) 0 0
\(371\) −60032.2 + 34659.6i −0.436151 + 0.251812i
\(372\) 0 0
\(373\) 61573.9 106649.i 0.442567 0.766548i −0.555312 0.831642i \(-0.687401\pi\)
0.997879 + 0.0650938i \(0.0207347\pi\)
\(374\) 0 0
\(375\) −5186.44 12591.6i −0.0368813 0.0895404i
\(376\) 0 0
\(377\) 57558.3i 0.404972i
\(378\) 0 0
\(379\) 116524. 0.811218 0.405609 0.914047i \(-0.367059\pi\)
0.405609 + 0.914047i \(0.367059\pi\)
\(380\) 0 0
\(381\) −159381. + 65648.4i −1.09796 + 0.452246i
\(382\) 0 0
\(383\) −156848. 90556.2i −1.06925 0.617334i −0.141277 0.989970i \(-0.545121\pi\)
−0.927978 + 0.372636i \(0.878454\pi\)
\(384\) 0 0
\(385\) −31292.5 54200.2i −0.211115 0.365661i
\(386\) 0 0
\(387\) 156321. + 42550.2i 1.04375 + 0.284106i
\(388\) 0 0
\(389\) −44017.9 + 25413.7i −0.290891 + 0.167946i −0.638343 0.769752i \(-0.720380\pi\)
0.347453 + 0.937697i \(0.387047\pi\)
\(390\) 0 0
\(391\) −106094. + 183760.i −0.693965 + 1.20198i
\(392\) 0 0
\(393\) −26940.4 3601.04i −0.174429 0.0233154i
\(394\) 0 0
\(395\) 41777.0i 0.267758i
\(396\) 0 0
\(397\) 228710. 1.45112 0.725561 0.688158i \(-0.241580\pi\)
0.725561 + 0.688158i \(0.241580\pi\)
\(398\) 0 0
\(399\) −25764.9 19851.3i −0.161839 0.124693i
\(400\) 0 0
\(401\) 163618. + 94464.9i 1.01752 + 0.587465i 0.913384 0.407099i \(-0.133459\pi\)
0.104134 + 0.994563i \(0.466793\pi\)
\(402\) 0 0
\(403\) −19963.3 34577.4i −0.122920 0.212903i
\(404\) 0 0
\(405\) −115504. + 196449.i −0.704182 + 1.19768i
\(406\) 0 0
\(407\) −44938.5 + 25945.2i −0.271287 + 0.156628i
\(408\) 0 0
\(409\) −138713. + 240258.i −0.829223 + 1.43626i 0.0694256 + 0.997587i \(0.477883\pi\)
−0.898649 + 0.438669i \(0.855450\pi\)
\(410\) 0 0
\(411\) 55741.9 72347.3i 0.329988 0.428291i
\(412\) 0 0
\(413\) 47198.8i 0.276714i
\(414\) 0 0
\(415\) 321508. 1.86679
\(416\) 0 0
\(417\) 41884.6 313351.i 0.240870 1.80201i
\(418\) 0 0
\(419\) 88699.9 + 51210.9i 0.505237 + 0.291699i 0.730874 0.682513i \(-0.239113\pi\)
−0.225637 + 0.974212i \(0.572446\pi\)
\(420\) 0 0
\(421\) −23567.9 40820.8i −0.132971 0.230312i 0.791850 0.610716i \(-0.209118\pi\)
−0.924821 + 0.380404i \(0.875785\pi\)
\(422\) 0 0
\(423\) 17260.5 63411.8i 0.0964658 0.354397i
\(424\) 0 0
\(425\) −194862. + 112504.i −1.07882 + 0.622857i
\(426\) 0 0
\(427\) 29690.3 51425.1i 0.162839 0.282046i
\(428\) 0 0
\(429\) −14504.1 35213.0i −0.0788090 0.191332i
\(430\) 0 0
\(431\) 45556.1i 0.245240i 0.992454 + 0.122620i \(0.0391297\pi\)
−0.992454 + 0.122620i \(0.960870\pi\)
\(432\) 0 0
\(433\) 209599. 1.11793 0.558965 0.829192i \(-0.311199\pi\)
0.558965 + 0.829192i \(0.311199\pi\)
\(434\) 0 0
\(435\) −226984. + 93493.9i −1.19955 + 0.494089i
\(436\) 0 0
\(437\) −54983.5 31744.8i −0.287919 0.166230i
\(438\) 0 0
\(439\) −91842.1 159075.i −0.476555 0.825417i 0.523084 0.852281i \(-0.324781\pi\)
−0.999639 + 0.0268637i \(0.991448\pi\)
\(440\) 0 0
\(441\) −29471.4 111757.i −0.151539 0.574641i
\(442\) 0 0
\(443\) 100410. 57971.5i 0.511644 0.295398i −0.221865 0.975077i \(-0.571215\pi\)
0.733509 + 0.679680i \(0.237881\pi\)
\(444\) 0 0
\(445\) 155086. 268617.i 0.783165 1.35648i
\(446\) 0 0
\(447\) −333472. 44574.2i −1.66895 0.223084i
\(448\) 0 0
\(449\) 328940.i 1.63164i −0.578305 0.815820i \(-0.696286\pi\)
