Properties

Label 1764.4.k.q.1549.1
Level $1764$
Weight $4$
Character 1764.1549
Analytic conductor $104.079$
Analytic rank $0$
Dimension $4$
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] = [1764,4,Mod(361,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), 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: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{193})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 49x^{2} + 48x + 2304 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1549.1
Root \(-3.22311 + 5.58259i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1549
Dual form 1764.4.k.q.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-6.22311 + 10.7787i) q^{5} +O(q^{10})\) \(q+(-6.22311 + 10.7787i) q^{5} +(-25.5618 - 44.2743i) q^{11} -37.2311 q^{13} +(-11.1076 - 19.2389i) q^{17} +(27.1693 - 47.0587i) q^{19} +(88.4622 - 153.221i) q^{23} +(-14.9542 - 25.9015i) q^{25} -61.0916 q^{29} +(159.962 + 277.063i) q^{31} +(157.540 - 272.867i) q^{37} -206.032 q^{41} +339.661 q^{43} +(-71.0320 + 123.031i) q^{47} +(155.008 + 268.482i) q^{53} +636.295 q^{55} +(140.825 + 243.916i) q^{59} +(-271.924 + 470.987i) q^{61} +(231.693 - 401.305i) q^{65} +(239.680 + 415.137i) q^{67} -1105.63 q^{71} +(119.675 + 207.283i) q^{73} +(-580.333 + 1005.17i) q^{79} -2.93158 q^{83} +276.494 q^{85} +(639.371 - 1107.42i) q^{89} +(338.156 + 585.703i) q^{95} -79.0596 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 11 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 11 q^{5} - 5 q^{11} - 10 q^{13} - 100 q^{17} + 67 q^{19} + 76 q^{23} + 93 q^{25} - 550 q^{29} + 362 q^{31} + 5 q^{37} - 324 q^{41} + 1442 q^{43} + 216 q^{47} + 495 q^{53} + 1406 q^{55} - 173 q^{59} - 532 q^{61} + 510 q^{65} - 111 q^{67} - 3200 q^{71} + 1215 q^{73} - 1460 q^{79} - 2818 q^{83} + 328 q^{85} + 1974 q^{89} + 658 q^{95} - 1122 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −6.22311 + 10.7787i −0.556612 + 0.964080i 0.441164 + 0.897426i \(0.354566\pi\)
−0.997776 + 0.0666538i \(0.978768\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −25.5618 44.2743i −0.700651 1.21356i −0.968238 0.250030i \(-0.919559\pi\)
0.267587 0.963534i \(-0.413774\pi\)
\(12\) 0 0
\(13\) −37.2311 −0.794312 −0.397156 0.917751i \(-0.630003\pi\)
−0.397156 + 0.917751i \(0.630003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −11.1076 19.2389i −0.158469 0.274477i 0.775848 0.630920i \(-0.217323\pi\)
−0.934317 + 0.356444i \(0.883989\pi\)
\(18\) 0 0
\(19\) 27.1693 47.0587i 0.328056 0.568210i −0.654070 0.756434i \(-0.726940\pi\)
0.982126 + 0.188224i \(0.0602730\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 88.4622 153.221i 0.801985 1.38908i −0.116323 0.993211i \(-0.537111\pi\)
0.918308 0.395867i \(-0.129556\pi\)
\(24\) 0 0
\(25\) −14.9542 25.9015i −0.119634 0.207212i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −61.0916 −0.391187 −0.195593 0.980685i \(-0.562663\pi\)
−0.195593 + 0.980685i \(0.562663\pi\)
\(30\) 0 0
\(31\) 159.962 + 277.063i 0.926776 + 1.60522i 0.788679 + 0.614805i \(0.210765\pi\)
0.138097 + 0.990419i \(0.455901\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 157.540 272.867i 0.699984 1.21241i −0.268487 0.963283i \(-0.586524\pi\)
0.968471 0.249125i \(-0.0801430\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −206.032 −0.784800 −0.392400 0.919795i \(-0.628355\pi\)
−0.392400 + 0.919795i \(0.628355\pi\)
\(42\) 0 0
\(43\) 339.661 1.20460 0.602301 0.798269i \(-0.294251\pi\)
0.602301 + 0.798269i \(0.294251\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −71.0320 + 123.031i −0.220449 + 0.381828i −0.954944 0.296785i \(-0.904085\pi\)
0.734496 + 0.678613i \(0.237419\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 155.008 + 268.482i 0.401736 + 0.695826i 0.993936 0.109965i \(-0.0350738\pi\)
−0.592200 + 0.805791i \(0.701740\pi\)
\(54\) 0 0
\(55\) 636.295 1.55996
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 140.825 + 243.916i 0.310743 + 0.538223i 0.978523 0.206136i \(-0.0660888\pi\)
−0.667780 + 0.744358i \(0.732755\pi\)
\(60\) 0 0
\(61\) −271.924 + 470.987i −0.570760 + 0.988585i 0.425728 + 0.904851i \(0.360018\pi\)
−0.996488 + 0.0837341i \(0.973315\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 231.693 401.305i 0.442123 0.765780i
\(66\) 0 0
\(67\) 239.680 + 415.137i 0.437038 + 0.756971i 0.997459 0.0712360i \(-0.0226943\pi\)
