Properties

Label 1764.4.t.a.1097.8
Level $1764$
Weight $4$
Character 1764.1097
Analytic conductor $104.079$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,4,Mod(521,1764)]
 
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, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.521");
 
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.t (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: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 42x^{12} + 1139x^{8} + 26250x^{4} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{16} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1097.8
Root \(-1.75687 + 1.38326i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1097
Dual form 1764.4.t.a.521.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(7.66648 - 13.2787i) q^{5} +O(q^{10})\) \(q+(7.66648 - 13.2787i) q^{5} +(17.6210 - 10.1735i) q^{11} -30.5385i q^{13} +(-1.72504 - 2.98786i) q^{17} +(-127.619 - 73.6807i) q^{19} +(-110.150 - 63.5954i) q^{23} +(-55.0500 - 95.3494i) q^{25} +216.198i q^{29} +(89.6694 - 51.7707i) q^{31} +(-118.325 + 204.945i) q^{37} -480.305 q^{41} +147.875 q^{43} +(-227.503 + 394.047i) q^{47} +(173.883 - 100.391i) q^{53} -311.980i q^{55} +(80.1149 + 138.763i) q^{59} +(-712.818 - 411.546i) q^{61} +(-405.513 - 234.123i) q^{65} +(203.550 + 352.559i) q^{67} -513.872i q^{71} +(410.478 - 236.989i) q^{73} +(60.6499 - 105.049i) q^{79} +536.269 q^{83} -52.9000 q^{85} +(115.956 - 200.842i) q^{89} +(-1956.77 + 1129.74i) q^{95} -734.188i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{25} - 512 q^{37} + 64 q^{43} + 2336 q^{67} - 1792 q^{79} - 2688 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(5\) 7.66648 13.2787i 0.685711 1.18769i −0.287501 0.957780i \(-0.592825\pi\)
0.973213 0.229907i \(-0.0738420\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 17.6210 10.1735i 0.482994 0.278857i −0.238669 0.971101i \(-0.576711\pi\)
0.721663 + 0.692244i \(0.243378\pi\)
\(12\) 0 0
\(13\) 30.5385i 0.651528i −0.945451 0.325764i \(-0.894379\pi\)
0.945451 0.325764i \(-0.105621\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.72504 2.98786i −0.0246108 0.0426272i 0.853458 0.521162i \(-0.174501\pi\)
−0.878069 + 0.478535i \(0.841168\pi\)
\(18\) 0 0
\(19\) −127.619 73.6807i −1.54093 0.889659i −0.998780 0.0493798i \(-0.984276\pi\)
−0.542154 0.840279i \(-0.682391\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −110.150 63.5954i −0.998607 0.576546i −0.0907711 0.995872i \(-0.528933\pi\)
−0.907836 + 0.419326i \(0.862266\pi\)
\(24\) 0 0
\(25\) −55.0500 95.3494i −0.440400 0.762795i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 216.198i 1.38438i 0.721717 + 0.692188i \(0.243353\pi\)
−0.721717 + 0.692188i \(0.756647\pi\)
\(30\) 0 0
\(31\) 89.6694 51.7707i 0.519519 0.299945i −0.217219 0.976123i \(-0.569698\pi\)
0.736738 + 0.676178i \(0.236365\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −118.325 + 204.945i −0.525743 + 0.910614i 0.473807 + 0.880629i \(0.342879\pi\)
−0.999550 + 0.0299855i \(0.990454\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −480.305 −1.82954 −0.914768 0.403979i \(-0.867627\pi\)
−0.914768 + 0.403979i \(0.867627\pi\)
\(42\) 0 0
\(43\) 147.875 0.524435 0.262218 0.965009i \(-0.415546\pi\)
0.262218 + 0.965009i \(0.415546\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −227.503 + 394.047i −0.706059 + 1.22293i 0.260250 + 0.965541i \(0.416195\pi\)
−0.966308 + 0.257388i \(0.917138\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 173.883 100.391i 0.450654 0.260185i −0.257452 0.966291i \(-0.582883\pi\)
0.708106 + 0.706106i \(0.249550\pi\)
\(54\) 0 0
\(55\) 311.980i 0.764861i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 80.1149 + 138.763i 0.176781 + 0.306194i 0.940776 0.339028i \(-0.110098\pi\)
−0.763995 + 0.645222i \(0.776765\pi\)
\(60\) 0 0
\(61\) −712.818 411.546i −1.49618 0.863820i −0.496190 0.868214i \(-0.665268\pi\)
−0.999990 + 0.00439331i \(0.998602\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −405.513 234.123i −0.773811 0.446760i
\(66\) 0 0
\(67\) 203.550 + 352.559i 0.371158 + 0.642865i 0.989744 0.142853i \(-0.0456275\pi\)
−0.618586 + 0.785717i \(0.712294\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 513.872i 0.858949i −0.903079 0.429475i \(-0.858699\pi\)
0.903079 0.429475i \(-0.141301\pi\)
\(72\) 0 0
\(73\) 410.478 236.989i 0.658120 0.379966i −0.133440 0.991057i \(-0.542602\pi\)