0.578305 0.815820i \(-0.303714\pi\)
\(450\) 0 0
\(451\) −149458. −0.734795
\(452\) 0 0
\(453\) −237192. 182751.i −1.15586 0.890562i
\(454\) 0 0
\(455\) 68812.3 + 39728.8i 0.332387 + 0.191904i
\(456\) 0 0
\(457\) 106369. + 184236.i 0.509308 + 0.882148i 0.999942 + 0.0107819i \(0.00343204\pi\)
−0.490634 + 0.871366i \(0.663235\pi\)
\(458\) 0 0
\(459\) −259992. 109506.i −1.23405 0.519770i
\(460\) 0 0
\(461\) −41957.2 + 24224.0i −0.197426 + 0.113984i −0.595454 0.803389i \(-0.703028\pi\)
0.398028 + 0.917373i \(0.369695\pi\)
\(462\) 0 0
\(463\) 84550.6 146446.i 0.394416 0.683149i −0.598610 0.801040i \(-0.704280\pi\)
0.993026 + 0.117892i \(0.0376135\pi\)
\(464\) 0 0
\(465\) 103931. 134891.i 0.480660 0.623848i
\(466\) 0 0
\(467\) 54646.4i 0.250569i −0.992121 0.125285i \(-0.960016\pi\)
0.992121 0.125285i \(-0.0399844\pi\)
\(468\) 0 0
\(469\) −140670. −0.639524
\(470\) 0 0
\(471\) 9634.97 72081.9i 0.0434319 0.324926i
\(472\) 0 0
\(473\) −99999.1 57734.5i −0.446965 0.258055i
\(474\) 0 0
\(475\) −33662.6 58305.3i −0.149197 0.258417i
\(476\) 0 0
\(477\) −126704. 127712.i −0.556871 0.561299i
\(478\) 0 0
\(479\) 120011. 69288.6i 0.523060 0.301989i −0.215126 0.976586i \(-0.569016\pi\)
0.738186 + 0.674597i \(0.235683\pi\)
\(480\) 0 0
\(481\) 32940.0 57053.7i 0.142375 0.246601i
\(482\) 0 0
\(483\) 58658.9 + 142412.i 0.251443 + 0.610453i
\(484\) 0 0
\(485\) 231684.i 0.984948i
\(486\) 0 0
\(487\) −23464.1 −0.0989340 −0.0494670 0.998776i \(-0.515752\pi\)
−0.0494670 + 0.998776i \(0.515752\pi\)
\(488\) 0 0
\(489\) −211597. + 87156.1i −0.884897 + 0.364485i
\(490\) 0 0
\(491\) −64368.8 37163.4i −0.267001 0.154153i 0.360523 0.932750i \(-0.382598\pi\)
−0.627524 + 0.778597i \(0.715932\pi\)
\(492\) 0 0
\(493\) −151948. 263181.i −0.625173 1.08283i
\(494\) 0 0
\(495\) 115305. 114395.i 0.470583 0.466872i
\(496\) 0 0
\(497\) 107943. 62321.0i 0.437001 0.252302i
\(498\) 0 0
\(499\) −9486.01 + 16430.2i −0.0380963 + 0.0659847i −0.884445 0.466645i \(-0.845463\pi\)
0.846349 + 0.532629i \(0.178796\pi\)
\(500\) 0 0
\(501\) −312694. 41796.9i −1.24579 0.166521i
\(502\) 0 0
\(503\) 117856.i 0.465818i −0.972498 0.232909i \(-0.925175\pi\)
0.972498 0.232909i \(-0.0748245\pi\)
\(504\) 0 0
\(505\) 317816. 1.24622
\(506\) 0 0
\(507\) −165320. 127375.i −0.643146 0.495529i
\(508\) 0 0
\(509\) 158881. + 91730.2i 0.613250 + 0.354060i 0.774236 0.632897i \(-0.218134\pi\)
−0.160986 + 0.986957i \(0.551468\pi\)
\(510\) 0 0
\(511\) 53631.2 + 92891.9i 0.205388 + 0.355743i
\(512\) 0 0
\(513\) 32765.5 77792.9i 0.124504 0.295601i
\(514\) 0 0
\(515\) −460602. + 265929.i −1.73665 + 1.00265i
\(516\) 0 0
\(517\) −23420.0 + 40564.7i −0.0876207 + 0.151763i
\(518\) 0 0
\(519\) −80749.2 + 104804.i −0.299781 + 0.389085i
\(520\) 0 0
\(521\) 409498.i 1.50861i 0.656526 + 0.754303i \(0.272025\pi\)
−0.656526 + 0.754303i \(0.727975\pi\)
\(522\) 0 0
\(523\) 211852. 0.774513 0.387256 0.921972i \(-0.373423\pi\)
0.387256 + 0.921972i \(0.373423\pi\)
\(524\) 0 0
\(525\) −21638.6 + 161885.i −0.0785075 + 0.587336i
\(526\) 0 0
\(527\) 182561. + 105402.i 0.657336 + 0.379513i
\(528\) 0 0
\(529\) 10402.4 + 18017.4i 0.0371724 + 0.0643845i