−0.560422 + 0.828207i \(0.689361\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1105.63 −1.84809 −0.924046 0.382280i \(-0.875139\pi\)
−0.924046 + 0.382280i \(0.875139\pi\)
\(72\) 0 0
\(73\) 119.675 + 207.283i 0.191876 + 0.332338i 0.945872 0.324541i \(-0.105210\pi\)
−0.753996 + 0.656879i \(0.771876\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −580.333 + 1005.17i −0.826488 + 1.43152i 0.0742888 + 0.997237i \(0.476331\pi\)
−0.900777 + 0.434282i \(0.857002\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.93158 −0.00387690 −0.00193845 0.999998i \(-0.500617\pi\)
−0.00193845 + 0.999998i \(0.500617\pi\)
\(84\) 0 0
\(85\) 276.494 0.352824
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 639.371 1107.42i 0.761496 1.31895i −0.180583 0.983560i \(-0.557798\pi\)
0.942079 0.335390i \(-0.108868\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 338.156 + 585.703i 0.365200 + 0.632545i
\(96\) 0 0
\(97\) −79.0596 −0.0827555 −0.0413777 0.999144i \(-0.513175\pi\)
−0.0413777 + 0.999144i \(0.513175\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −686.052 1188.28i −0.675889 1.17067i −0.976208 0.216835i \(-0.930427\pi\)
0.300319 0.953839i \(-0.402907\pi\)
\(102\) 0 0
\(103\) 129.265 223.894i 0.123659 0.214184i −0.797549 0.603254i \(-0.793870\pi\)
0.921208 + 0.389070i \(0.127204\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −631.184 + 1093.24i −0.570270 + 0.987736i 0.426268 + 0.904597i \(0.359828\pi\)
−0.996538 + 0.0831393i \(0.973505\pi\)
\(108\) 0 0
\(109\) −138.416 239.744i −0.121632 0.210673i 0.798779 0.601624i \(-0.205479\pi\)
−0.920411 + 0.390951i \(0.872146\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −52.9156 −0.0440520 −0.0220260 0.999757i \(-0.507012\pi\)
−0.0220260 + 0.999757i \(0.507012\pi\)
\(114\) 0 0
\(115\) 1101.02 + 1907.02i 0.892789 + 1.54636i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −641.309 + 1110.78i −0.481825 + 0.834545i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1183.53 −0.846866
\(126\) 0 0
\(127\) 443.700 0.310016 0.155008 0.987913i \(-0.450460\pi\)
0.155008 + 0.987913i \(0.450460\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1076.74 1864.97i 0.718133 1.24384i −0.243606 0.969874i \(-0.578330\pi\)
0.961739 0.273968i \(-0.0883363\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 131.554 + 227.858i 0.0820394 + 0.142096i 0.904126 0.427266i \(-0.140523\pi\)
−0.822086 + 0.569363i \(0.807190\pi\)
\(138\) 0 0
\(139\) −1165.77 −0.711360 −0.355680 0.934608i \(-0.615751\pi\)
−0.355680 + 0.934608i \(0.615751\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 951.693 + 1648.38i 0.556536 + 0.963948i
\(144\) 0 0
\(145\) 380.180 658.490i 0.217739 0.377135i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −882.597 + 1528.70i −0.485270 + 0.840512i −0.999857 0.0169263i \(-0.994612\pi\)
0.514587 + 0.857438i \(0.327945\pi\)
\(150\) 0 0
\(151\) −1346.16 2331.61i −0.725488 1.25658i −0.958773 0.284173i \(-0.908281\pi\)
0.233285 0.972408i \(-0.425052\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3981.85 −2.06342
\(156\) 0 0
\(157\) −970.691 1681.29i −0.493437 0.854657i 0.506535 0.862220i \(-0.330926\pi\)
−0.999971 + 0.00756226i \(0.997593\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1051.42 + 1821.11i −0.505236 + 0.875094i 0.494746 + 0.869038i \(0.335261\pi\)
−0.999982 + 0.00605658i \(0.998072\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2344.22 −1.08623 −0.543116 0.839658i \(-0.682756\pi\)
−0.543116 + 0.839658i \(0.682756\pi\)
\(168\) 0 0
\(169\) −810.844 −0.369069
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1735.32 + 3005.67i −0.762626 + 1.32091i 0.178867 + 0.983873i \(0.442757\pi\)
−0.941493 + 0.337033i \(0.890576\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −477.885 827.722i −0.199547 0.345625i 0.748835 0.662757i \(-0.230614\pi\)
−0.948381 + 0.317132i \(0.897280\pi\)
\(180\) 0 0
\(181\) −4220.26 −1.73309 −0.866546 0.499098i \(-0.833665\pi\)
−0.866546 + 0.499098i \(0.833665\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1960.78 + 3396.17i 0.779239 + 1.34968i
\(186\) 0 0
\(187\) −567.858 + 983.558i −0.222063 + 0.384625i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1759.02 + 3046.71i −0.666377 + 1.15420i 0.312534 + 0.949907i \(0.398822\pi\)
−0.978910 + 0.204291i \(0.934511\pi\)
\(192\) 0 0
\(193\) 2508.84 + 4345.43i 0.935699 + 1.62068i 0.773382 + 0.633940i \(0.218563\pi\)