0.791560 + 0.611091i \(0.209269\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 60.6499 105.049i 0.0863753 0.149606i −0.819601 0.572935i \(-0.805805\pi\)
0.905976 + 0.423328i \(0.139138\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 536.269 0.709195 0.354597 0.935019i \(-0.384618\pi\)
0.354597 + 0.935019i \(0.384618\pi\)
\(84\) 0 0
\(85\) −52.9000 −0.0675037
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 115.956 200.842i 0.138105 0.239204i −0.788675 0.614811i \(-0.789232\pi\)
0.926779 + 0.375607i \(0.122566\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1956.77 + 1129.74i −2.11327 + 1.22010i
\(96\) 0 0
\(97\) 734.188i 0.768510i −0.923227 0.384255i \(-0.874458\pi\)
0.923227 0.384255i \(-0.125542\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −363.775 630.077i −0.358386 0.620742i 0.629306 0.777158i \(-0.283339\pi\)
−0.987691 + 0.156416i \(0.950006\pi\)
\(102\) 0 0
\(103\) 703.585 + 406.215i 0.673070 + 0.388597i 0.797239 0.603664i \(-0.206293\pi\)
−0.124169 + 0.992261i \(0.539626\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 766.093 + 442.304i 0.692159 + 0.399618i 0.804420 0.594061i \(-0.202476\pi\)
−0.112262 + 0.993679i \(0.535809\pi\)
\(108\) 0 0
\(109\) 800.525 + 1386.55i 0.703453 + 1.21842i 0.967247 + 0.253837i \(0.0816928\pi\)
−0.263794 + 0.964579i \(0.584974\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 740.553i 0.616508i −0.951304 0.308254i \(-0.900255\pi\)
0.951304 0.308254i \(-0.0997446\pi\)
\(114\) 0 0
\(115\) −1688.93 + 975.106i −1.36951 + 0.790688i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −458.500 + 794.145i −0.344478 + 0.596653i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 228.462 0.163474
\(126\) 0 0
\(127\) −2428.70 −1.69695 −0.848473 0.529238i \(-0.822478\pi\)
−0.848473 + 0.529238i \(0.822478\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −254.334 + 440.520i −0.169628 + 0.293805i −0.938289 0.345852i \(-0.887590\pi\)
0.768661 + 0.639656i \(0.220923\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1748.26 + 1009.36i −1.09025 + 0.629456i −0.933643 0.358206i \(-0.883389\pi\)
−0.156606 + 0.987661i \(0.550055\pi\)
\(138\) 0 0
\(139\) 3146.55i 1.92005i 0.279920 + 0.960023i \(0.409692\pi\)
−0.279920 + 0.960023i \(0.590308\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −310.683 538.119i −0.181683 0.314684i
\(144\) 0 0
\(145\) 2870.83 + 1657.48i 1.64421 + 0.949283i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2647.01 1528.25i −1.45538 0.840264i −0.456601 0.889672i \(-0.650933\pi\)
−0.998779 + 0.0494081i \(0.984267\pi\)
\(150\) 0 0
\(151\) 900.513 + 1559.73i 0.485316 + 0.840591i 0.999858 0.0168738i \(-0.00537134\pi\)
−0.514542 + 0.857465i \(0.672038\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1587.60i 0.822702i
\(156\) 0 0
\(157\) 1165.70 673.019i 0.592568 0.342119i −0.173544 0.984826i \(-0.555522\pi\)
0.766112 + 0.642707i \(0.222189\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 373.250 646.487i 0.179357 0.310655i −0.762304 0.647220i \(-0.775932\pi\)
0.941660 + 0.336564i \(0.109265\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2625.39 1.21652 0.608259 0.793738i \(-0.291868\pi\)
0.608259 + 0.793738i \(0.291868\pi\)
\(168\) 0 0
\(169\) 1264.40 0.575512
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1838.42 3184.24i 0.807934 1.39938i −0.106359 0.994328i \(-0.533919\pi\)
0.914293 0.405054i \(-0.132747\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3447.70 + 1990.53i −1.43963 + 0.831170i −0.997823 0.0659418i \(-0.978995\pi\)
−0.441804 + 0.897111i \(0.645661\pi\)
\(180\) 0 0
\(181\) 1230.75i 0.505421i −0.967542 0.252711i \(-0.918678\pi\)
0.967542 0.252711i \(-0.0813221\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1814.27 + 3142.41i 0.721016 + 1.24884i
\(186\) 0 0
\(187\) −60.7940 35.0994i −0.0237738 0.0137258i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2746.14 1585.48i −1.04033 0.600636i −0.120405 0.992725i \(-0.538419\pi\)
−0.919928 + 0.392088i \(0.871753\pi\)
\(192\) 0 0
\(193\) 120.100 + 208.020i 0.0447928 + 0.0775834i 0.887553 0.460706i \(-0.152404\pi\)
−0.842760 + 0.538290i \(0.819071\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2565.35i 0.927784i 0.885892 + 0.463892i \(0.153547\pi\)
−0.885892 + 0.463892i \(0.846453\pi\)
\(198\) 0 0
\(199\) −1129.94 + 652.372i −0.402510 + 0.232389i −0.687566 0.726122i \(-0.741321\pi\)