\(530\) 0 0
\(531\) −118444. + 31234.7i −0.420071 + 0.110777i
\(532\) 0 0
\(533\) 164330. 94875.7i 0.578444 0.333965i
\(534\) 0 0
\(535\) 105926. 183468.i 0.370078 0.640994i
\(536\) 0 0
\(537\) 127004. + 308341.i 0.440423 + 1.06926i
\(538\) 0 0
\(539\) 82375.9i 0.283545i
\(540\) 0 0
\(541\) 44016.5 0.150391 0.0751954 0.997169i \(-0.476042\pi\)
0.0751954 + 0.997169i \(0.476042\pi\)
\(542\) 0 0
\(543\) 330078. 135958.i 1.11948 0.461111i
\(544\) 0 0
\(545\) −456304. 263447.i −1.53625 0.886953i
\(546\) 0 0
\(547\) −21429.6 37117.1i −0.0716207 0.124051i 0.827991 0.560741i \(-0.189484\pi\)
−0.899612 + 0.436691i \(0.856150\pi\)
\(548\) 0 0
\(549\) 148697. + 40475.0i 0.493354 + 0.134290i
\(550\) 0 0
\(551\) 78747.3 45464.8i 0.259378 0.149752i
\(552\) 0 0
\(553\) 18769.8 32510.3i 0.0613776 0.106309i
\(554\) 0 0
\(555\) 278500. + 37226.3i 0.904148 + 0.120855i
\(556\) 0 0
\(557\) 311007.i 1.00244i 0.865319 + 0.501222i \(0.167116\pi\)
−0.865319 + 0.501222i \(0.832884\pi\)
\(558\) 0 0
\(559\) 146599. 0.469145
\(560\) 0 0
\(561\) 159277. + 122719.i 0.506090 + 0.389931i
\(562\) 0 0
\(563\) 247482. + 142884.i 0.780777 + 0.450782i 0.836705 0.547653i \(-0.184479\pi\)
−0.0559288 + 0.998435i \(0.517812\pi\)
\(564\) 0 0
\(565\) 23845.9 + 41302.2i 0.0746992 + 0.129383i
\(566\) 0 0
\(567\) −178145. + 100980.i −0.554125 + 0.314100i
\(568\) 0 0
\(569\) −118371. + 68341.3i −0.365611 + 0.211086i −0.671539 0.740969i \(-0.734367\pi\)
0.305928 + 0.952055i \(0.401033\pi\)
\(570\) 0 0
\(571\) −14255.0 + 24690.4i −0.0437215 + 0.0757279i −0.887058 0.461658i \(-0.847255\pi\)
0.843337 + 0.537386i \(0.180588\pi\)
\(572\) 0 0
\(573\) 316599. 410912.i 0.964273 1.25153i
\(574\) 0 0
\(575\) 318809.i 0.964261i
\(576\) 0 0
\(577\) −293742. −0.882297 −0.441149 0.897434i \(-0.645429\pi\)
−0.441149 + 0.897434i \(0.645429\pi\)
\(578\) 0 0
\(579\) 4983.78 37285.1i 0.0148663 0.111219i
\(580\) 0 0
\(581\) 250193. + 144449.i 0.741179 + 0.427920i
\(582\) 0 0
\(583\) 64110.6 + 111043.i 0.188622 + 0.326703i
\(584\) 0 0
\(585\) −54160.0 + 198973.i −0.158258 + 0.581410i
\(586\) 0 0
\(587\) −334334. + 193028.i −0.970295 + 0.560200i −0.899326 0.437279i \(-0.855942\pi\)
−0.0709687 + 0.997479i \(0.522609\pi\)
\(588\) 0 0
\(589\) −31537.6 + 54624.7i −0.0909072 + 0.157456i
\(590\) 0 0
\(591\) −76065.3 184671.i −0.217777 0.528718i
\(592\) 0 0
\(593\) 305581.i 0.868995i −0.900673 0.434497i \(-0.856926\pi\)
0.900673 0.434497i \(-0.143074\pi\)
\(594\) 0 0
\(595\) −419519. −1.18500
\(596\) 0 0
\(597\) 421152. 173471.i 1.18165 0.486718i
\(598\) 0 0
\(599\) −229270. 132369.i −0.638988 0.368920i 0.145236 0.989397i \(-0.453606\pi\)
−0.784225 + 0.620477i \(0.786939\pi\)
\(600\) 0 0
\(601\) 66101.5 + 114491.i 0.183005 + 0.316973i 0.942902 0.333069i \(-0.108084\pi\)
−0.759898 + 0.650043i \(0.774751\pi\)
\(602\) 0 0
\(603\) −93091.3 353006.i −0.256020 0.970841i
\(604\) 0 0
\(605\) 340151. 196386.i 0.929312 0.536538i
\(606\) 0 0
\(607\) −127073. + 220097.i −0.344886 + 0.597360i −0.985333 0.170642i \(-0.945416\pi\)
0.640447 + 0.768002i \(0.278749\pi\)
\(608\) 0 0
\(609\) −218642. 29225.2i −0.589520 0.0787994i
\(610\) 0 0
