Properties

Label 324.5.g.d.269.1
Level $324$
Weight $5$
Character 324.269
Analytic conductor $33.492$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [324,5,Mod(53,324)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(324, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("324.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 324.g (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: \(33.4918680392\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 269.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 324.269
Dual form 324.5.g.d.53.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-7.79423 + 4.50000i) q^{5} +(-2.50000 + 4.33013i) q^{7} +O(q^{10})\) \(q+(-7.79423 + 4.50000i) q^{5} +(-2.50000 + 4.33013i) q^{7} +(101.325 + 58.5000i) q^{11} +(17.0000 + 29.4449i) q^{13} -450.000i q^{17} -64.0000 q^{19} +(-530.008 + 306.000i) q^{23} +(-272.000 + 471.118i) q^{25} +(919.719 + 531.000i) q^{29} +(348.500 + 603.620i) q^{31} -45.0000i q^{35} -748.000 q^{37} +(-592.361 + 342.000i) q^{41} +(-1309.00 + 2267.25i) q^{43} +(-2291.50 - 1323.00i) q^{47} +(1188.00 + 2057.68i) q^{49} +1071.00i q^{53} -1053.00 q^{55} +(5035.07 - 2907.00i) q^{59} +(-3202.00 + 5546.03i) q^{61} +(-265.004 - 153.000i) q^{65} +(2609.00 + 4518.92i) q^{67} +6570.00i q^{71} -4519.00 q^{73} +(-506.625 + 292.500i) q^{77} +(-3751.00 + 6496.92i) q^{79} +(-4746.69 - 2740.50i) q^{83} +(2025.00 + 3507.40i) q^{85} -8874.00i q^{89} -170.000 q^{91} +(498.831 - 288.000i) q^{95} +(-5285.50 + 9154.75i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{7} + 68 q^{13} - 256 q^{19} - 1088 q^{25} + 1394 q^{31} - 2992 q^{37} - 5236 q^{43} + 4752 q^{49} - 4212 q^{55} - 12808 q^{61} + 10436 q^{67} - 18076 q^{73} - 15004 q^{79} + 8100 q^{85} - 680 q^{91} - 21142 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −7.79423 + 4.50000i −0.311769 + 0.180000i −0.647718 0.761880i \(-0.724276\pi\)
0.335949 + 0.941880i \(0.390943\pi\)
\(6\) 0 0
\(7\) −2.50000 + 4.33013i −0.0510204 + 0.0883699i −0.890408 0.455164i \(-0.849581\pi\)
0.839387 + 0.543534i \(0.182914\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 101.325 + 58.5000i 0.837396 + 0.483471i 0.856378 0.516349i \(-0.172709\pi\)
−0.0189819 + 0.999820i \(0.506043\pi\)
\(12\) 0 0
\(13\) 17.0000 + 29.4449i 0.100592 + 0.174230i 0.911929 0.410349i \(-0.134593\pi\)
−0.811337 + 0.584579i \(0.801260\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 450.000i 1.55709i −0.627587 0.778547i \(-0.715957\pi\)
0.627587 0.778547i \(-0.284043\pi\)
\(18\) 0 0
\(19\) −64.0000 −0.177285 −0.0886427 0.996063i \(-0.528253\pi\)
−0.0886427 + 0.996063i \(0.528253\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −530.008 + 306.000i −1.00190 + 0.578450i −0.908811 0.417207i \(-0.863009\pi\)
−0.0930934 + 0.995657i \(0.529676\pi\)
\(24\) 0 0
\(25\) −272.000 + 471.118i −0.435200 + 0.753789i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 919.719 + 531.000i 1.09360 + 0.631391i 0.934533 0.355876i \(-0.115818\pi\)
0.159069 + 0.987268i \(0.449151\pi\)
\(30\) 0 0
\(31\) 348.500 + 603.620i 0.362643 + 0.628116i 0.988395 0.151906i \(-0.0485411\pi\)
−0.625752 + 0.780022i \(0.715208\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 45.0000i 0.0367347i
\(36\) 0 0
\(37\) −748.000 −0.546384 −0.273192 0.961959i \(-0.588079\pi\)
−0.273192 + 0.961959i \(0.588079\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −592.361 + 342.000i −0.352386 + 0.203450i −0.665736 0.746188i \(-0.731882\pi\)
0.313349 + 0.949638i \(0.398549\pi\)
\(42\) 0 0
\(43\) −1309.00 + 2267.25i −0.707950 + 1.22621i 0.257666 + 0.966234i \(0.417047\pi\)
−0.965616 + 0.259972i \(0.916287\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2291.50 1323.00i −1.03735 0.598914i −0.118267 0.992982i \(-0.537734\pi\)
−0.919081 + 0.394068i \(0.871067\pi\)
\(48\) 0 0
\(49\) 1188.00 + 2057.68i 0.494794 + 0.857008i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1071.00i 0.381274i 0.981661 + 0.190637i \(0.0610554\pi\)
−0.981661 + 0.190637i \(0.938945\pi\)
\(54\) 0 0
\(55\) −1053.00 −0.348099
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5035.07 2907.00i 1.44644 0.835105i 0.448177 0.893945i \(-0.352074\pi\)
0.998267 + 0.0588402i \(0.0187402\pi\)
\(60\) 0 0
\(61\) −3202.00 + 5546.03i −0.860521 + 1.49047i 0.0109051 + 0.999941i \(0.496529\pi\)
−0.871426 + 0.490526i \(0.836805\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −265.004 153.000i −0.0627228 0.0362130i
\(66\) 0 0
\(67\) 2609.00 + 4518.92i 0.581198 + 1.00667i 0.995338 + 0.0964517i \(0.0307493\pi\)
−0.414139 + 0.910214i \(0.635917\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6570.00i 1.30331i 0.758514 + 0.651656i \(0.225926\pi\)