0.285057 + 0.958511i \(0.407988\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3682.25 + 6377.84i −1.25453 + 2.17292i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2998.36 −0.992349
\(210\) 0 0
\(211\) 1773.22 0.578549 0.289274 0.957246i \(-0.406586\pi\)
0.289274 + 0.957246i \(0.406586\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1133.68 1963.59i 0.359611 0.622865i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −91.2448 + 52.6802i −0.0277728 + 0.0160346i
\(222\) 0 0
\(223\) 3314.68i 0.995369i 0.867358 + 0.497684i \(0.165816\pi\)
−0.867358 + 0.497684i \(0.834184\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 658.933 + 1141.31i 0.192665 + 0.333705i 0.946132 0.323780i \(-0.104954\pi\)
−0.753468 + 0.657485i \(0.771620\pi\)
\(228\) 0 0
\(229\) −1826.72 1054.66i −0.527131 0.304339i 0.212716 0.977114i \(-0.431769\pi\)
−0.739848 + 0.672775i \(0.765102\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5715.21 3299.68i −1.60694 0.927764i −0.990051 0.140711i \(-0.955061\pi\)
−0.616884 0.787054i \(-0.711605\pi\)
\(234\) 0 0
\(235\) 3488.30 + 6041.91i 0.968305 + 1.67715i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 300.361i 0.0812919i −0.999174 0.0406459i \(-0.987058\pi\)
0.999174 0.0406459i \(-0.0129416\pi\)
\(240\) 0 0
\(241\) 3960.64 2286.68i 1.05862 0.611194i 0.133569 0.991040i \(-0.457356\pi\)
0.925050 + 0.379846i \(0.124023\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2250.10 + 3897.29i −0.579637 + 1.00396i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 165.982 0.0417399 0.0208699 0.999782i \(-0.493356\pi\)
0.0208699 + 0.999782i \(0.493356\pi\)
\(252\) 0 0
\(253\) −2587.95 −0.643095
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3308.66 + 5730.77i −0.803069 + 1.39096i 0.114518 + 0.993421i \(0.463468\pi\)
−0.917587 + 0.397535i \(0.869866\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3956.13 + 2284.07i −0.927548 + 0.535520i −0.886035 0.463618i \(-0.846551\pi\)
−0.0415128 + 0.999138i \(0.513218\pi\)
\(264\) 0 0
\(265\) 3078.60i 0.713648i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3839.76 6650.66i −0.870314 1.50743i −0.861673 0.507465i \(-0.830583\pi\)
−0.00864111 0.999963i \(-0.502751\pi\)
\(270\) 0 0
\(271\) −2481.49 1432.69i −0.556236 0.321143i 0.195397 0.980724i \(-0.437400\pi\)
−0.751633 + 0.659581i \(0.770734\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1940.07 1120.10i −0.425421 0.245617i
\(276\) 0 0
\(277\) −84.3492 146.097i −0.0182962 0.0316900i 0.856732 0.515761i \(-0.172491\pi\)
−0.875029 + 0.484071i \(0.839158\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5412.27i 1.14900i −0.818505 0.574499i \(-0.805197\pi\)
0.818505 0.574499i \(-0.194803\pi\)
\(282\) 0 0
\(283\) −2834.46 + 1636.48i −0.595375 + 0.343740i −0.767220 0.641384i \(-0.778361\pi\)
0.171845 + 0.985124i \(0.445027\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2450.55 4244.47i 0.498789 0.863927i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7800.65 −1.55535 −0.777677 0.628664i \(-0.783602\pi\)
−0.777677 + 0.628664i \(0.783602\pi\)
\(294\) 0 0
\(295\) 2456.80 0.484883
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1942.11 + 3363.83i −0.375636 + 0.650620i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −10929.6 + 6310.22i −2.05190 + 1.18466i
\(306\) 0 0
\(307\) 7079.58i 1.31613i −0.752960 0.658066i \(-0.771375\pi\)
0.752960 0.658066i \(-0.228625\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2905.22 + 5031.98i 0.529710 + 0.917484i 0.999399 + 0.0346525i \(0.0110324\pi\)
−0.469690 + 0.882832i \(0.655634\pi\)
\(312\) 0 0
\(313\) −4042.49 2333.93i −0.730016 0.421475i 0.0884122 0.996084i \(-0.471821\pi\)
−0.818428 + 0.574609i \(0.805154\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4416.52 2549.88i −0.782513 0.451784i 0.0548072 0.998497i \(-0.482546\pi\)
−0.837320 + 0.546713i \(0.815879\pi\)
\(318\) 0 0
\(319\) 2199.49 + 3809.62i 0.386043 + 0.668646i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 508.409i 0.0875810i
\(324\) 0 0
\(325\) −2911.83 + 1681.14i −0.496982 + 0.286933i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2036.04 + 3526.52i −0.338099 + 0.585604i −0.984075 0.177753i \(-0.943117\pi\)
0.645976 + 0.763357i \(0.276450\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6242.05 1.01803
\(336\) 0 0
\(337\) −8456.90 −1.36699 −0.683496 0.729954i \(-0.739542\pi\)