0.162317 + 0.986739i \(0.448103\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2838.14 1.02644 0.513221 0.858257i \(-0.328452\pi\)
0.513221 + 0.858257i \(0.328452\pi\)
\(198\) 0 0
\(199\) 177.451 + 307.354i 0.0632118 + 0.109486i 0.895899 0.444257i \(-0.146532\pi\)
−0.832688 + 0.553743i \(0.813199\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1282.16 2220.77i 0.436829 0.756610i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2777.99 −0.919413
\(210\) 0 0
\(211\) 752.672 0.245574 0.122787 0.992433i \(-0.460817\pi\)
0.122787 + 0.992433i \(0.460817\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2113.75 + 3661.12i −0.670496 + 1.16133i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 413.547 + 716.284i 0.125874 + 0.218020i
\(222\) 0 0
\(223\) 3077.75 0.924221 0.462111 0.886822i \(-0.347092\pi\)
0.462111 + 0.886822i \(0.347092\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3108.72 + 5384.46i 0.908955 + 1.57436i 0.815518 + 0.578731i \(0.196452\pi\)
0.0934368 + 0.995625i \(0.470215\pi\)
\(228\) 0 0
\(229\) 251.627 435.831i 0.0726113 0.125766i −0.827434 0.561563i \(-0.810200\pi\)
0.900045 + 0.435797i \(0.143533\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −134.170 + 232.389i −0.0377243 + 0.0653404i −0.884271 0.466974i \(-0.845344\pi\)
0.846547 + 0.532314i \(0.178678\pi\)
\(234\) 0 0
\(235\) −884.080 1531.27i −0.245409 0.425060i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5189.77 1.40459 0.702297 0.711884i \(-0.252158\pi\)
0.702297 + 0.711884i \(0.252158\pi\)
\(240\) 0 0
\(241\) 3085.47 + 5344.19i 0.824699 + 1.42842i 0.902149 + 0.431425i \(0.141989\pi\)
−0.0774495 + 0.996996i \(0.524678\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1011.54 + 1752.05i −0.260579 + 0.451336i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1891.91 0.475763 0.237882 0.971294i \(-0.423547\pi\)
0.237882 + 0.971294i \(0.423547\pi\)
\(252\) 0 0
\(253\) −9045.01 −2.24765
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3269.97 5663.75i 0.793676 1.37469i −0.130000 0.991514i \(-0.541498\pi\)
0.923676 0.383174i \(-0.125169\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2687.95 + 4655.67i 0.630214 + 1.09156i 0.987508 + 0.157570i \(0.0503661\pi\)
−0.357294 + 0.933992i \(0.616301\pi\)
\(264\) 0 0
\(265\) −3858.53 −0.894443
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1619.44 + 2804.95i 0.367060 + 0.635766i 0.989104 0.147216i \(-0.0470311\pi\)
−0.622045 + 0.782982i \(0.713698\pi\)
\(270\) 0 0
\(271\) 678.729 1175.59i 0.152140 0.263514i −0.779874 0.625936i \(-0.784717\pi\)
0.932014 + 0.362423i \(0.118050\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −764.513 + 1324.18i −0.167643 + 0.290366i
\(276\) 0 0
\(277\) 1280.82 + 2218.44i 0.277823 + 0.481203i 0.970843 0.239715i \(-0.0770539\pi\)
−0.693021 + 0.720918i \(0.743721\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1786.17 −0.379196 −0.189598 0.981862i \(-0.560718\pi\)
−0.189598 + 0.981862i \(0.560718\pi\)
\(282\) 0 0
\(283\) −3694.14 6398.44i −0.775950 1.34398i −0.934259 0.356595i \(-0.883937\pi\)
0.158309 0.987390i \(-0.449396\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2209.74 3827.39i 0.449775 0.779033i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 492.981 0.0982945 0.0491472 0.998792i \(-0.484350\pi\)
0.0491472 + 0.998792i \(0.484350\pi\)
\(294\) 0 0
\(295\) −3505.48 −0.691853
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3293.55 + 5704.59i −0.637026 + 1.10336i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3384.43 5862.01i −0.635384 1.10052i
\(306\) 0 0
\(307\) 988.810 0.183825 0.0919126 0.995767i \(-0.470702\pi\)
0.0919126 + 0.995767i \(0.470702\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4798.28 + 8310.87i 0.874874 + 1.51533i 0.856897 + 0.515487i \(0.172389\pi\)
0.0179763 + 0.999838i \(0.494278\pi\)
\(312\) 0 0
\(313\) 482.856 836.332i 0.0871970 0.151030i −0.819128 0.573610i \(-0.805542\pi\)
0.906325 + 0.422581i \(0.138876\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4492.99 + 7782.08i −0.796061 + 1.37882i 0.126103 + 0.992017i \(0.459753\pi\)
−0.922163 + 0.386801i \(0.873580\pi\)
\(318\) 0 0
\(319\) 1561.61 + 2704.79i 0.274086 + 0.474730i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1207.14 −0.207947
\(324\) 0 0
\(325\) 556.762 + 964.340i 0.0950265 + 0.164591i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1903.65 3297.22i 0.316115 0.547527i −0.663559 0.748124i \(-0.730955\pi\)