−0.758514 + 0.651656i \(0.774074\pi\)
\(72\) 0 0
\(73\) −4519.00 −0.848002 −0.424001 0.905662i \(-0.639375\pi\)
−0.424001 + 0.905662i \(0.639375\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −506.625 + 292.500i −0.0854486 + 0.0493338i
\(78\) 0 0
\(79\) −3751.00 + 6496.92i −0.601025 + 1.04101i 0.391641 + 0.920118i \(0.371908\pi\)
−0.992666 + 0.120888i \(0.961426\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4746.69 2740.50i −0.689024 0.397808i 0.114222 0.993455i \(-0.463562\pi\)
−0.803246 + 0.595647i \(0.796896\pi\)
\(84\) 0 0
\(85\) 2025.00 + 3507.40i 0.280277 + 0.485454i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8874.00i 1.12031i −0.828387 0.560157i \(-0.810741\pi\)
0.828387 0.560157i \(-0.189259\pi\)
\(90\) 0 0
\(91\) −170.000 −0.0205289
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 498.831 288.000i 0.0552721 0.0319114i
\(96\) 0 0
\(97\) −5285.50 + 9154.75i −0.561749 + 0.972978i 0.435595 + 0.900143i \(0.356538\pi\)
−0.997344 + 0.0728355i \(0.976795\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11356.2 + 6556.50i 1.11324 + 0.642731i 0.939667 0.342089i \(-0.111135\pi\)
0.173576 + 0.984821i \(0.444468\pi\)
\(102\) 0 0
\(103\) 2915.00 + 5048.93i 0.274767 + 0.475910i 0.970076 0.242801i \(-0.0780660\pi\)
−0.695310 + 0.718710i \(0.744733\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1089.00i 0.0951175i −0.998868 0.0475587i \(-0.984856\pi\)
0.998868 0.0475587i \(-0.0151441\pi\)
\(108\) 0 0
\(109\) −5020.00 −0.422523 −0.211262 0.977430i \(-0.567757\pi\)
−0.211262 + 0.977430i \(0.567757\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18519.1 + 10692.0i −1.45032 + 0.837340i −0.998499 0.0547701i \(-0.982557\pi\)
−0.451817 + 0.892111i \(0.649224\pi\)
\(114\) 0 0
\(115\) 2754.00 4770.07i 0.208242 0.360686i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1948.56 + 1125.00i 0.137600 + 0.0794435i
\(120\) 0 0
\(121\) −476.000 824.456i −0.0325114 0.0563115i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 10521.0i 0.673344i
\(126\) 0 0
\(127\) 9227.00 0.572075 0.286038 0.958218i \(-0.407662\pi\)
0.286038 + 0.958218i \(0.407662\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3702.26 + 2137.50i −0.215737 + 0.124556i −0.603975 0.797003i \(-0.706417\pi\)
0.388238 + 0.921559i \(0.373084\pi\)
\(132\) 0 0
\(133\) 160.000 277.128i 0.00904517 0.0156667i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9805.14 + 5661.00i 0.522411 + 0.301614i 0.737921 0.674887i \(-0.235808\pi\)
−0.215509 + 0.976502i \(0.569141\pi\)
\(138\) 0 0
\(139\) −4906.00 8497.44i −0.253921 0.439803i 0.710681 0.703514i \(-0.248387\pi\)
−0.964602 + 0.263711i \(0.915054\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3978.00i 0.194533i
\(144\) 0 0
\(145\) −9558.00 −0.454602
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14271.2 8239.50i 0.642819 0.371132i −0.142880 0.989740i \(-0.545636\pi\)
0.785700 + 0.618608i \(0.212303\pi\)
\(150\) 0 0
\(151\) 12777.5 22131.3i 0.560392 0.970628i −0.437070 0.899428i \(-0.643984\pi\)
0.997462 0.0712001i \(-0.0226829\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5432.58 3136.50i −0.226122 0.130552i
\(156\) 0 0
\(157\) −10582.0 18328.6i −0.429307 0.743582i 0.567504 0.823370i \(-0.307909\pi\)
−0.996812 + 0.0797880i \(0.974576\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3060.00i 0.118051i
\(162\) 0 0
\(163\) 33830.0 1.27329 0.636644 0.771158i \(-0.280322\pi\)
0.636644 + 0.771158i \(0.280322\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 23710.0 13689.0i 0.850158 0.490839i −0.0105465 0.999944i \(-0.503357\pi\)
0.860704 + 0.509106i \(0.170024\pi\)
\(168\) 0 0
\(169\) 13702.5 23733.4i 0.479763 0.830973i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −40007.8 23098.5i −1.33676 0.771777i −0.350431 0.936588i \(-0.613965\pi\)
−0.986325 + 0.164812i \(0.947298\pi\)
\(174\) 0 0
\(175\) −1360.00 2355.59i −0.0444082 0.0769172i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20781.0i 0.648575i 0.945959 + 0.324288i \(0.105125\pi\)
−0.945959 + 0.324288i \(0.894875\pi\)
\(180\) 0 0
\(181\) −19504.0 −0.595342 −0.297671 0.954669i \(-0.596210\pi\)
−0.297671 + 0.954669i \(0.596210\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5830.08 3366.00i 0.170346 0.0983492i
\(186\) 0 0
\(187\) 26325.0 45596.2i 0.752810 1.30390i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6095.09 + 3519.00i 0.167076 + 0.0964612i 0.581206 0.813756i \(-0.302581\pi\)
−0.414131 + 0.910217i \(0.635914\pi\)
\(192\) 0 0
\(193\) −25763.5 44623.7i −0.691656 1.19798i −0.971295 0.237878i \(-0.923548\pi\)
0.279639 0.960105i \(-0.409785\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 60435.0i 1.55724i 0.627495 + 0.778621i \(0.284080\pi\)