−0.683496 + 0.729954i \(0.739542\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1053.38 1824.50i 0.167283 0.289743i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6815.08 + 3934.69i −1.05433 + 0.608718i −0.923858 0.382734i \(-0.874982\pi\)
−0.130472 + 0.991452i \(0.541649\pi\)
\(348\) 0 0
\(349\) 3417.23i 0.524127i −0.965051 0.262063i \(-0.915597\pi\)
0.965051 0.262063i \(-0.0844030\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3998.64 6925.85i −0.602907 1.04427i −0.992379 0.123227i \(-0.960676\pi\)
0.389472 0.921039i \(-0.372658\pi\)
\(354\) 0 0
\(355\) −6823.57 3939.59i −1.02016 0.588991i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4233.62 + 2444.28i 0.622401 + 0.359343i 0.777803 0.628508i \(-0.216334\pi\)
−0.155402 + 0.987851i \(0.549667\pi\)
\(360\) 0 0
\(361\) 7428.20 + 12866.0i 1.08299 + 1.87579i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 7267.50i 1.04219i
\(366\) 0 0
\(367\) 9088.63 5247.32i 1.29271 0.746344i 0.313572 0.949564i \(-0.398474\pi\)
0.979133 + 0.203221i \(0.0651409\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2698.30 + 4673.59i −0.374565 + 0.648765i −0.990262 0.139218i \(-0.955541\pi\)
0.615697 + 0.787983i \(0.288875\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6602.36 0.901960
\(378\) 0 0
\(379\) −10159.5 −1.37693 −0.688466 0.725269i \(-0.741715\pi\)
−0.688466 + 0.725269i \(0.741715\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3503.40 + 6068.06i −0.467403 + 0.809565i −0.999306 0.0372396i \(-0.988144\pi\)
0.531904 + 0.846805i \(0.321477\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7529.03 + 4346.88i −0.981328 + 0.566570i −0.902671 0.430331i \(-0.858397\pi\)
−0.0786575 + 0.996902i \(0.525063\pi\)
\(390\) 0 0
\(391\) 438.819i 0.0567571i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −929.944 1610.71i −0.118457 0.205174i
\(396\) 0 0
\(397\) 6502.69 + 3754.33i 0.822067 + 0.474620i 0.851129 0.524957i \(-0.175919\pi\)
−0.0290620 + 0.999578i \(0.509252\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1927.81 + 1113.02i 0.240075 + 0.138607i 0.615211 0.788362i \(-0.289071\pi\)
−0.375136 + 0.926970i \(0.622404\pi\)
\(402\) 0 0
\(403\) −1581.00 2738.37i −0.195422 0.338481i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4815.11i 0.586428i
\(408\) 0 0
\(409\) −3966.88 + 2290.28i −0.479584 + 0.276888i −0.720243 0.693722i \(-0.755970\pi\)
0.240659 + 0.970610i \(0.422636\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4111.30 7120.98i 0.486303 0.842301i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −9723.41 −1.13370 −0.566849 0.823821i \(-0.691838\pi\)
−0.566849 + 0.823821i \(0.691838\pi\)
\(420\) 0 0
\(421\) 6827.70 0.790408 0.395204 0.918593i \(-0.370674\pi\)
0.395204 + 0.918593i \(0.370674\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −189.927 + 328.963i −0.0216772 + 0.0375460i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4019.87 2320.87i 0.449258 0.259379i −0.258259 0.966076i \(-0.583149\pi\)
0.707517 + 0.706696i \(0.249815\pi\)
\(432\) 0 0
\(433\) 3332.86i 0.369900i 0.982748 + 0.184950i \(0.0592124\pi\)
−0.982748 + 0.184950i \(0.940788\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9371.51 + 16231.9i 1.02586 + 1.77684i
\(438\) 0 0
\(439\) 13274.2 + 7663.84i 1.44315 + 0.833201i 0.998058 0.0622858i \(-0.0198390\pi\)
0.445088 + 0.895487i \(0.353172\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4878.42 2816.55i −0.523207 0.302073i 0.215039 0.976605i \(-0.431012\pi\)
−0.738246 + 0.674532i \(0.764345\pi\)
\(444\) 0 0
\(445\) −1777.95 3079.50i −0.189400 0.328050i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 17327.5i 1.82124i −0.413248 0.910619i \(-0.635606\pi\)
0.413248 0.910619i \(-0.364394\pi\)
\(450\) 0 0
\(451\) −8463.46 + 4886.38i −0.883655 + 0.510179i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2456.00 + 4253.92i −0.251394 + 0.435426i −0.963910 0.266229i \(-0.914222\pi\)
0.712516 + 0.701656i \(0.247555\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6760.33 0.682993 0.341497 0.939883i \(-0.389066\pi\)
0.341497 + 0.939883i \(0.389066\pi\)
\(462\) 0 0
\(463\) 16397.3 1.64589 0.822945 0.568121i \(-0.192330\pi\)
0.822945 + 0.568121i \(0.192330\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6290.15 10894.9i 0.623284 1.07956i −0.365587 0.930777i \(-0.619132\pi\)
0.988870 0.148781i \(-0.0475351\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2605.71 1504.41i 0.253299 0.146242i