\(611\) 59468.0i 0.159295i
\(612\) 0 0
\(613\) −492878. −1.31165 −0.655826 0.754912i \(-0.727680\pi\)
−0.655826 + 0.754912i \(0.727680\pi\)
\(614\) 0 0
\(615\) 641074. + 493933.i 1.69495 + 1.30592i
\(616\) 0 0
\(617\) −105029. 60638.8i −0.275893 0.159287i 0.355670 0.934612i \(-0.384253\pi\)
−0.631563 + 0.775325i \(0.717586\pi\)
\(618\) 0 0
\(619\) 153682. + 266185.i 0.401090 + 0.694709i 0.993858 0.110665i \(-0.0352981\pi\)
−0.592768 + 0.805374i \(0.701965\pi\)
\(620\) 0 0
\(621\) −318558. + 241446.i −0.826049 + 0.626090i
\(622\) 0 0
\(623\) 241372. 139356.i 0.621886 0.359046i
\(624\) 0 0
\(625\) 207977. 360227.i 0.532421 0.922181i
\(626\) 0 0
\(627\) −36719.3 + 47657.9i −0.0934026 + 0.121227i
\(628\) 0 0
\(629\) 347832.i 0.879161i
\(630\) 0 0
\(631\) 254196. 0.638425 0.319212 0.947683i \(-0.396582\pi\)
0.319212 + 0.947683i \(0.396582\pi\)
\(632\) 0 0
\(633\) 10633.9 79554.9i 0.0265390 0.198545i
\(634\) 0 0
\(635\) −576112. 332618.i −1.42876 0.824895i
\(636\) 0 0
\(637\) −52292.1 90572.5i −0.128872 0.223212i
\(638\) 0 0
\(639\) 227825. + 229637.i 0.557957 + 0.562393i
\(640\) 0 0
\(641\) 316954. 182993.i 0.771400 0.445368i −0.0619737 0.998078i \(-0.519739\pi\)
0.833374 + 0.552710i \(0.186406\pi\)
\(642\) 0 0
\(643\) 115391. 199864.i 0.279094 0.483406i −0.692066 0.721835i \(-0.743299\pi\)
0.971160 + 0.238429i \(0.0766324\pi\)
\(644\) 0 0
\(645\) 238126. + 578121.i 0.572383 + 1.38963i
\(646\) 0 0
\(647\) 278596.i 0.665529i 0.943010 + 0.332764i \(0.107981\pi\)
−0.943010 + 0.332764i \(0.892019\pi\)
\(648\) 0 0
\(649\) 87304.7 0.207276
\(650\) 0 0
\(651\) 141483. 58276.1i 0.333842 0.137508i
\(652\) 0 0
\(653\) −67988.8 39253.4i −0.159445 0.0920557i 0.418154 0.908376i \(-0.362677\pi\)
−0.577600 + 0.816320i \(0.696010\pi\)
\(654\) 0 0
\(655\) −52448.0 90842.6i −0.122249 0.211742i
\(656\) 0 0
\(657\) −197617. + 196058.i −0.457819 + 0.454208i
\(658\) 0 0
\(659\) 742420. 428637.i 1.70954 0.987003i 0.774418 0.632675i \(-0.218043\pi\)
0.935121 0.354328i \(-0.115290\pi\)
\(660\) 0 0
\(661\) −11172.0 + 19350.4i −0.0255698 + 0.0442882i −0.878527 0.477692i \(-0.841473\pi\)
0.852957 + 0.521981i \(0.174807\pi\)
\(662\) 0 0
\(663\) −253028. 33821.5i −0.575627 0.0769424i
\(664\) 0 0
\(665\) 125526.i 0.283850i
\(666\) 0 0
\(667\) −430584. −0.967846
\(668\) 0 0
\(669\) 60581.6 + 46676.7i 0.135359 + 0.104291i
\(670\) 0 0
\(671\) −95122.1 54918.8i −0.211269 0.121976i
\(672\) 0 0
\(673\) 331739. + 574588.i 0.732430 + 1.26861i 0.955842 + 0.293882i \(0.0949472\pi\)
−0.223412 + 0.974724i \(0.571719\pi\)
\(674\) 0 0
\(675\) −420563. + 52829.0i −0.923046 + 0.115948i
\(676\) 0 0
\(677\) −197558. + 114060.i −0.431040 + 0.248861i −0.699790 0.714349i \(-0.746723\pi\)
0.268749 + 0.963210i \(0.413390\pi\)
\(678\) 0 0
\(679\) −104093. + 180294.i −0.225777 + 0.391058i
\(680\) 0 0
\(681\) 104931. 136190.i 0.226261 0.293664i
\(682\) 0 0
\(683\) 53606.6i 0.114915i 0.998348 + 0.0574575i \(0.0182994\pi\)
−0.998348 + 0.0574575i \(0.981701\pi\)
\(684\) 0 0
\(685\) 352474. 0.751182
\(686\) 0 0
\(687\) −12786.1 + 95656.4i −0.0270910 + 0.202675i
\(688\) 0 0
\(689\) −140980. 81394.6i −0.296974 0.171458i
\(690\) 0 0