−0.627495 + 0.778621i \(0.715920\pi\)
\(198\) 0 0
\(199\) 45665.0 1.15313 0.576564 0.817052i \(-0.304393\pi\)
0.576564 + 0.817052i \(0.304393\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4598.59 + 2655.00i −0.111592 + 0.0644277i
\(204\) 0 0
\(205\) 3078.00 5331.25i 0.0732421 0.126859i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6484.80 3744.00i −0.148458 0.0857123i
\(210\) 0 0
\(211\) −6562.00 11365.7i −0.147391 0.255289i 0.782871 0.622184i \(-0.213754\pi\)
−0.930262 + 0.366895i \(0.880421\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 23562.0i 0.509724i
\(216\) 0 0
\(217\) −3485.00 −0.0740088
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13250.2 7650.00i 0.271292 0.156631i
\(222\) 0 0
\(223\) 39425.0 68286.1i 0.792797 1.37316i −0.131432 0.991325i \(-0.541957\pi\)
0.924229 0.381839i \(-0.124709\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 19314.1 + 11151.0i 0.374820 + 0.216402i 0.675562 0.737303i \(-0.263901\pi\)
−0.300742 + 0.953705i \(0.597234\pi\)
\(228\) 0 0
\(229\) −4435.00 7681.65i −0.0845712 0.146482i 0.820637 0.571450i \(-0.193619\pi\)
−0.905208 + 0.424968i \(0.860285\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26010.0i 0.479103i −0.970884 0.239551i \(-0.923000\pi\)
0.970884 0.239551i \(-0.0770003\pi\)
\(234\) 0 0
\(235\) 23814.0 0.431218
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −55557.3 + 32076.0i −0.972624 + 0.561545i −0.900035 0.435817i \(-0.856459\pi\)
−0.0725889 + 0.997362i \(0.523126\pi\)
\(240\) 0 0
\(241\) −16711.0 + 28944.3i −0.287719 + 0.498344i −0.973265 0.229686i \(-0.926230\pi\)
0.685546 + 0.728029i \(0.259564\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −18519.1 10692.0i −0.308523 0.178126i
\(246\) 0 0
\(247\) −1088.00 1884.47i −0.0178334 0.0308884i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 110466.i 1.75340i −0.481037 0.876700i \(-0.659740\pi\)
0.481037 0.876700i \(-0.340260\pi\)
\(252\) 0 0
\(253\) −71604.0 −1.11866
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13141.1 + 7587.00i −0.198959 + 0.114869i −0.596170 0.802858i \(-0.703312\pi\)
0.397211 + 0.917727i \(0.369978\pi\)
\(258\) 0 0
\(259\) 1870.00 3238.94i 0.0278767 0.0482839i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 60576.7 + 34974.0i 0.875779 + 0.505631i 0.869264 0.494348i \(-0.164593\pi\)
0.00651453 + 0.999979i \(0.497926\pi\)
\(264\) 0 0
\(265\) −4819.50 8347.62i −0.0686294 0.118870i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 50346.0i 0.695762i 0.937539 + 0.347881i \(0.113099\pi\)
−0.937539 + 0.347881i \(0.886901\pi\)
\(270\) 0 0
\(271\) 108323. 1.47497 0.737483 0.675366i \(-0.236014\pi\)
0.737483 + 0.675366i \(0.236014\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −55120.8 + 31824.0i −0.728870 + 0.420813i
\(276\) 0 0
\(277\) −15358.0 + 26600.8i −0.200159 + 0.346686i −0.948580 0.316539i \(-0.897479\pi\)
0.748421 + 0.663224i \(0.230813\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 111941. + 64629.0i 1.41767 + 0.818493i 0.996094 0.0882999i \(-0.0281434\pi\)
0.421577 + 0.906793i \(0.361477\pi\)
\(282\) 0 0
\(283\) 31988.0 + 55404.8i 0.399406 + 0.691791i 0.993653 0.112492i \(-0.0358832\pi\)
−0.594247 + 0.804283i \(0.702550\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3420.00i 0.0415205i
\(288\) 0 0
\(289\) −118979. −1.42454
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 94076.3 54315.0i 1.09584 0.632681i 0.160711 0.987001i \(-0.448621\pi\)
0.935124 + 0.354321i \(0.115288\pi\)
\(294\) 0 0
\(295\) −26163.0 + 45315.6i −0.300638 + 0.520720i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −18020.3 10404.0i −0.201567 0.116375i
\(300\) 0 0
\(301\) −6545.00 11336.3i −0.0722398 0.125123i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 57636.0i 0.619575i
\(306\) 0 0
\(307\) 6410.00 0.0680113 0.0340057 0.999422i \(-0.489174\pi\)
0.0340057 + 0.999422i \(0.489174\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −61480.9 + 35496.0i −0.635652 + 0.366994i −0.782938 0.622100i \(-0.786280\pi\)
0.147286 + 0.989094i \(0.452946\pi\)
\(312\) 0 0
\(313\) −80480.5 + 139396.i −0.821489 + 1.42286i 0.0830835 + 0.996543i \(0.473523\pi\)
−0.904573 + 0.426319i \(0.859810\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −168644. 97366.5i −1.67823 0.968927i −0.962789 0.270255i \(-0.912892\pi\)
−0.715442 0.698672i \(-0.753775\pi\)
\(318\) 0 0
\(319\) 62127.0 + 107607.i 0.610519 + 1.05745i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 28800.0i 0.276050i
\(324\) 0 0
\(325\) −18496.0 −0.175110
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 11457.5 6615.00i 0.105852 0.0611136i
\(330\) 0 0
\(331\) 16643.0 28826.5i 0.151906 0.263109i −0.780022 0.625752i \(-0.784792\pi\)