\(474\) 0 0
\(475\) 16224.5i 1.56722i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6373.15 11038.6i −0.607927 1.05296i −0.991582 0.129483i \(-0.958668\pi\)
0.383655 0.923477i \(-0.374665\pi\)
\(480\) 0 0
\(481\) 6258.71 + 3613.47i 0.593290 + 0.342536i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9749.09 5628.64i −0.912749 0.526976i
\(486\) 0 0
\(487\) −786.712 1362.63i −0.0732019 0.126789i 0.827101 0.562053i \(-0.189988\pi\)
−0.900303 + 0.435264i \(0.856655\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17035.7i 1.56581i 0.622142 + 0.782904i \(0.286263\pi\)
−0.622142 + 0.782904i \(0.713737\pi\)
\(492\) 0 0
\(493\) 645.969 372.950i 0.0590121 0.0340707i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −20.6112 + 35.6996i −0.00184907 + 0.00320268i −0.866948 0.498398i \(-0.833922\pi\)
0.865099 + 0.501600i \(0.167255\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6871.06 −0.609076 −0.304538 0.952500i \(-0.598502\pi\)
−0.304538 + 0.952500i \(0.598502\pi\)
\(504\) 0 0
\(505\) −11155.5 −0.982996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1102.82 1910.14i 0.0960349 0.166337i −0.814005 0.580858i \(-0.802717\pi\)
0.910040 + 0.414520i \(0.136051\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10788.0 6228.48i 0.923064 0.532931i
\(516\) 0 0
\(517\) 9258.01i 0.787557i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3862.18 6689.50i −0.324770 0.562519i 0.656696 0.754156i \(-0.271954\pi\)
−0.981466 + 0.191637i \(0.938620\pi\)
\(522\) 0 0
\(523\) −1917.64 1107.15i −0.160330 0.0925667i 0.417688 0.908590i \(-0.362840\pi\)
−0.578018 + 0.816024i \(0.696174\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −309.367 178.613i −0.0255716 0.0147638i
\(528\) 0 0
\(529\) 2005.25 + 3473.19i 0.164810 + 0.285460i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 14667.8i 1.19199i
\(534\) 0 0
\(535\) 11746.5 6781.83i 0.949242 0.548045i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −6535.15 + 11319.2i −0.519349 + 0.899539i 0.480398 + 0.877051i \(0.340492\pi\)
−0.999747 + 0.0224886i \(0.992841\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24548.8 1.92946
\(546\) 0 0
\(547\) 19391.3 1.51574 0.757872 0.652403i \(-0.226239\pi\)
0.757872 + 0.652403i \(0.226239\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15929.6 27590.9i 1.23162 2.13323i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20477.8 11822.9i 1.55776 0.899374i 0.560291 0.828296i \(-0.310689\pi\)
0.997471 0.0710781i \(-0.0226439\pi\)
\(558\) 0 0
\(559\) 4515.88i 0.341684i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 11635.2 + 20152.8i 0.870988 + 1.50859i 0.860976 + 0.508645i \(0.169853\pi\)
0.0100112 + 0.999950i \(0.496813\pi\)
\(564\) 0 0
\(565\) −9833.61 5677.44i −0.732218 0.422746i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 11785.6 + 6804.42i 0.868326 + 0.501329i 0.866792 0.498670i \(-0.166178\pi\)
0.00153471 + 0.999999i \(0.499511\pi\)
\(570\) 0 0
\(571\) −10558.1 18287.2i −0.773809 1.34028i −0.935462 0.353428i \(-0.885016\pi\)
0.161653 0.986848i \(-0.448317\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14003.7i 1.01564i
\(576\) 0 0
\(577\) −8887.46 + 5131.18i −0.641230 + 0.370215i −0.785088 0.619384i \(-0.787383\pi\)
0.143858 + 0.989598i \(0.454049\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2042.66 3538.00i 0.145109 0.251336i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13227.4 −0.930072 −0.465036 0.885292i \(-0.653959\pi\)
−0.465036 + 0.885292i \(0.653959\pi\)
\(588\) 0 0
\(589\) −15258.0 −1.06739
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4129.36 7152.26i 0.285957 0.495292i −0.686884 0.726767i \(-0.741022\pi\)
0.972841 + 0.231475i \(0.0743552\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3549.97 + 2049.57i −0.242150 + 0.139805i −0.616164 0.787618i \(-0.711314\pi\)
0.374015 + 0.927423i \(0.377981\pi\)
\(600\) 0 0
\(601\) 480.297i 0.0325985i 0.999867 + 0.0162993i \(0.00518844\pi\)
−0.999867 + 0.0162993i \(0.994812\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7030.17 + 12176.6i 0.472425 + 0.818264i
\(606\) 0 0
\(607\) 10878.6 + 6280.76i 0.727428 + 0.419981i 0.817481 0.575956i \(-0.195370\pi\)
−0.0900524 + 0.995937i \(0.528703\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12033.6 + 6947.61i 0.796772 + 0.460017i
\(612\) 0 0
\(613\) −15062.7 26089.4i −0.992458 1.71899i −0.602392 0.798200i \(-0.705786\pi\)