\(691\) 378663. + 655863.i 0.793043 + 1.37359i 0.924074 + 0.382213i \(0.124838\pi\)
−0.131031 + 0.991378i \(0.541829\pi\)
\(692\) 0 0
\(693\) 141125. 37216.0i 0.293858 0.0774932i
\(694\) 0 0
\(695\) 1.05661e6 610036.i 2.18749 1.26295i
\(696\) 0 0
\(697\) −500923. + 867624.i −1.03111 + 1.78594i
\(698\) 0 0
\(699\) 350308. + 850477.i 0.716962 + 1.74064i
\(700\) 0 0
\(701\) 506359.i 1.03044i −0.857058 0.515220i \(-0.827710\pi\)
0.857058 0.515220i \(-0.172290\pi\)
\(702\) 0 0
\(703\) −104076. −0.210591
\(704\) 0 0
\(705\) 234515. 96595.9i 0.471838 0.194348i
\(706\) 0 0
\(707\) 247321. + 142791.i 0.494791 + 0.285668i
\(708\) 0 0
\(709\) 325622. + 563993.i 0.647770 + 1.12197i 0.983654 + 0.180067i \(0.0576314\pi\)
−0.335885 + 0.941903i \(0.609035\pi\)
\(710\) 0 0
\(711\) 94004.6 + 25587.8i 0.185956 + 0.0506167i
\(712\) 0 0
\(713\) 258668. 149342.i 0.508819 0.293767i
\(714\) 0 0
\(715\) 73487.2 127284.i 0.143747 0.248978i
\(716\) 0 0
\(717\) 138450. + 18506.3i 0.269312 + 0.0359981i
\(718\) 0 0
\(719\) 69724.6i 0.134874i −0.997724 0.0674370i \(-0.978518\pi\)
0.997724 0.0674370i \(-0.0214822\pi\)
\(720\) 0 0
\(721\) −477913. −0.919345
\(722\) 0 0
\(723\) −513010. 395263.i −0.981407 0.756152i
\(724\) 0 0
\(725\) −395425. 228299.i −0.752294 0.434337i
\(726\) 0 0
\(727\) −373864. 647552.i −0.707367 1.22520i −0.965830 0.259175i \(-0.916549\pi\)
0.258463 0.966021i \(-0.416784\pi\)
\(728\) 0 0
\(729\) −371295. 380223.i −0.698658 0.715456i
\(730\) 0 0
\(731\) −670313. + 387006.i −1.25442 + 0.724240i
\(732\) 0 0
\(733\) −32869.3 + 56931.2i −0.0611761 + 0.105960i −0.894991 0.446083i \(-0.852818\pi\)
0.833815 + 0.552044i \(0.186152\pi\)
\(734\) 0 0
\(735\) 272238. 353337.i 0.503934 0.654054i
\(736\) 0 0
\(737\) 260201.i 0.479042i
\(738\) 0 0
\(739\) −977921. −1.79067 −0.895334 0.445395i \(-0.853063\pi\)
−0.895334 + 0.445395i \(0.853063\pi\)
\(740\) 0 0
\(741\) 10119.8 75709.3i 0.0184305 0.137884i
\(742\) 0 0
\(743\) −437233. 252437.i −0.792019 0.457272i 0.0486539 0.998816i \(-0.484507\pi\)
−0.840673 + 0.541543i \(0.817840\pi\)
\(744\) 0 0
\(745\) −649209. 1.12446e6i −1.16969 2.02597i
\(746\) 0 0
\(747\) −196919. + 723442.i −0.352896 + 1.29647i
\(748\) 0 0
\(749\) 164860. 95181.9i 0.293867 0.169664i
\(750\) 0 0
\(751\) −266395. + 461411.i −0.472332 + 0.818102i −0.999499 0.0316593i \(-0.989921\pi\)
0.527167 + 0.849762i \(0.323254\pi\)
\(752\) 0 0
\(753\) 142207. + 345250.i 0.250802 + 0.608896i
\(754\) 0 0
\(755\) 1.15559e6i 2.02727i
\(756\) 0 0
\(757\) −293571. −0.512297 −0.256148 0.966637i \(-0.582454\pi\)
−0.256148 + 0.966637i \(0.582454\pi\)
\(758\) 0 0
\(759\) 263422. 108503.i 0.457266 0.188346i
\(760\) 0 0
\(761\) −22249.2 12845.6i −0.0384190 0.0221812i 0.480667 0.876903i \(-0.340394\pi\)
−0.519086 + 0.854722i \(0.673728\pi\)
\(762\) 0 0
\(763\) −236727. 410022.i −0.406629 0.704301i
\(764\) 0 0
\(765\) −277625. 1.05277e6i −0.474390 1.79891i
\(766\) 0 0
\(767\) −95991.7 + 55420.9i −0.163171 + 0.0942069i
\(768\) 0 0
\(769\) 323572. 560443.i 0.547165 0.947717i −0.451302 0.892371i \(-0.649040\pi\)
0.998467 0.0553463i \(-0.0176263\pi\)
\(770\) 0 0
\(771\) 647009. + 86483.8i 1.08843 + 0.145488i
\(772\) 0 0