0.931928 + 0.362643i \(0.118125\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −40670.3 23481.0i −0.362400 0.209231i
\(336\) 0 0
\(337\) 63845.0 + 110583.i 0.562169 + 0.973706i 0.997307 + 0.0733418i \(0.0233664\pi\)
−0.435138 + 0.900364i \(0.643300\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 81549.0i 0.701310i
\(342\) 0 0
\(343\) −23885.0 −0.203019
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 120210. 69403.5i 0.998351 0.576398i 0.0905907 0.995888i \(-0.471125\pi\)
0.907760 + 0.419490i \(0.137791\pi\)
\(348\) 0 0
\(349\) −101896. + 176489.i −0.836578 + 1.44900i 0.0561617 + 0.998422i \(0.482114\pi\)
−0.892739 + 0.450573i \(0.851220\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27996.9 + 16164.0i 0.224678 + 0.129718i 0.608114 0.793849i \(-0.291926\pi\)
−0.383437 + 0.923567i \(0.625260\pi\)
\(354\) 0 0
\(355\) −29565.0 51208.1i −0.234596 0.406333i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 185922.i 1.44259i 0.692630 + 0.721293i \(0.256452\pi\)
−0.692630 + 0.721293i \(0.743548\pi\)
\(360\) 0 0
\(361\) −126225. −0.968570
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 35222.1 20335.5i 0.264381 0.152640i
\(366\) 0 0
\(367\) −75539.5 + 130838.i −0.560844 + 0.971410i 0.436579 + 0.899666i \(0.356190\pi\)
−0.997423 + 0.0717442i \(0.977143\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4637.57 2677.50i −0.0336932 0.0194528i
\(372\) 0 0
\(373\) 3239.00 + 5610.11i 0.0232806 + 0.0403231i 0.877431 0.479703i \(-0.159256\pi\)
−0.854150 + 0.520026i \(0.825922\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 36108.0i 0.254051i
\(378\) 0 0
\(379\) 99008.0 0.689274 0.344637 0.938736i \(-0.388002\pi\)
0.344637 + 0.938736i \(0.388002\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 78862.0 45531.0i 0.537614 0.310391i −0.206498 0.978447i \(-0.566207\pi\)
0.744111 + 0.668056i \(0.232873\pi\)
\(384\) 0 0
\(385\) 2632.50 4559.62i 0.0177602 0.0307615i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −82548.7 47659.5i −0.545520 0.314956i 0.201793 0.979428i \(-0.435323\pi\)
−0.747313 + 0.664472i \(0.768657\pi\)
\(390\) 0 0
\(391\) 137700. + 238503.i 0.900701 + 1.56006i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 67518.0i 0.432738i
\(396\) 0 0
\(397\) −163438. −1.03698 −0.518492 0.855083i \(-0.673506\pi\)
−0.518492 + 0.855083i \(0.673506\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −246485. + 142308.i −1.53286 + 0.884994i −0.533627 + 0.845720i \(0.679171\pi\)
−0.999228 + 0.0392745i \(0.987495\pi\)
\(402\) 0 0
\(403\) −11849.0 + 20523.1i −0.0729578 + 0.126367i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −75791.1 43758.0i −0.457540 0.264161i
\(408\) 0 0
\(409\) −53762.5 93119.4i −0.321390 0.556664i 0.659385 0.751806i \(-0.270817\pi\)
−0.980775 + 0.195141i \(0.937483\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 29070.0i 0.170430i
\(414\) 0 0
\(415\) 49329.0 0.286422
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 130148. 75141.0i 0.741327 0.428005i −0.0812249 0.996696i \(-0.525883\pi\)
0.822551 + 0.568691i \(0.192550\pi\)
\(420\) 0 0
\(421\) −67210.0 + 116411.i −0.379201 + 0.656796i −0.990946 0.134259i \(-0.957135\pi\)
0.611745 + 0.791055i \(0.290468\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 212003. + 122400.i 1.17372 + 0.677647i
\(426\) 0 0
\(427\) −16010.0 27730.1i −0.0878083 0.152088i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 237726.i 1.27974i −0.768483 0.639871i \(-0.778988\pi\)
0.768483 0.639871i \(-0.221012\pi\)
\(432\) 0 0
\(433\) 112187. 0.598366 0.299183 0.954196i \(-0.403286\pi\)
0.299183 + 0.954196i \(0.403286\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 33920.5 19584.0i 0.177623 0.102551i
\(438\) 0 0
\(439\) 102322. 177226.i 0.530931 0.919599i −0.468418 0.883507i \(-0.655176\pi\)
0.999348 0.0360919i \(-0.0114909\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 43725.6 + 25245.0i 0.222807 + 0.128638i 0.607249 0.794511i \(-0.292273\pi\)
−0.384442 + 0.923149i \(0.625606\pi\)
\(444\) 0 0
\(445\) 39933.0 + 69166.0i 0.201656 + 0.349279i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 363528.i 1.80321i −0.432565 0.901603i \(-0.642391\pi\)
0.432565 0.901603i \(-0.357609\pi\)
\(450\) 0 0
\(451\) −80028.0 −0.393449
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1325.02 765.000i 0.00640028 0.00369521i
\(456\) 0 0
\(457\) −3338.50 + 5782.45i −0.0159852 + 0.0276872i −0.873907 0.486092i \(-0.838422\pi\)
0.857922 + 0.513780i \(0.171755\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 198620. + 114674.i 0.934592 + 0.539587i 0.888261 0.459339i \(-0.151914\pi\)
0.0463307 + 0.998926i \(0.485247\pi\)
\(462\) 0 0