−0.390065 0.920787i \(-0.627548\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13044.4i 0.851132i −0.904927 0.425566i \(-0.860075\pi\)
0.904927 0.425566i \(-0.139925\pi\)
\(618\) 0 0
\(619\) −7878.85 + 4548.85i −0.511595 + 0.295370i −0.733489 0.679701i \(-0.762110\pi\)
0.221894 + 0.975071i \(0.428776\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 8632.75 14952.4i 0.552496 0.956951i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 816.462 0.0517559
\(630\) 0 0
\(631\) 1769.90 0.111662 0.0558310 0.998440i \(-0.482219\pi\)
0.0558310 + 0.998440i \(0.482219\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −18619.6 + 32250.1i −1.16362 + 2.01544i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7935.19 4581.38i 0.488956 0.282299i −0.235185 0.971951i \(-0.575570\pi\)
0.724141 + 0.689652i \(0.242236\pi\)
\(642\) 0 0
\(643\) 24337.4i 1.49265i −0.665581 0.746326i \(-0.731816\pi\)
0.665581 0.746326i \(-0.268184\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −15143.6 26229.5i −0.920180 1.59380i −0.799135 0.601151i \(-0.794709\pi\)
−0.121044 0.992647i \(-0.538624\pi\)
\(648\) 0 0
\(649\) 2823.41 + 1630.10i 0.170768 + 0.0985931i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14499.5 8371.28i −0.868925 0.501674i −0.00193444 0.999998i \(-0.500616\pi\)
−0.866991 + 0.498324i \(0.833949\pi\)
\(654\) 0 0
\(655\) 3899.70 + 6754.48i 0.232632 + 0.402930i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9709.90i 0.573966i −0.957936 0.286983i \(-0.907348\pi\)
0.957936 0.286983i \(-0.0926524\pi\)
\(660\) 0 0
\(661\) 10470.1 6044.90i 0.616095 0.355703i −0.159252 0.987238i \(-0.550908\pi\)
0.775347 + 0.631535i \(0.217575\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13749.2 23814.3i 0.798157 1.38245i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −16747.4 −0.963529
\(672\) 0 0
\(673\) 1800.90 0.103150 0.0515748 0.998669i \(-0.483576\pi\)
0.0515748 + 0.998669i \(0.483576\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −14153.9 + 24515.2i −0.803512 + 1.39172i 0.113779 + 0.993506i \(0.463705\pi\)
−0.917291 + 0.398218i \(0.869629\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10251.0 5918.41i 0.574294 0.331569i −0.184569 0.982820i \(-0.559089\pi\)
0.758863 + 0.651251i \(0.225755\pi\)
\(684\) 0 0
\(685\) 30953.0i 1.72650i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −3065.80 5310.13i −0.169518 0.293614i
\(690\) 0 0
\(691\) −16366.4 9449.13i −0.901022 0.520205i −0.0234902 0.999724i \(-0.507478\pi\)
−0.877532 + 0.479519i \(0.840811\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 41782.2 + 24122.9i 2.28041 + 1.31660i
\(696\) 0 0
\(697\) 828.546 + 1435.08i 0.0450264 + 0.0779881i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3733.84i 0.201177i 0.994928 + 0.100589i \(0.0320726\pi\)
−0.994928 + 0.100589i \(0.967927\pi\)
\(702\) 0 0
\(703\) 30201.0 17436.5i 1.62027 0.935464i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1672.07 2896.12i 0.0885699 0.153408i −0.818337 0.574739i \(-0.805104\pi\)
0.906907 + 0.421331i \(0.138437\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −13169.5 −0.691728
\(714\) 0 0
\(715\) −9527.40 −0.498328
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −7627.74 + 13211.6i −0.395642 + 0.685273i −0.993183 0.116566i \(-0.962811\pi\)
0.597541 + 0.801839i \(0.296145\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 20614.3 11901.7i 1.05600 0.609679i
\(726\) 0 0
\(727\) 31743.3i 1.61939i −0.586854 0.809693i \(-0.699634\pi\)
0.586854 0.809693i \(-0.300366\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −255.091 441.830i −0.0129068 0.0223552i
\(732\) 0 0
\(733\) −9653.79 5573.62i −0.486454 0.280854i 0.236648 0.971595i \(-0.423951\pi\)
−0.723102 + 0.690741i \(0.757284\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7173.51 + 4141.63i 0.358534 + 0.207000i
\(738\) 0 0
\(739\) 1875.05 + 3247.68i 0.0933352 + 0.161661i 0.908913 0.416987i \(-0.136914\pi\)
−0.815577 + 0.578648i \(0.803581\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12014.8i 0.593245i 0.954995 + 0.296623i \(0.0958603\pi\)
−0.954995 + 0.296623i \(0.904140\pi\)
\(744\) 0 0
\(745\) −40586.5 + 23432.6i −1.99594 + 1.15236i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 11928.6 20661.0i 0.579603 1.00390i −0.415921 0.909401i \(-0.636541\pi\)