0.979674 + 0.200597i \(0.0642882\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5966.21 −0.973041
\(336\) 0 0
\(337\) −1649.82 −0.266681 −0.133340 0.991070i \(-0.542570\pi\)
−0.133340 + 0.991070i \(0.542570\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8177.84 14164.4i 1.29869 2.24940i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2855.00 + 4945.00i 0.441684 + 0.765019i 0.997815 0.0660760i \(-0.0210480\pi\)
−0.556131 + 0.831095i \(0.687715\pi\)
\(348\) 0 0
\(349\) 447.244 0.0685973 0.0342986 0.999412i \(-0.489080\pi\)
0.0342986 + 0.999412i \(0.489080\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5322.85 + 9219.45i 0.802569 + 1.39009i 0.917920 + 0.396765i \(0.129867\pi\)
−0.115352 + 0.993325i \(0.536799\pi\)
\(354\) 0 0
\(355\) 6880.48 11917.3i 1.02867 1.78171i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4548.73 + 7878.64i −0.668727 + 1.15827i 0.309533 + 0.950889i \(0.399827\pi\)
−0.978260 + 0.207381i \(0.933506\pi\)
\(360\) 0 0
\(361\) 1953.15 + 3382.96i 0.284758 + 0.493215i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2979.01 −0.427201
\(366\) 0 0
\(367\) 2643.91 + 4579.39i 0.376052 + 0.651341i 0.990484 0.137629i \(-0.0439480\pi\)
−0.614432 + 0.788970i \(0.710615\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2947.51 + 5105.23i −0.409159 + 0.708683i −0.994796 0.101890i \(-0.967511\pi\)
0.585637 + 0.810573i \(0.300844\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2274.51 0.310724
\(378\) 0 0
\(379\) 3842.41 0.520769 0.260384 0.965505i \(-0.416151\pi\)
0.260384 + 0.965505i \(0.416151\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −2506.87 + 4342.02i −0.334452 + 0.579287i −0.983379 0.181563i \(-0.941884\pi\)
0.648928 + 0.760850i \(0.275218\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5591.24 + 9684.31i 0.728758 + 1.26225i 0.957408 + 0.288738i \(0.0932355\pi\)
−0.228650 + 0.973509i \(0.573431\pi\)
\(390\) 0 0
\(391\) −3930.40 −0.508360
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7222.95 12510.5i −0.920066 1.59360i
\(396\) 0 0
\(397\) −1703.40 + 2950.37i −0.215343 + 0.372985i −0.953379 0.301777i \(-0.902420\pi\)
0.738036 + 0.674762i \(0.235754\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 41.5201 71.9149i 0.00517061 0.00895575i −0.863429 0.504471i \(-0.831687\pi\)
0.868599 + 0.495515i \(0.165021\pi\)
\(402\) 0 0
\(403\) −5955.57 10315.4i −0.736149 1.27505i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −16108.0 −1.96178
\(408\) 0 0
\(409\) 1228.10 + 2127.13i 0.148473 + 0.257164i 0.930663 0.365876i \(-0.119231\pi\)
−0.782190 + 0.623040i \(0.785897\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 18.2435 31.5987i 0.00215793 0.00373764i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3437.96 −0.400848 −0.200424 0.979709i \(-0.564232\pi\)
−0.200424 + 0.979709i \(0.564232\pi\)
\(420\) 0 0
\(421\) −5347.62 −0.619067 −0.309533 0.950889i \(-0.600173\pi\)
−0.309533 + 0.950889i \(0.600173\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −332.210 + 575.404i −0.0379166 + 0.0656734i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 425.821 + 737.544i 0.0475895 + 0.0824275i 0.888839 0.458220i \(-0.151513\pi\)
−0.841249 + 0.540647i \(0.818179\pi\)
\(432\) 0 0
\(433\) 3433.42 0.381061 0.190531 0.981681i \(-0.438979\pi\)
0.190531 + 0.981681i \(0.438979\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4806.92 8325.83i −0.526192 0.911392i
\(438\) 0 0
\(439\) −4869.20 + 8433.70i −0.529371 + 0.916898i 0.470042 + 0.882644i \(0.344239\pi\)
−0.999413 + 0.0342540i \(0.989094\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4967.48 8603.93i 0.532759 0.922765i −0.466509 0.884516i \(-0.654489\pi\)
0.999268 0.0382491i \(-0.0121780\pi\)
\(444\) 0 0
\(445\) 7957.75 + 13783.2i 0.847716 + 1.46829i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7557.33 0.794327 0.397163 0.917748i \(-0.369995\pi\)
0.397163 + 0.917748i \(0.369995\pi\)
\(450\) 0 0
\(451\) 5266.54 + 9121.92i 0.549871 + 0.952405i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7005.92 + 12134.6i −0.717118 + 1.24209i 0.245018 + 0.969518i \(0.421206\pi\)
−0.962137 + 0.272567i \(0.912127\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1669.61 −0.168680 −0.0843399 0.996437i \(-0.526878\pi\)
−0.0843399 + 0.996437i \(0.526878\pi\)
\(462\) 0 0
\(463\) 14785.4 1.48409 0.742046 0.670349i \(-0.233856\pi\)