\(463\) −119400. 206806.i −0.556981 0.964720i −0.997746 0.0670973i \(-0.978626\pi\)
0.440765 0.897622i \(-0.354707\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 263133.i 1.20654i 0.797537 + 0.603270i \(0.206136\pi\)
−0.797537 + 0.603270i \(0.793864\pi\)
\(468\) 0 0
\(469\) −26090.0 −0.118612
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −265269. + 153153.i −1.18567 + 0.684547i
\(474\) 0 0
\(475\) 17408.0 30151.5i 0.0771546 0.133636i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8090.41 + 4671.00i 0.0352614 + 0.0203582i 0.517527 0.855667i \(-0.326853\pi\)
−0.482266 + 0.876025i \(0.660186\pi\)
\(480\) 0 0
\(481\) −12716.0 22024.8i −0.0549617 0.0951965i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 95139.0i 0.404460i
\(486\) 0 0
\(487\) 331262. 1.39673 0.698367 0.715740i \(-0.253910\pi\)
0.698367 + 0.715740i \(0.253910\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −343203. + 198148.i −1.42360 + 0.821917i −0.996605 0.0823364i \(-0.973762\pi\)
−0.426997 + 0.904253i \(0.640428\pi\)
\(492\) 0 0
\(493\) 238950. 413874.i 0.983135 1.70284i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −28448.9 16425.0i −0.115174 0.0664956i
\(498\) 0 0
\(499\) −22525.0 39014.4i −0.0904615 0.156684i 0.817244 0.576292i \(-0.195501\pi\)
−0.907705 + 0.419608i \(0.862168\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 233172.i 0.921596i 0.887505 + 0.460798i \(0.152437\pi\)
−0.887505 + 0.460798i \(0.847563\pi\)
\(504\) 0 0
\(505\) −118017. −0.462766
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 141551. 81724.5i 0.546358 0.315440i −0.201294 0.979531i \(-0.564515\pi\)
0.747652 + 0.664091i \(0.231181\pi\)
\(510\) 0 0
\(511\) 11297.5 19567.8i 0.0432654 0.0749378i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −45440.4 26235.0i −0.171328 0.0989160i
\(516\) 0 0
\(517\) −154791. 268106.i −0.579115 1.00306i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 384606.i 1.41690i −0.705759 0.708452i \(-0.749394\pi\)
0.705759 0.708452i \(-0.250606\pi\)
\(522\) 0 0
\(523\) −214642. −0.784714 −0.392357 0.919813i \(-0.628340\pi\)
−0.392357 + 0.919813i \(0.628340\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 271629. 156825.i 0.978036 0.564669i
\(528\) 0 0
\(529\) 47351.5 82015.2i 0.169209 0.293078i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −20140.3 11628.0i −0.0708943 0.0409308i
\(534\) 0 0
\(535\) 4900.50 + 8487.91i 0.0171211 + 0.0296547i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 277992.i 0.956874i
\(540\) 0 0
\(541\) 545156. 1.86263 0.931314 0.364217i \(-0.118663\pi\)
0.931314 + 0.364217i \(0.118663\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 39127.0 22590.0i 0.131730 0.0760542i
\(546\) 0 0
\(547\) 245711. 425584.i 0.821202 1.42236i −0.0835858 0.996501i \(-0.526637\pi\)
0.904788 0.425863i \(-0.140029\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −58862.0 33984.0i −0.193880 0.111936i
\(552\) 0 0
\(553\) −18755.0 32484.6i −0.0613291 0.106225i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 445077.i 1.43458i 0.696775 + 0.717290i \(0.254618\pi\)
−0.696775 + 0.717290i \(0.745382\pi\)
\(558\) 0 0
\(559\) −89012.0 −0.284856
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 133164. 76882.5i 0.420118 0.242555i −0.275010 0.961441i \(-0.588681\pi\)
0.695128 + 0.718886i \(0.255348\pi\)
\(564\) 0 0
\(565\) 96228.0 166672.i 0.301443 0.522114i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −233577. 134856.i −0.721450 0.416529i 0.0938360 0.995588i \(-0.470087\pi\)
−0.815286 + 0.579058i \(0.803420\pi\)
\(570\) 0 0
\(571\) −294859. 510711.i −0.904362 1.56640i −0.821772 0.569817i \(-0.807014\pi\)
−0.0825900 0.996584i \(-0.526319\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 332928.i 1.00697i
\(576\) 0 0
\(577\) 184094. 0.552953 0.276476 0.961021i \(-0.410833\pi\)
0.276476 + 0.961021i \(0.410833\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 23733.4 13702.5i 0.0703086 0.0405927i
\(582\) 0 0
\(583\) −62653.5 + 108519.i −0.184335 + 0.319278i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −356407. 205772.i −1.03436 0.597185i −0.116126 0.993235i \(-0.537048\pi\)
−0.918229 + 0.396049i \(0.870381\pi\)
\(588\) 0 0
\(589\) −22304.0 38631.7i −0.0642913 0.111356i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 339966.i 0.966777i 0.875406 + 0.483388i \(0.160594\pi\)
−0.875406 + 0.483388i \(0.839406\pi\)
\(594\) 0 0
\(595\) −20250.0 −0.0571994
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 259439. 149787.i 0.723071 0.417465i −0.0928108 0.995684i \(-0.529585\pi\)
0.815882 + 0.578218i \(0.196252\pi\)
\(600\) 0 0
\(601\) 258058. 446969.i 0.714443 1.23745i −0.248731 0.968573i \(-0.580014\pi\)