0.995525 0.0945019i \(-0.0301259\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 27615.1 1.33115
\(756\) 0 0
\(757\) 18603.5 0.893206 0.446603 0.894732i \(-0.352634\pi\)
0.446603 + 0.894732i \(0.352634\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1982.35 + 3433.53i −0.0944286 + 0.163555i −0.909370 0.415988i \(-0.863436\pi\)
0.814941 + 0.579543i \(0.196769\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4237.62 2446.59i 0.199494 0.115178i
\(768\) 0 0
\(769\) 20811.6i 0.975925i −0.872865 0.487962i \(-0.837740\pi\)
0.872865 0.487962i \(-0.162260\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2025.85 + 3508.87i 0.0942621 + 0.163267i 0.909300 0.416140i \(-0.136617\pi\)
−0.815038 + 0.579407i \(0.803284\pi\)
\(774\) 0 0
\(775\) −9872.60 5699.95i −0.457592 0.264191i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 61295.9 + 35389.2i 2.81920 + 1.62766i
\(780\) 0 0
\(781\) −5227.87 9054.94i −0.239524 0.414867i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 20638.7i 0.938380i
\(786\) 0 0
\(787\) −6765.77 + 3906.22i −0.306447 + 0.176927i −0.645335 0.763899i \(-0.723282\pi\)
0.338889 + 0.940826i \(0.389949\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −12568.0 + 21768.4i −0.562803 + 0.974803i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −20518.5 −0.911925 −0.455962 0.889999i \(-0.650705\pi\)
−0.455962 + 0.889999i \(0.650705\pi\)
\(798\) 0 0
\(799\) 1569.81 0.0695068
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4822.02 8351.98i 0.211912 0.367042i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5160.47 2979.40i 0.224267 0.129481i −0.383657 0.923476i \(-0.625336\pi\)
0.607925 + 0.793995i \(0.292002\pi\)
\(810\) 0 0
\(811\) 35305.0i 1.52864i −0.644838 0.764319i \(-0.723075\pi\)
0.644838 0.764319i \(-0.276925\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5723.03 9912.57i −0.245974 0.426040i
\(816\) 0 0
\(817\) −18871.6 10895.5i −0.808120 0.466568i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23053.5 13310.0i −0.979992 0.565799i −0.0777242 0.996975i \(-0.524765\pi\)
−0.902268 + 0.431176i \(0.858099\pi\)
\(822\) 0 0
\(823\) −13046.9 22598.0i −0.552598 0.957128i −0.998086 0.0618398i \(-0.980303\pi\)
0.445488 0.895288i \(-0.353030\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 264.904i 0.0111386i 0.999984 + 0.00556930i \(0.00177277\pi\)
−0.999984 + 0.00556930i \(0.998227\pi\)
\(828\) 0 0
\(829\) 19446.4 11227.4i 0.814718 0.470377i −0.0338739 0.999426i \(-0.510784\pi\)
0.848591 + 0.529049i \(0.177451\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 20127.5 34861.8i 0.834181 1.44484i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 31288.8 1.28750 0.643748 0.765238i \(-0.277379\pi\)
0.643748 + 0.765238i \(0.277379\pi\)
\(840\) 0 0
\(841\) −22352.5 −0.916499
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9693.50 16789.6i 0.394635 0.683528i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 26067.1 15049.8i 1.05002 0.606230i
\(852\) 0 0
\(853\) 4559.95i 0.183036i 0.995803 + 0.0915180i \(0.0291719\pi\)
−0.995803 + 0.0915180i \(0.970828\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8935.12 15476.1i −0.356147 0.616865i 0.631167 0.775647i \(-0.282576\pi\)
−0.987314 + 0.158783i \(0.949243\pi\)
\(858\) 0 0
\(859\) −6639.43 3833.28i −0.263719 0.152258i 0.362311 0.932057i \(-0.381988\pi\)
−0.626030 + 0.779799i \(0.715321\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4683.45 + 2703.99i 0.184735 + 0.106657i 0.589515 0.807757i \(-0.299319\pi\)
−0.404780 + 0.914414i \(0.632652\pi\)
\(864\) 0 0
\(865\) −28188.4 48823.8i −1.10802 1.91914i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2468.09i 0.0963454i
\(870\) 0 0
\(871\) 10766.6 6216.11i 0.418844 0.241820i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2847.40 + 4931.84i −0.109635 + 0.189893i −0.915622 0.402039i \(-0.868302\pi\)
0.805987 + 0.591933i \(0.201635\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −17761.3 −0.679221 −0.339611 0.940566i \(-0.610295\pi\)
−0.339611 + 0.940566i \(0.610295\pi\)
\(882\) 0 0
\(883\) 27753.8 1.05774 0.528872 0.848701i \(-0.322615\pi\)
0.528872 + 0.848701i \(0.322615\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 11406.0 19755.8i 0.431766 0.747841i −0.565259 0.824913i \(-0.691224\pi\)
0.997025 + 0.0770725i \(0.0245573\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 58067.3 33525.2i 2.17598 1.25630i
\(894\) 0 0