0.742046 + 0.670349i \(0.233856\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2301.58 + 3986.46i −0.228061 + 0.395014i −0.957233 0.289317i \(-0.906572\pi\)
0.729172 + 0.684330i \(0.239905\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8682.35 15038.3i −0.844006 1.46186i
\(474\) 0 0
\(475\) −1625.18 −0.156987
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1738.68 3011.47i −0.165850 0.287261i 0.771107 0.636706i \(-0.219703\pi\)
−0.936957 + 0.349445i \(0.886370\pi\)
\(480\) 0 0
\(481\) −5865.39 + 10159.2i −0.556006 + 0.963030i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 491.996 852.163i 0.0460627 0.0797829i
\(486\) 0 0
\(487\) −2172.08 3762.16i −0.202108 0.350061i 0.747100 0.664712i \(-0.231446\pi\)
−0.949207 + 0.314651i \(0.898112\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −4982.89 −0.457993 −0.228997 0.973427i \(-0.573544\pi\)
−0.228997 + 0.973427i \(0.573544\pi\)
\(492\) 0 0
\(493\) 678.578 + 1175.33i 0.0619911 + 0.107372i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −7663.08 + 13272.8i −0.687468 + 1.19073i 0.285187 + 0.958472i \(0.407944\pi\)
−0.972654 + 0.232257i \(0.925389\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −1516.04 −0.134387 −0.0671936 0.997740i \(-0.521405\pi\)
−0.0671936 + 0.997740i \(0.521405\pi\)
\(504\) 0 0
\(505\) 17077.5 1.50483
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1326.82 + 2298.12i −0.115541 + 0.200122i −0.917996 0.396590i \(-0.870193\pi\)
0.802455 + 0.596713i \(0.203527\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1608.86 + 2786.64i 0.137660 + 0.238435i
\(516\) 0 0
\(517\) 7262.82 0.617830
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6566.06 + 11372.8i 0.552139 + 0.956333i 0.998120 + 0.0612905i \(0.0195216\pi\)
−0.445981 + 0.895042i \(0.647145\pi\)
\(522\) 0 0
\(523\) −1543.17 + 2672.85i −0.129021 + 0.223471i −0.923298 0.384085i \(-0.874517\pi\)
0.794276 + 0.607557i \(0.207850\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3553.58 6154.98i 0.293731 0.508757i
\(528\) 0 0
\(529\) −9567.63 16571.6i −0.786359 1.36201i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7670.80 0.623376
\(534\) 0 0
\(535\) −7855.86 13606.7i −0.634838 1.09957i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −463.047 + 802.022i −0.0367985 + 0.0637368i −0.883838 0.467793i \(-0.845049\pi\)
0.847040 + 0.531530i \(0.178383\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 3445.52 0.270807
\(546\) 0 0
\(547\) 592.871 0.0463425 0.0231712 0.999732i \(-0.492624\pi\)
0.0231712 + 0.999732i \(0.492624\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1659.82 + 2874.89i −0.128331 + 0.222276i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6122.67 10604.8i −0.465756 0.806713i 0.533480 0.845813i \(-0.320884\pi\)
−0.999235 + 0.0391003i \(0.987551\pi\)
\(558\) 0 0
\(559\) −12646.0 −0.956829
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7297.31 12639.3i −0.546261 0.946152i −0.998526 0.0542682i \(-0.982717\pi\)
0.452266 0.891883i \(-0.350616\pi\)
\(564\) 0 0
\(565\) 329.300 570.364i 0.0245199 0.0424697i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11455.6 19841.7i 0.844015 1.46188i −0.0424590 0.999098i \(-0.513519\pi\)
0.886474 0.462779i \(-0.153147\pi\)
\(570\) 0 0
\(571\) −2952.32 5113.57i −0.216376 0.374774i 0.737321 0.675542i \(-0.236090\pi\)
−0.953697 + 0.300768i \(0.902757\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5291.53 −0.383778
\(576\) 0 0
\(577\) 4756.61 + 8238.68i 0.343189 + 0.594421i 0.985023 0.172423i \(-0.0551595\pi\)
−0.641834 + 0.766844i \(0.721826\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 7924.56 13725.7i 0.562953 0.975064i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −22790.6 −1.60250 −0.801252 0.598327i \(-0.795833\pi\)
−0.801252 + 0.598327i \(0.795833\pi\)
\(588\) 0 0
\(589\) 17384.3 1.21614
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 9131.39 15816.0i 0.632346 1.09526i −0.354724 0.934971i \(-0.615425\pi\)
0.987071 0.160285i \(-0.0512414\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3479.05 + 6025.88i 0.237312 + 0.411037i 0.959942 0.280198i \(-0.0904003\pi\)
−0.722630 + 0.691235i \(0.757067\pi\)
\(600\) 0 0
\(601\) −2305.39 −0.156471 −0.0782353 0.996935i \(-0.524929\pi\)
−0.0782353 + 0.996935i \(0.524929\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7981.87 13825.0i −0.536379 0.929036i
\(606\) 0 0
\(607\) −8089.62 + 14011.6i −0.540935 + 0.936927i 0.457916 + 0.888996i \(0.348596\pi\)