0.963174 0.268879i \(-0.0866531\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7420.11 + 4284.00i 0.0202721 + 0.0117041i
\(606\) 0 0
\(607\) 243287. + 421385.i 0.660300 + 1.14367i 0.980537 + 0.196336i \(0.0629043\pi\)
−0.320236 + 0.947338i \(0.603762\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 89964.0i 0.240983i
\(612\) 0 0
\(613\) −189550. −0.504432 −0.252216 0.967671i \(-0.581159\pi\)
−0.252216 + 0.967671i \(0.581159\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 128387. 74124.0i 0.337248 0.194710i −0.321807 0.946805i \(-0.604290\pi\)
0.659054 + 0.752095i \(0.270957\pi\)
\(618\) 0 0
\(619\) −5695.00 + 9864.03i −0.0148632 + 0.0257438i −0.873361 0.487073i \(-0.838065\pi\)
0.858498 + 0.512817i \(0.171398\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 38425.5 + 22185.0i 0.0990020 + 0.0571588i
\(624\) 0 0
\(625\) −122656. 212446.i −0.313998 0.543861i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 336600.i 0.850771i
\(630\) 0 0
\(631\) −6715.00 −0.0168650 −0.00843252 0.999964i \(-0.502684\pi\)
−0.00843252 + 0.999964i \(0.502684\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −71917.3 + 41521.5i −0.178355 + 0.102974i
\(636\) 0 0
\(637\) −40392.0 + 69961.0i −0.0995443 + 0.172416i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 382011. + 220554.i 0.929736 + 0.536783i 0.886728 0.462292i \(-0.152973\pi\)
0.0430077 + 0.999075i \(0.486306\pi\)
\(642\) 0 0
\(643\) −273724. 474104.i −0.662050 1.14670i −0.980076 0.198623i \(-0.936353\pi\)
0.318026 0.948082i \(-0.396980\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 42228.0i 0.100877i −0.998727 0.0504385i \(-0.983938\pi\)
0.998727 0.0504385i \(-0.0160619\pi\)
\(648\) 0 0
\(649\) 680238. 1.61500
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 224295. 129496.i 0.526008 0.303691i −0.213381 0.976969i \(-0.568448\pi\)
0.739389 + 0.673278i \(0.235114\pi\)
\(654\) 0 0
\(655\) 19237.5 33320.3i 0.0448400 0.0776652i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 271138. + 156542.i 0.624337 + 0.360461i 0.778556 0.627575i \(-0.215953\pi\)
−0.154218 + 0.988037i \(0.549286\pi\)
\(660\) 0 0
\(661\) 343160. + 594371.i 0.785405 + 1.36036i 0.928757 + 0.370690i \(0.120879\pi\)
−0.143352 + 0.989672i \(0.545788\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2880.00i 0.00651252i
\(666\) 0 0
\(667\) −649944. −1.46091
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −648885. + 374634.i −1.44120 + 0.832074i
\(672\) 0 0
\(673\) −107364. + 185959.i −0.237043 + 0.410570i −0.959864 0.280465i \(-0.909511\pi\)
0.722822 + 0.691035i \(0.242845\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −297599. 171819.i −0.649314 0.374881i 0.138880 0.990309i \(-0.455650\pi\)
−0.788193 + 0.615428i \(0.788983\pi\)
\(678\) 0 0
\(679\) −26427.5 45773.8i −0.0573214 0.0992835i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 626238.i 1.34245i −0.741254 0.671225i \(-0.765769\pi\)
0.741254 0.671225i \(-0.234231\pi\)
\(684\) 0 0
\(685\) −101898. −0.217162
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −31535.4 + 18207.0i −0.0664294 + 0.0383531i
\(690\) 0 0
\(691\) −342238. + 592774.i −0.716757 + 1.24146i 0.245521 + 0.969391i \(0.421041\pi\)
−0.962278 + 0.272069i \(0.912292\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 76477.0 + 44154.0i 0.158329 + 0.0914114i
\(696\) 0 0
\(697\) 153900. + 266563.i 0.316791 + 0.548698i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 809523.i 1.64738i −0.567042 0.823689i \(-0.691912\pi\)
0.567042 0.823689i \(-0.308088\pi\)
\(702\) 0 0
\(703\) 47872.0 0.0968659
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −56781.0 + 32782.5i −0.113596 + 0.0655848i
\(708\) 0 0
\(709\) 159824. 276823.i 0.317943 0.550694i −0.662116 0.749402i \(-0.730341\pi\)
0.980059 + 0.198708i \(0.0636745\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −369415. 213282.i −0.726668 0.419542i
\(714\) 0 0
\(715\) −17901.0 31005.4i −0.0350159 0.0606493i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 55836.0i 0.108008i −0.998541 0.0540041i \(-0.982802\pi\)
0.998541 0.0540041i \(-0.0171984\pi\)
\(720\) 0 0
\(721\) −29150.0 −0.0560748
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −500327. + 288864.i −0.951871 + 0.549563i
\(726\) 0 0
\(727\) −431936. + 748136.i −0.817243 + 1.41551i 0.0904635 + 0.995900i \(0.471165\pi\)
−0.907706 + 0.419606i \(0.862168\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.02026e6 + 589050.i 1.90932 + 1.10234i
\(732\) 0 0
\(733\) −103804. 179794.i −0.193200 0.334631i 0.753109 0.657895i \(-0.228553\pi\)
−0.946309 + 0.323264i \(0.895220\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 610506.i 1.12397i
\(738\) 0 0
\(739\) −803590. −1.47145 −0.735725 0.677280i \(-0.763159\pi\)