\(773\) 610238.i 1.02127i −0.859798 0.510634i \(-0.829411\pi\)
0.859798 0.510634i \(-0.170589\pi\)
\(774\) 0 0
\(775\) 316728. 0.527331
\(776\) 0 0
\(777\) 200000. + 154095.i 0.331274 + 0.255239i
\(778\) 0 0
\(779\) −259605. 149883.i −0.427797 0.246989i
\(780\) 0 0
\(781\) −115276. 199665.i −0.188990 0.327340i
\(782\) 0 0
\(783\) −71351.0 568013.i −0.116379 0.926477i
\(784\) 0 0
\(785\) 243059. 140330.i 0.394432 0.227726i
\(786\) 0 0
\(787\) −290160. + 502572.i −0.468477 + 0.811427i −0.999351 0.0360244i \(-0.988531\pi\)
0.530874 + 0.847451i \(0.321864\pi\)
\(788\) 0 0
\(789\) −471793. + 612339.i −0.757875 + 0.983644i
\(790\) 0 0
\(791\) 42854.5i 0.0684926i
\(792\) 0 0
\(793\) 139449. 0.221753
\(794\) 0 0
\(795\) 91986.0 688172.i 0.145542 1.08884i
\(796\) 0 0
\(797\) −915508. 528569.i −1.44127 0.832118i −0.443336 0.896355i \(-0.646205\pi\)
−0.997935 + 0.0642373i \(0.979539\pi\)
\(798\) 0 0
\(799\) 156989. + 271913.i 0.245910 + 0.425928i
\(800\) 0 0
\(801\) 509442. + 513492.i 0.794016 + 0.800329i
\(802\) 0 0
\(803\) 171824. 99202.7i 0.266473 0.153848i
\(804\) 0 0
\(805\) −297205. + 514774.i −0.458631 + 0.794373i
\(806\) 0 0
\(807\) 304953. + 740363.i 0.468258 + 1.13684i
\(808\) 0 0
\(809\) 355969.i 0.543895i 0.962312 + 0.271947i \(0.0876677\pi\)
−0.962312 + 0.271947i \(0.912332\pi\)
\(810\) 0 0
\(811\) 136417. 0.207409 0.103704 0.994608i \(-0.466930\pi\)
0.103704 + 0.994608i \(0.466930\pi\)
\(812\) 0 0
\(813\) 803392. 330914.i 1.21548 0.500650i
\(814\) 0 0
\(815\) −764857. 441590.i −1.15150 0.664820i
\(816\) 0 0
\(817\) −115797. 200567.i −0.173482 0.300479i
\(818\) 0 0
\(819\) −131543. + 130505.i −0.196109 + 0.194563i
\(820\) 0 0
\(821\) −394889. + 227989.i −0.585854 + 0.338243i −0.763456 0.645860i \(-0.776499\pi\)
0.177603 + 0.984102i \(0.443166\pi\)
\(822\) 0 0
\(823\) 251795. 436122.i 0.371747 0.643885i −0.618087 0.786109i \(-0.712092\pi\)
0.989834 + 0.142225i \(0.0454255\pi\)
\(824\) 0 0
\(825\) 299441. + 40025.4i 0.439951 + 0.0588069i
\(826\) 0 0
\(827\) 551861.i 0.806898i 0.915002 + 0.403449i \(0.132189\pi\)
−0.915002 + 0.403449i \(0.867811\pi\)
\(828\) 0 0
\(829\) −182308. −0.265275 −0.132638 0.991165i \(-0.542345\pi\)
−0.132638 + 0.991165i \(0.542345\pi\)
\(830\) 0 0
\(831\) −167836. 129314.i −0.243042 0.187259i
\(832\) 0 0
\(833\) 478203. + 276091.i 0.689164 + 0.397889i
\(834\) 0 0
\(835\) −608759. 1.05440e6i −0.873117 1.51228i
\(836\) 0 0
\(837\) 239870. + 316479.i 0.342394 + 0.451746i
\(838\) 0 0
\(839\) 359993. 207842.i 0.511410 0.295263i −0.222003 0.975046i \(-0.571259\pi\)
0.733413 + 0.679783i \(0.237926\pi\)
\(840\) 0 0
\(841\) −45299.7 + 78461.4i −0.0640477 + 0.110934i
\(842\) 0 0
\(843\) 320844. 416423.i 0.451480 0.585975i
\(844\) 0 0
\(845\) 805434.i 1.12802i
\(846\) 0 0
\(847\) 352935. 0.491958
\(848\) 0 0
\(849\) 90762.1 679016.i 0.125918 0.942030i
\(850\) 0 0
\(851\) 426809. + 246419.i 0.589352 + 0.340263i
\(852\) 0 0
\(853\) −75135.3 130138.i −0.103263 0.178857i 0.809764 0.586756i \(-0.199595\pi\)
−0.913027 + 0.407898i \(0.866262\pi\)
\(854\) 0 0
\(855\) 315002. 83069.1i 0.430904 0.113634i
\(856\) 0 0
\(857\) 125373. 72383.9i 0.170703 0.0985554i −0.412214 0.911087i \(-0.635245\pi\)