−0.998851 + 0.0479312i \(0.984737\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2644.60 4580.58i 0.175105 0.303291i
\(612\) 0 0
\(613\) −10270.4 17788.8i −0.676699 1.17208i −0.975969 0.217908i \(-0.930077\pi\)
0.299271 0.954168i \(-0.403257\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6918.19 −0.451403 −0.225702 0.974196i \(-0.572467\pi\)
−0.225702 + 0.974196i \(0.572467\pi\)
\(618\) 0 0
\(619\) −4040.81 6998.89i −0.262381 0.454457i 0.704493 0.709711i \(-0.251174\pi\)
−0.966874 + 0.255254i \(0.917841\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9234.52 15994.7i 0.591009 1.02366i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −6999.54 −0.443704
\(630\) 0 0
\(631\) −27293.3 −1.72191 −0.860957 0.508677i \(-0.830135\pi\)
−0.860957 + 0.508677i \(0.830135\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2761.20 + 4782.53i −0.172559 + 0.298880i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9627.82 + 16675.9i 0.593254 + 1.02755i 0.993791 + 0.111266i \(0.0354905\pi\)
−0.400536 + 0.916281i \(0.631176\pi\)
\(642\) 0 0
\(643\) 19996.4 1.22641 0.613204 0.789925i \(-0.289880\pi\)
0.613204 + 0.789925i \(0.289880\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −3532.10 6117.77i −0.214623 0.371738i 0.738533 0.674218i \(-0.235519\pi\)
−0.953156 + 0.302479i \(0.902186\pi\)
\(648\) 0 0
\(649\) 7199.47 12469.8i 0.435445 0.754213i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6910.53 + 11969.4i −0.414134 + 0.717302i −0.995337 0.0964570i \(-0.969249\pi\)
0.581203 + 0.813759i \(0.302582\pi\)
\(654\) 0 0
\(655\) 13401.4 + 23211.9i 0.799443 + 1.38468i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7802.80 0.461235 0.230617 0.973044i \(-0.425925\pi\)
0.230617 + 0.973044i \(0.425925\pi\)
\(660\) 0 0
\(661\) 7908.65 + 13698.2i 0.465372 + 0.806048i 0.999218 0.0395338i \(-0.0125873\pi\)
−0.533846 + 0.845581i \(0.679254\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5404.29 + 9360.51i −0.313726 + 0.543389i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 27803.5 1.59962
\(672\) 0 0
\(673\) 2943.30 0.168582 0.0842911 0.996441i \(-0.473137\pi\)
0.0842911 + 0.996441i \(0.473137\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1585.74 2746.58i 0.0900220 0.155923i −0.817498 0.575931i \(-0.804640\pi\)
0.907520 + 0.420009i \(0.137973\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −12226.6 21177.1i −0.684975 1.18641i −0.973445 0.228923i \(-0.926480\pi\)
0.288470 0.957489i \(-0.406854\pi\)
\(684\) 0 0
\(685\) −3274.70 −0.182656
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5771.12 9995.87i −0.319103 0.552703i
\(690\) 0 0
\(691\) 4297.95 7444.26i 0.236616 0.409831i −0.723125 0.690717i \(-0.757295\pi\)
0.959741 + 0.280886i \(0.0906284\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7254.69 12565.5i 0.395951 0.685808i
\(696\) 0 0
\(697\) 2288.51 + 3963.82i 0.124367 + 0.215409i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 21476.1 1.15712 0.578561 0.815639i \(-0.303615\pi\)
0.578561 + 0.815639i \(0.303615\pi\)
\(702\) 0 0
\(703\) −8560.51 14827.2i −0.459269 0.795477i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6769.46 11725.0i 0.358579 0.621077i −0.629145 0.777288i \(-0.716595\pi\)
0.987724 + 0.156211i \(0.0499281\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 56602.5 2.97304
\(714\) 0 0
\(715\) −23690.0 −1.23910
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6941.84 + 12023.6i −0.360065 + 0.623652i −0.987971 0.154638i \(-0.950579\pi\)
0.627906 + 0.778289i \(0.283912\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 913.577 + 1582.36i 0.0467992 + 0.0810585i
\(726\) 0 0
\(727\) 18292.9 0.933215 0.466607 0.884465i \(-0.345476\pi\)
0.466607 + 0.884465i \(0.345476\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3772.81 6534.69i −0.190892 0.330635i
\(732\) 0 0
\(733\) 7122.92 12337.3i 0.358924 0.621674i −0.628857 0.777521i \(-0.716477\pi\)
0.987781 + 0.155846i \(0.0498104\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12253.3 21223.3i 0.612422 1.06075i
\(738\) 0 0
\(739\) 681.947 + 1181.17i 0.0339456 + 0.0587956i 0.882499 0.470314i \(-0.155859\pi\)
−0.848553 + 0.529110i \(0.822526\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −21789.4 −1.07588 −0.537938 0.842984i \(-0.680797\pi\)
−0.537938 + 0.842984i \(0.680797\pi\)
\(744\) 0 0