−0.735725 + 0.677280i \(0.763159\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −536633. + 309825.i −0.972074 + 0.561227i −0.899868 0.436162i \(-0.856337\pi\)
−0.0722063 + 0.997390i \(0.523004\pi\)
\(744\) 0 0
\(745\) −74155.5 + 128441.i −0.133607 + 0.231415i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4715.51 + 2722.50i 0.00840553 + 0.00485293i
\(750\) 0 0
\(751\) 532992. + 923170.i 0.945020 + 1.63682i 0.755710 + 0.654906i \(0.227292\pi\)
0.189310 + 0.981917i \(0.439375\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 229995.i 0.403482i
\(756\) 0 0
\(757\) 750410. 1.30950 0.654752 0.755844i \(-0.272773\pi\)
0.654752 + 0.755844i \(0.272773\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 357849. 206604.i 0.617917 0.356754i −0.158141 0.987417i \(-0.550550\pi\)
0.776058 + 0.630662i \(0.217217\pi\)
\(762\) 0 0
\(763\) 12550.0 21737.2i 0.0215573 0.0373384i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 171192. + 98838.0i 0.291001 + 0.168009i
\(768\) 0 0
\(769\) 366690. + 635127.i 0.620079 + 1.07401i 0.989471 + 0.144734i \(0.0462328\pi\)
−0.369392 + 0.929274i \(0.620434\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 231822.i 0.387968i −0.981005 0.193984i \(-0.937859\pi\)
0.981005 0.193984i \(-0.0621410\pi\)
\(774\) 0 0
\(775\) −379168. −0.631289
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 37911.1 21888.0i 0.0624729 0.0360688i
\(780\) 0 0
\(781\) −384345. + 665705.i −0.630114 + 1.09139i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 164957. + 95238.0i 0.267690 + 0.154551i
\(786\) 0 0
\(787\) −251362. 435372.i −0.405836 0.702928i 0.588583 0.808437i \(-0.299686\pi\)
−0.994418 + 0.105509i \(0.966353\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 106920.i 0.170886i
\(792\) 0 0
\(793\) −217736. −0.346245
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 516360. 298120.i 0.812898 0.469327i −0.0350635 0.999385i \(-0.511163\pi\)
0.847961 + 0.530058i \(0.177830\pi\)
\(798\) 0 0
\(799\) −595350. + 1.03118e6i −0.932564 + 1.61525i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −457888. 264362.i −0.710113 0.409984i
\(804\) 0 0
\(805\) 13770.0 + 23850.3i 0.0212492 + 0.0368047i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 376038.i 0.574559i 0.957847 + 0.287280i \(0.0927509\pi\)
−0.957847 + 0.287280i \(0.907249\pi\)
\(810\) 0 0
\(811\) −331072. −0.503362 −0.251681 0.967810i \(-0.580983\pi\)
−0.251681 + 0.967810i \(0.580983\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −263679. + 152235.i −0.396972 + 0.229192i
\(816\) 0 0
\(817\) 83776.0 145104.i 0.125509 0.217388i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −237833. 137313.i −0.352847 0.203716i 0.313092 0.949723i \(-0.398635\pi\)
−0.665938 + 0.746007i \(0.731969\pi\)
\(822\) 0 0
\(823\) 270598. + 468689.i 0.399507 + 0.691966i 0.993665 0.112382i \(-0.0358481\pi\)
−0.594158 + 0.804348i \(0.702515\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 358362.i 0.523975i −0.965071 0.261988i \(-0.915622\pi\)
0.965071 0.261988i \(-0.0843780\pi\)
\(828\) 0 0
\(829\) 39626.0 0.0576595 0.0288298 0.999584i \(-0.490822\pi\)
0.0288298 + 0.999584i \(0.490822\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 925954. 534600.i 1.33444 0.770440i
\(834\) 0 0
\(835\) −123201. + 213390.i −0.176702 + 0.306057i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −914029. 527715.i −1.29848 0.749679i −0.318341 0.947976i \(-0.603126\pi\)
−0.980142 + 0.198297i \(0.936459\pi\)
\(840\) 0 0
\(841\) 210282. + 364218.i 0.297310 + 0.514956i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 246645.i 0.345429i
\(846\) 0 0
\(847\) 4760.00 0.00663499
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 396446. 228888.i 0.547425 0.316056i
\(852\) 0 0
\(853\) 16517.0 28608.3i 0.0227004 0.0393182i −0.854452 0.519530i \(-0.826107\pi\)
0.877152 + 0.480212i \(0.159440\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −395121. 228123.i −0.537982 0.310604i 0.206278 0.978493i \(-0.433865\pi\)
−0.744261 + 0.667889i \(0.767198\pi\)
\(858\) 0 0
\(859\) −171802. 297570.i −0.232832 0.403276i 0.725809 0.687897i \(-0.241466\pi\)
−0.958640 + 0.284620i \(0.908132\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 74556.0i 0.100106i 0.998747 + 0.0500531i \(0.0159391\pi\)
−0.998747 + 0.0500531i \(0.984061\pi\)
\(864\) 0 0
\(865\) 415773. 0.555679
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −760140. + 438867.i −1.00659 + 0.581157i
\(870\) 0 0
\(871\) −88706.0 + 153643.i −0.116928 + 0.202524i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 45557.3 + 26302.5i 0.0595034 + 0.0343543i
\(876\) 0 0
\(877\) 348047. + 602835.i 0.452521 + 0.783789i 0.998542 0.0539822i \(-0.0171914\pi\)