0.582917 + 0.812532i \(0.301911\pi\)
\(858\) 0 0
\(859\) 166948. 289163.i 0.226254 0.391883i −0.730441 0.682976i \(-0.760685\pi\)
0.956695 + 0.291093i \(0.0940188\pi\)
\(860\) 0 0
\(861\) 276958. + 672398.i 0.373601 + 0.907026i
\(862\) 0 0
\(863\) 1.13280e6i 1.52101i 0.649330 + 0.760507i \(0.275049\pi\)
−0.649330 + 0.760507i \(0.724951\pi\)
\(864\) 0 0
\(865\) −510603. −0.682418
\(866\) 0 0
\(867\) 551198. 227036.i 0.733279 0.302035i
\(868\) 0 0
\(869\) −60135.0 34718.9i −0.0796320 0.0459755i
\(870\) 0 0
\(871\) −165175. 286092.i −0.217725 0.377111i
\(872\) 0 0
\(873\) −521325. 141903.i −0.684038 0.186193i
\(874\) 0 0
\(875\) 40898.3 23612.6i 0.0534181 0.0308410i
\(876\) 0 0
\(877\) −248271. + 430017.i −0.322795 + 0.559097i −0.981064 0.193686i \(-0.937956\pi\)
0.658269 + 0.752783i \(0.271289\pi\)
\(878\) 0 0
\(879\) −1.24810e6 166830.i −1.61537 0.215921i
\(880\) 0 0
\(881\) 1.21533e6i 1.56583i 0.622131 + 0.782913i \(0.286267\pi\)
−0.622131 + 0.782913i \(0.713733\pi\)
\(882\) 0 0
\(883\) 999070. 1.28137 0.640685 0.767804i \(-0.278651\pi\)
0.640685 + 0.767804i \(0.278651\pi\)
\(884\) 0 0
\(885\) −374478. 288527.i −0.478123 0.368383i
\(886\) 0 0
\(887\) −938048. 541582.i −1.19228 0.688362i −0.233456 0.972367i \(-0.575003\pi\)
−0.958823 + 0.284005i \(0.908337\pi\)
\(888\) 0 0
\(889\) −298882. 517678.i −0.378178 0.655023i
\(890\) 0 0
\(891\) 186784. + 329519.i 0.235280 + 0.415073i
\(892\) 0 0
\(893\) −81360.0 + 46973.2i −0.102025 + 0.0589044i
\(894\) 0 0
\(895\) −643488. + 1.11455e6i −0.803330 + 1.39141i
\(896\) 0 0
\(897\) −220756. + 286519.i −0.274365 + 0.356097i
\(898\) 0 0
\(899\) 427774.i 0.529292i
\(900\) 0 0
\(901\) 859491. 1.05875
\(902\) 0 0
\(903\) −74435.6 + 556873.i −0.0912862 + 0.682937i
\(904\) 0 0
\(905\) 1.19313e6 + 688853.i 1.45677 + 0.841065i
\(906\) 0 0
\(907\) 638426. + 1.10579e6i 0.776061 + 1.34418i 0.934196 + 0.356760i \(0.116118\pi\)
−0.158135 + 0.987417i \(0.550548\pi\)
\(908\) 0 0
\(909\) −194658. + 715136.i −0.235583 + 0.865487i
\(910\) 0 0
\(911\) −236846. + 136743.i −0.285383 + 0.164766i −0.635858 0.771806i \(-0.719354\pi\)
0.350475 + 0.936572i \(0.386020\pi\)
\(912\) 0 0
\(913\) 267191. 462788.i 0.320538 0.555188i
\(914\) 0 0
\(915\) 226512. + 549926.i 0.270551 + 0.656844i
\(916\) 0 0
\(917\) 94256.7i 0.112092i
\(918\) 0 0
\(919\) −1.15882e6 −1.37210 −0.686051 0.727553i \(-0.740657\pi\)
−0.686051 + 0.727553i \(0.740657\pi\)
\(920\) 0 0
\(921\) −680690. + 280373.i −0.802472 + 0.330535i
\(922\) 0 0
\(923\) 253493. + 146355.i 0.297552 + 0.171792i
\(924\) 0 0
\(925\) 261306. + 452595.i 0.305397 + 0.528964i
\(926\) 0 0
\(927\) −316268. 1.19930e6i −0.368041 1.39563i
\(928\) 0 0
\(929\) 738465. 426353.i 0.855655 0.494012i −0.00690018 0.999976i \(-0.502196\pi\)
0.862555 + 0.505964i \(0.168863\pi\)
\(930\) 0 0
\(931\) −82610.0 + 143085.i −0.0953089 + 0.165080i
\(932\) 0 0
\(933\) −15189.3 2030.31i −0.0174491 0.00233237i
\(934\) 0 0
\(935\) 775993.i 0.887636i
\(936\) 0 0
\(937\) 629989. 0.717552 0.358776 0.933424i \(-0.383194\pi\)
0.358776 + 0.933424i \(0.383194\pi\)
\(938\) 0 0
\(939\) −498768. 384290.i −0.565676 0.435840i