\(745\) −10985.0 19026.6i −0.540214 0.935678i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1059.78 1835.59i 0.0514937 0.0891898i −0.839130 0.543932i \(-0.816935\pi\)
0.890623 + 0.454742i \(0.150268\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 33509.1 1.61526
\(756\) 0 0
\(757\) 28202.4 1.35408 0.677038 0.735948i \(-0.263263\pi\)
0.677038 + 0.735948i \(0.263263\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −5573.52 + 9653.62i −0.265493 + 0.459847i −0.967693 0.252133i \(-0.918868\pi\)
0.702200 + 0.711980i \(0.252201\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5243.07 9081.26i −0.246827 0.427517i
\(768\) 0 0
\(769\) 4109.29 0.192698 0.0963491 0.995348i \(-0.469283\pi\)
0.0963491 + 0.995348i \(0.469283\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6945.68 12030.3i −0.323181 0.559766i 0.657962 0.753052i \(-0.271419\pi\)
−0.981143 + 0.193286i \(0.938086\pi\)
\(774\) 0 0
\(775\) 4784.22 8286.51i 0.221747 0.384078i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5597.75 + 9695.59i −0.257459 + 0.445931i
\(780\) 0 0
\(781\) 28262.0 + 48951.2i 1.29487 + 2.24278i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 24162.9 1.09861
\(786\) 0 0
\(787\) −4671.18 8090.71i −0.211575 0.366458i 0.740633 0.671910i \(-0.234526\pi\)
−0.952208 + 0.305452i \(0.901193\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 10124.0 17535.4i 0.453361 0.785245i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24324.3 −1.08107 −0.540534 0.841322i \(-0.681778\pi\)
−0.540534 + 0.841322i \(0.681778\pi\)
\(798\) 0 0
\(799\) 3155.97 0.139737
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6118.22 10597.1i 0.268876 0.465706i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17227.5 + 29838.8i 0.748684 + 1.29676i 0.948454 + 0.316915i \(0.102647\pi\)
−0.199770 + 0.979843i \(0.564020\pi\)
\(810\) 0 0
\(811\) 8350.13 0.361545 0.180772 0.983525i \(-0.442140\pi\)
0.180772 + 0.983525i \(0.442140\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −13086.2 22665.9i −0.562441 0.974176i
\(816\) 0 0
\(817\) 9228.37 15984.0i 0.395177 0.684467i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −9417.38 + 16311.4i −0.400327 + 0.693387i −0.993765 0.111492i \(-0.964437\pi\)
0.593438 + 0.804880i \(0.297770\pi\)
\(822\) 0 0
\(823\) −4828.38 8363.00i −0.204504 0.354211i 0.745471 0.666538i \(-0.232225\pi\)
−0.949975 + 0.312327i \(0.898891\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20759.6 0.872892 0.436446 0.899731i \(-0.356237\pi\)
0.436446 + 0.899731i \(0.356237\pi\)
\(828\) 0 0
\(829\) −7808.05 13523.9i −0.327123 0.566593i 0.654817 0.755787i \(-0.272746\pi\)
−0.981940 + 0.189194i \(0.939412\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 14588.3 25267.7i 0.604610 1.04722i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −417.027 −0.0171601 −0.00858007 0.999963i \(-0.502731\pi\)
−0.00858007 + 0.999963i \(0.502731\pi\)
\(840\) 0 0
\(841\) −20656.8 −0.846973
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5045.98 8739.89i 0.205428 0.355812i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −27872.7 48276.9i −1.12275 1.94467i
\(852\) 0 0
\(853\) 24917.4 1.00018 0.500092 0.865972i \(-0.333300\pi\)
0.500092 + 0.865972i \(0.333300\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22261.9 + 38558.7i 0.887342 + 1.53692i 0.843006 + 0.537904i \(0.180784\pi\)
0.0443361 + 0.999017i \(0.485883\pi\)
\(858\) 0 0
\(859\) −12073.2 + 20911.4i −0.479548 + 0.830602i −0.999725 0.0234566i \(-0.992533\pi\)
0.520176 + 0.854059i \(0.325866\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −13221.3 + 22900.0i −0.521505 + 0.903273i 0.478182 + 0.878261i \(0.341296\pi\)
−0.999687 + 0.0250123i \(0.992037\pi\)
\(864\) 0 0
\(865\) −21598.2 37409.2i −0.848973 1.47046i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 59337.4 2.31632
\(870\) 0 0
\(871\) −8923.54 15456.0i −0.347144 0.601271i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11258.1 19499.5i 0.433475 0.750801i −0.563695 0.825983i \(-0.690621\pi\)
0.997170 + 0.0751826i \(0.0239540\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10120.6 0.387027 0.193514 0.981098i \(-0.438012\pi\)
0.193514 + 0.981098i \(0.438012\pi\)
\(882\) 0 0
\(883\) −20748.5 −0.790761 −0.395380 0.918517i \(-0.629387\pi\)
−0.395380 + 0.918517i \(0.629387\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −12749.7 + 22083.1i −0.482630 + 0.835939i −0.999801 0.0199428i \(-0.993652\pi\)