−0.546021 + 0.837772i \(0.683858\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 672426.i 0.866349i 0.901310 + 0.433174i \(0.142607\pi\)
−0.901310 + 0.433174i \(0.857393\pi\)
\(882\) 0 0
\(883\) −1.44813e6 −1.85731 −0.928657 0.370938i \(-0.879036\pi\)
−0.928657 + 0.370938i \(0.879036\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.30739e6 754821.i 1.66172 0.959393i 0.689823 0.723978i \(-0.257688\pi\)
0.971895 0.235416i \(-0.0756452\pi\)
\(888\) 0 0
\(889\) −23067.5 + 39954.1i −0.0291875 + 0.0505542i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 146656. + 84672.0i 0.183907 + 0.106179i
\(894\) 0 0
\(895\) −93514.5 161972.i −0.116744 0.202206i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 740214.i 0.915879i
\(900\) 0 0
\(901\) 481950. 0.593680
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 152019. 87768.0i 0.185609 0.107162i
\(906\) 0 0
\(907\) −187414. + 324611.i −0.227818 + 0.394592i −0.957161 0.289556i \(-0.906492\pi\)
0.729343 + 0.684148i \(0.239826\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.04918e6 + 605745.i 1.26419 + 0.729883i 0.973883 0.227050i \(-0.0729080\pi\)
0.290311 + 0.956932i \(0.406241\pi\)
\(912\) 0 0
\(913\) −320638. 555362.i −0.384657 0.666246i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21375.0i 0.0254195i
\(918\) 0 0
\(919\) 656777. 0.777655 0.388827 0.921311i \(-0.372880\pi\)
0.388827 + 0.921311i \(0.372880\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −193453. + 111690.i −0.227076 + 0.131102i
\(924\) 0 0
\(925\) 203456. 352396.i 0.237786 0.411858i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −501418. 289494.i −0.580990 0.335435i 0.180537 0.983568i \(-0.442217\pi\)
−0.761527 + 0.648133i \(0.775550\pi\)
\(930\) 0 0
\(931\) −76032.0 131691.i −0.0877197 0.151935i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 473850.i 0.542023i
\(936\) 0 0
\(937\) 195173. 0.222301 0.111150 0.993804i \(-0.464547\pi\)
0.111150 + 0.993804i \(0.464547\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.14256e6 + 659660.i −1.29033 + 0.744973i −0.978713 0.205234i \(-0.934204\pi\)
−0.311618 + 0.950207i \(0.600871\pi\)
\(942\) 0 0
\(943\) 209304. 362525.i 0.235372 0.407676i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −842314. 486310.i −0.939235 0.542268i −0.0495146 0.998773i \(-0.515767\pi\)
−0.889720 + 0.456506i \(0.849101\pi\)
\(948\) 0 0
\(949\) −76823.0 133061.i −0.0853019 0.147747i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 534384.i 0.588393i −0.955745 0.294197i \(-0.904948\pi\)
0.955745 0.294197i \(-0.0950520\pi\)
\(954\) 0 0
\(955\) −63342.0 −0.0694520
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −49025.7 + 28305.0i −0.0533073 + 0.0307770i
\(960\) 0 0
\(961\) 218856. 379070.i 0.236980 0.410461i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 401613. + 231872.i 0.431274 + 0.248996i
\(966\) 0 0
\(967\) 391810. + 678634.i 0.419008 + 0.725743i 0.995840 0.0911200i \(-0.0290447\pi\)
−0.576832 + 0.816863i \(0.695711\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 276777.i 0.293556i −0.989169 0.146778i \(-0.953110\pi\)
0.989169 0.146778i \(-0.0468904\pi\)
\(972\) 0 0
\(973\) 49060.0 0.0518205
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −561387. + 324117.i −0.588130 + 0.339557i −0.764358 0.644792i \(-0.776944\pi\)
0.176228 + 0.984349i \(0.443610\pi\)
\(978\) 0 0
\(979\) 519129. 899158.i 0.541639 0.938146i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 251894. + 145431.i 0.260682 + 0.150505i 0.624646 0.780908i \(-0.285243\pi\)
−0.363964 + 0.931413i \(0.618577\pi\)
\(984\) 0 0
\(985\) −271958. 471044.i −0.280304 0.485500i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.60222e6i 1.63806i
\(990\) 0 0
\(991\) 881243. 0.897322 0.448661 0.893702i \(-0.351901\pi\)
0.448661 + 0.893702i \(0.351901\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −355923. + 205492.i −0.359510 + 0.207563i
\(996\) 0 0
\(997\) 345083. 597701.i 0.347163 0.601304i −0.638581 0.769554i \(-0.720478\pi\)
0.985744 + 0.168251i \(0.0538118\pi\)
\(998\) 0 0
\(999\) 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 324.5.g.d.269.1 4
3.2 odd 2 inner 324.5.g.d.269.2 4
9.2 odd 6 108.5.c.c.53.2 yes 2
9.4 even 3 inner 324.5.g.d.53.2 4
9.5 odd 6 inner 324.5.g.d.53.1 4
9.7 even 3 108.5.c.c.53.1 2
36.7 odd 6 432.5.e.d.161.1 2
36.11 even 6 432.5.e.d.161.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.5.c.c.53.1 2 9.7 even 3
108.5.c.c.53.2 yes 2 9.2 odd 6
324.5.g.d.53.1 4 9.5 odd 6 inner
324.5.g.d.53.2 4 9.4 even 3 inner
324.5.g.d.269.1 4 1.1 even 1 trivial
324.5.g.d.269.2 4 3.2 odd 2 inner
432.5.e.d.161.1 2 36.7 odd 6
432.5.e.d.161.2 2 36.11 even 6