\(895\) 61041.5i 2.27977i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 11192.7 + 19386.3i 0.415236 + 0.719211i
\(900\) 0 0
\(901\) −599.911 346.359i −0.0221820 0.0128068i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −16342.9 9435.56i −0.600282 0.346573i
\(906\) 0 0
\(907\) 12499.0 + 21648.8i 0.457576 + 0.792544i 0.998832 0.0483132i \(-0.0153846\pi\)
−0.541257 + 0.840857i \(0.682051\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 49221.0i 1.79008i 0.445985 + 0.895040i \(0.352853\pi\)
−0.445985 + 0.895040i \(0.647147\pi\)
\(912\) 0 0
\(913\) 9449.60 5455.73i 0.342537 0.197764i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 11943.1 20686.0i 0.428690 0.742513i −0.568067 0.822982i \(-0.692309\pi\)
0.996757 + 0.0804694i \(0.0256419\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −15692.9 −0.559629
\(924\) 0 0
\(925\) 26055.1 0.926149
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −10399.8 + 18013.0i −0.367283 + 0.636153i −0.989140 0.146978i \(-0.953045\pi\)
0.621857 + 0.783131i \(0.286379\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −932.152 + 538.178i −0.0326039 + 0.0188239i
\(936\) 0 0
\(937\) 29654.7i 1.03391i 0.856011 + 0.516957i \(0.172935\pi\)
−0.856011 + 0.516957i \(0.827065\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 7850.46 + 13597.4i 0.271963 + 0.471054i 0.969365 0.245627i \(-0.0789937\pi\)
−0.697401 + 0.716681i \(0.745660\pi\)
\(942\) 0 0
\(943\) 52905.8 + 30545.2i 1.82699 + 1.05481i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19293.1 + 11138.9i 0.662030 + 0.382223i 0.793050 0.609157i \(-0.208492\pi\)
−0.131020 + 0.991380i \(0.541825\pi\)
\(948\) 0 0
\(949\) −7237.30 12535.4i −0.247558 0.428783i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26599.9i 0.904149i 0.891980 + 0.452074i \(0.149316\pi\)
−0.891980 + 0.452074i \(0.850684\pi\)
\(954\) 0 0
\(955\) −42106.4 + 24310.2i −1.42674 + 0.823726i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −9535.10 + 16515.3i −0.320066 + 0.554371i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3682.99 0.122860
\(966\) 0 0
\(967\) −482.698 −0.0160523 −0.00802613 0.999968i \(-0.502555\pi\)
−0.00802613 + 0.999968i \(0.502555\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −12535.9 + 21712.8i −0.414310 + 0.717607i −0.995356 0.0962646i \(-0.969311\pi\)
0.581045 + 0.813871i \(0.302644\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29435.3 16994.5i 0.963887 0.556500i 0.0665199 0.997785i \(-0.478810\pi\)
0.897367 + 0.441285i \(0.145477\pi\)
\(978\) 0 0
\(979\) 4718.71i 0.154046i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2763.57 + 4786.64i 0.0896685 + 0.155310i 0.907371 0.420331i \(-0.138086\pi\)
−0.817703 + 0.575641i \(0.804753\pi\)
\(984\) 0 0
\(985\) 34064.6 + 19667.2i 1.10192 + 0.636192i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16288.5 9404.17i −0.523705 0.302361i
\(990\) 0 0
\(991\) −16600.4 28752.8i −0.532119 0.921658i −0.999297 0.0374940i \(-0.988062\pi\)
0.467178 0.884163i \(-0.345271\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20005.6i 0.637407i
\(996\) 0 0
\(997\) 20369.1 11760.1i 0.647037 0.373567i −0.140283 0.990111i \(-0.544801\pi\)
0.787320 + 0.616545i \(0.211468\pi\)
\(998\) 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.t.a.1097.8 16
3.2 odd 2 inner 1764.4.t.a.1097.1 16
7.2 even 3 252.4.f.a.125.1 8
7.3 odd 6 inner 1764.4.t.a.521.1 16
7.4 even 3 inner 1764.4.t.a.521.7 16
7.5 odd 6 252.4.f.a.125.8 yes 8
7.6 odd 2 inner 1764.4.t.a.1097.2 16
21.2 odd 6 252.4.f.a.125.7 yes 8
21.5 even 6 252.4.f.a.125.2 yes 8
21.11 odd 6 inner 1764.4.t.a.521.2 16
21.17 even 6 inner 1764.4.t.a.521.8 16
21.20 even 2 inner 1764.4.t.a.1097.7 16
28.19 even 6 1008.4.k.d.881.7 8
28.23 odd 6 1008.4.k.d.881.2 8
84.23 even 6 1008.4.k.d.881.8 8
84.47 odd 6 1008.4.k.d.881.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.f.a.125.1 8 7.2 even 3
252.4.f.a.125.2 yes 8 21.5 even 6
252.4.f.a.125.7 yes 8 21.2 odd 6
252.4.f.a.125.8 yes 8 7.5 odd 6
1008.4.k.d.881.1 8 84.47 odd 6
1008.4.k.d.881.2 8 28.23 odd 6
1008.4.k.d.881.7 8 28.19 even 6
1008.4.k.d.881.8 8 84.23 even 6
1764.4.t.a.521.1 16 7.3 odd 6 inner
1764.4.t.a.521.2 16 21.11 odd 6 inner
1764.4.t.a.521.7 16 7.4 even 3 inner
1764.4.t.a.521.8 16 21.17 even 6 inner
1764.4.t.a.1097.1 16 3.2 odd 2 inner
1764.4.t.a.1097.2 16 7.6 odd 2 inner
1764.4.t.a.1097.7 16 21.20 even 2 inner
1764.4.t.a.1097.8 16 1.1 even 1 trivial