0.517172 + 0.855882i \(0.326985\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3859.78 + 6685.34i 0.144639 + 0.250522i
\(894\) 0 0
\(895\) 11895.7 0.444280
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −9772.34 16926.2i −0.362543 0.627942i
\(900\) 0 0
\(901\) 3443.52 5964.35i 0.127326 0.220534i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26263.1 45489.1i 0.964659 1.67084i
\(906\) 0 0
\(907\) −18903.5 32741.8i −0.692040 1.19865i −0.971168 0.238395i \(-0.923379\pi\)
0.279128 0.960254i \(-0.409954\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3230.08 0.117472 0.0587362 0.998274i \(-0.481293\pi\)
0.0587362 + 0.998274i \(0.481293\pi\)
\(912\) 0 0
\(913\) 74.9364 + 129.794i 0.00271635 + 0.00470486i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 17671.6 30608.0i 0.634310 1.09866i −0.352351 0.935868i \(-0.614618\pi\)
0.986661 0.162789i \(-0.0520490\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 41164.0 1.46796
\(924\) 0 0
\(925\) −9423.55 −0.334967
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −16515.1 + 28605.0i −0.583254 + 1.01023i 0.411836 + 0.911258i \(0.364888\pi\)
−0.995091 + 0.0989684i \(0.968446\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7067.68 12241.6i −0.247206 0.428174i
\(936\) 0 0
\(937\) −54695.9 −1.90698 −0.953488 0.301430i \(-0.902536\pi\)
−0.953488 + 0.301430i \(0.902536\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9127.64 + 15809.5i 0.316209 + 0.547690i 0.979694 0.200500i \(-0.0642566\pi\)
−0.663485 + 0.748190i \(0.730923\pi\)
\(942\) 0 0
\(943\) −18226.0 + 31568.4i −0.629397 + 1.09015i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3107.88 5383.00i 0.106645 0.184714i −0.807764 0.589506i \(-0.799323\pi\)
0.914409 + 0.404792i \(0.132656\pi\)
\(948\) 0 0
\(949\) −4455.64 7717.39i −0.152409 0.263980i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −8594.53 −0.292135 −0.146067 0.989275i \(-0.546662\pi\)
−0.146067 + 0.989275i \(0.546662\pi\)
\(954\) 0 0
\(955\) −21893.1 37920.0i −0.741826 1.28488i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −36280.3 + 62839.4i −1.21783 + 2.10934i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −62451.1 −2.08329
\(966\) 0 0
\(967\) −17168.1 −0.570929 −0.285464 0.958389i \(-0.592148\pi\)
−0.285464 + 0.958389i \(0.592148\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −5963.11 + 10328.4i −0.197081 + 0.341354i −0.947581 0.319517i \(-0.896479\pi\)
0.750500 + 0.660871i \(0.229813\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7019.12 + 12157.5i 0.229848 + 0.398108i 0.957763 0.287559i \(-0.0928438\pi\)
−0.727915 + 0.685667i \(0.759510\pi\)
\(978\) 0 0
\(979\) −65373.8 −2.13417
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −10175.7 17624.8i −0.330167 0.571866i 0.652378 0.757894i \(-0.273772\pi\)
−0.982544 + 0.186028i \(0.940438\pi\)
\(984\) 0 0
\(985\) −17662.0 + 30591.6i −0.571329 + 0.989571i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 30047.2 52043.3i 0.966072 1.67329i
\(990\) 0 0
\(991\) 16252.9 + 28150.8i 0.520978 + 0.902360i 0.999702 + 0.0243951i \(0.00776598\pi\)
−0.478724 + 0.877965i \(0.658901\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4417.18 −0.140738
\(996\) 0 0
\(997\) −12423.5 21518.2i −0.394641 0.683539i 0.598414 0.801187i \(-0.295798\pi\)
−0.993055 + 0.117648i \(0.962464\pi\)
\(998\) 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 1764.4.k.q.1549.1 4
3.2 odd 2 588.4.i.j.373.2 4
7.2 even 3 1764.4.a.y.1.2 2
7.3 odd 6 252.4.k.f.109.2 4
7.4 even 3 inner 1764.4.k.q.361.1 4
7.5 odd 6 1764.4.a.o.1.1 2
7.6 odd 2 252.4.k.f.37.2 4
21.2 odd 6 588.4.a.f.1.1 2
21.5 even 6 588.4.a.i.1.2 2
21.11 odd 6 588.4.i.j.361.2 4
21.17 even 6 84.4.i.a.25.1 4
21.20 even 2 84.4.i.a.37.1 yes 4
84.23 even 6 2352.4.a.bx.1.1 2
84.47 odd 6 2352.4.a.bt.1.2 2
84.59 odd 6 336.4.q.i.193.1 4
84.83 odd 2 336.4.q.i.289.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.i.a.25.1 4 21.17 even 6
84.4.i.a.37.1 yes 4 21.20 even 2
252.4.k.f.37.2 4 7.6 odd 2
252.4.k.f.109.2 4 7.3 odd 6
336.4.q.i.193.1 4 84.59 odd 6
336.4.q.i.289.1 4 84.83 odd 2
588.4.a.f.1.1 2 21.2 odd 6
588.4.a.i.1.2 2 21.5 even 6
588.4.i.j.361.2 4 21.11 odd 6
588.4.i.j.373.2 4 3.2 odd 2
1764.4.a.o.1.1 2 7.5 odd 6
1764.4.a.y.1.2 2 7.2 even 3
1764.4.k.q.361.1 4 7.4 even 3 inner
1764.4.k.q.1549.1 4 1.1 even 1 trivial
2352.4.a.bt.1.2 2 84.47 odd 6
2352.4.a.bx.1.1 2 84.23 even 6