\(940\) 0 0
\(941\) −366547. 211626.i −0.413953 0.238996i 0.278534 0.960426i \(-0.410152\pi\)
−0.692487 + 0.721431i \(0.743485\pi\)
\(942\) 0 0
\(943\) 709749. + 1.22932e6i 0.798145 + 1.38243i
\(944\) 0 0
\(945\) −728322. 306761.i −0.815568 0.343508i
\(946\) 0 0
\(947\) 1.20762e6 697219.i 1.34657 0.777444i 0.358810 0.933411i \(-0.383183\pi\)
0.987762 + 0.155967i \(0.0498493\pi\)
\(948\) 0 0
\(949\) −125947. + 218147.i −0.139848 + 0.242224i
\(950\) 0 0
\(951\) −640642. + 831488.i −0.708361 + 0.919379i
\(952\) 0 0
\(953\) 24001.0i 0.0264267i −0.999913 0.0132134i \(-0.995794\pi\)
0.999913 0.0132134i \(-0.00420607\pi\)
\(954\) 0 0
\(955\) 2.00195e6 2.19506
\(956\) 0 0
\(957\) −54058.5 + 404426.i −0.0590255 + 0.441586i
\(958\) 0 0
\(959\) 274290. + 158362.i 0.298245 + 0.172192i
\(960\) 0 0
\(961\) 313393. + 542813.i 0.339346 + 0.587765i
\(962\) 0 0
\(963\) 347954. + 350721.i 0.375206 + 0.378189i
\(964\) 0 0
\(965\) 125725. 72587.2i 0.135010 0.0779481i
\(966\) 0 0
\(967\) 522031. 904183.i 0.558268 0.966949i −0.439373 0.898305i \(-0.644799\pi\)
0.997641 0.0686443i \(-0.0218673\pi\)
\(968\) 0 0
\(969\) 153592. + 372890.i 0.163577 + 0.397131i
\(970\) 0 0
\(971\) 605213.i 0.641904i −0.947096 0.320952i \(-0.895997\pi\)
0.947096 0.320952i \(-0.104003\pi\)
\(972\) 0 0
\(973\) 1.09632e6 1.15801
\(974\) 0 0
\(975\) −354645. + 146077.i −0.373065 + 0.153664i
\(976\) 0 0
\(977\) 41660.2 + 24052.5i 0.0436448 + 0.0251983i 0.521664 0.853151i \(-0.325312\pi\)
−0.478019 + 0.878350i \(0.658645\pi\)
\(978\) 0 0
\(979\) −257770. 446471.i −0.268947 0.465831i
\(980\) 0 0
\(981\) 872276. 865396.i 0.906392 0.899243i
\(982\) 0 0
\(983\) 107286. 61941.5i 0.111029 0.0641024i −0.443457 0.896295i \(-0.646248\pi\)
0.554486 + 0.832193i \(0.312915\pi\)
\(984\) 0 0
\(985\) 385397. 667527.i 0.397224 0.688012i
\(986\) 0 0
\(987\) 225896. + 30194.9i 0.231886 + 0.0309955i
\(988\) 0 0
\(989\) 1.09668e6i 1.12121i
\(990\) 0 0
\(991\) −851418. −0.866953 −0.433477 0.901165i \(-0.642713\pi\)
−0.433477 + 0.901165i \(0.642713\pi\)
\(992\) 0 0
\(993\) −718894. 553891.i −0.729065 0.561728i
\(994\) 0 0
\(995\) 1.52233e6 + 878917.i 1.53767 + 0.887773i
\(996\) 0 0
\(997\) 801524. + 1.38828e6i 0.806355 + 1.39665i 0.915373 + 0.402607i \(0.131896\pi\)
−0.109018 + 0.994040i \(0.534771\pi\)
\(998\) 0 0
\(999\) −254342. + 603867.i −0.254852 + 0.605077i
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 144.5.q.a.113.1 6
3.2 odd 2 432.5.q.a.17.3 6
4.3 odd 2 9.5.d.a.5.2 yes 6
9.2 odd 6 inner 144.5.q.a.65.1 6
9.4 even 3 1296.5.e.c.161.6 6
9.5 odd 6 1296.5.e.c.161.1 6
9.7 even 3 432.5.q.a.305.3 6
12.11 even 2 27.5.d.a.17.2 6
36.7 odd 6 27.5.d.a.8.2 6
36.11 even 6 9.5.d.a.2.2 6
36.23 even 6 81.5.b.a.80.3 6
36.31 odd 6 81.5.b.a.80.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.5.d.a.2.2 6 36.11 even 6
9.5.d.a.5.2 yes 6 4.3 odd 2
27.5.d.a.8.2 6 36.7 odd 6
27.5.d.a.17.2 6 12.11 even 2
81.5.b.a.80.3 6 36.23 even 6
81.5.b.a.80.4 6 36.31 odd 6
144.5.q.a.65.1 6 9.2 odd 6 inner
144.5.q.a.113.1 6 1.1 even 1 trivial
432.5.q.a.17.3 6 3.2 odd 2
432.5.q.a.305.3 6 9.7 even 3
1296.5.e.c.161.1 6 9.5 odd 6
1296.5.e.c.161.6 6 9.4 even 3