Properties

Label 504.4.s.e.361.1
Level $504$
Weight $4$
Character 504.361
Analytic conductor $29.737$
Analytic rank $0$
Dimension $2$
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] = [504,4,Mod(289,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.289");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\), degree \(2\), 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: \(29.7369626429\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 504.361
Dual form 504.4.s.e.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.50000 + 6.06218i) q^{5} +(-17.5000 + 6.06218i) q^{7} +O(q^{10})\) \(q+(3.50000 + 6.06218i) q^{5} +(-17.5000 + 6.06218i) q^{7} +(3.50000 - 6.06218i) q^{11} -52.0000 q^{13} +(36.0000 - 62.3538i) q^{17} +(-10.0000 - 17.3205i) q^{19} +(-24.0000 - 41.5692i) q^{23} +(38.0000 - 65.8179i) q^{25} +243.000 q^{29} +(-47.5000 + 82.2724i) q^{31} +(-98.0000 - 84.8705i) q^{35} +(-176.000 - 304.841i) q^{37} +296.000 q^{41} +158.000 q^{43} +(-71.0000 - 122.976i) q^{47} +(269.500 - 212.176i) q^{49} +(-187.500 + 324.760i) q^{53} +49.0000 q^{55} +(139.500 - 241.621i) q^{59} +(-123.000 - 213.042i) q^{61} +(-182.000 - 315.233i) q^{65} +(365.000 - 632.199i) q^{67} -338.000 q^{71} +(271.000 - 469.386i) q^{73} +(-24.5000 + 127.306i) q^{77} +(152.500 + 264.138i) q^{79} -1123.00 q^{83} +504.000 q^{85} +(-213.000 - 368.927i) q^{89} +(910.000 - 315.233i) q^{91} +(70.0000 - 121.244i) q^{95} -369.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{5} - 35 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 7 q^{5} - 35 q^{7} + 7 q^{11} - 104 q^{13} + 72 q^{17} - 20 q^{19} - 48 q^{23} + 76 q^{25} + 486 q^{29} - 95 q^{31} - 196 q^{35} - 352 q^{37} + 592 q^{41} + 316 q^{43} - 142 q^{47} + 539 q^{49} - 375 q^{53} + 98 q^{55} + 279 q^{59} - 246 q^{61} - 364 q^{65} + 730 q^{67} - 676 q^{71} + 542 q^{73} - 49 q^{77} + 305 q^{79} - 2246 q^{83} + 1008 q^{85} - 426 q^{89} + 1820 q^{91} + 140 q^{95} - 738 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.50000 + 6.06218i 0.313050 + 0.542218i 0.979021 0.203760i \(-0.0653161\pi\)
−0.665971 + 0.745977i \(0.731983\pi\)
\(6\) 0 0
\(7\) −17.5000 + 6.06218i −0.944911 + 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.50000 6.06218i 0.0959354 0.166165i −0.814063 0.580776i \(-0.802749\pi\)
0.909999 + 0.414611i \(0.136082\pi\)
\(12\) 0 0
\(13\) −52.0000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 36.0000 62.3538i 0.513605 0.889590i −0.486271 0.873808i \(-0.661643\pi\)
0.999875 0.0157814i \(-0.00502359\pi\)
\(18\) 0 0
\(19\) −10.0000 17.3205i −0.120745 0.209137i 0.799317 0.600910i \(-0.205195\pi\)
−0.920062 + 0.391773i \(0.871862\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −24.0000 41.5692i −0.217580 0.376860i 0.736487 0.676451i \(-0.236483\pi\)
−0.954068 + 0.299591i \(0.903150\pi\)
\(24\) 0 0
\(25\) 38.0000 65.8179i 0.304000 0.526543i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 243.000 1.55600 0.777999 0.628265i \(-0.216235\pi\)
0.777999 + 0.628265i \(0.216235\pi\)
\(30\) 0 0
\(31\) −47.5000 + 82.2724i −0.275202 + 0.476663i −0.970186 0.242362i \(-0.922078\pi\)
0.694984 + 0.719025i \(0.255411\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −98.0000 84.8705i −0.473286 0.409878i
\(36\) 0 0
\(37\) −176.000 304.841i −0.782006 1.35447i −0.930771 0.365602i \(-0.880863\pi\)
0.148765 0.988873i \(-0.452470\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 296.000 1.12750 0.563749 0.825946i \(-0.309358\pi\)
0.563749 + 0.825946i \(0.309358\pi\)
\(42\) 0 0
\(43\) 158.000 0.560344 0.280172 0.959950i \(-0.409609\pi\)
0.280172 + 0.959950i \(0.409609\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −71.0000 122.976i −0.220349 0.381656i 0.734565 0.678539i \(-0.237386\pi\)
−0.954914 + 0.296882i \(0.904053\pi\)
\(48\) 0 0
\(49\) 269.500 212.176i 0.785714 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −187.500 + 324.760i −0.485945 + 0.841682i −0.999870 0.0161535i \(-0.994858\pi\)
0.513924 + 0.857836i \(0.328191\pi\)
\(54\) 0 0
\(55\) 49.0000 0.120130
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 139.500 241.621i 0.307820 0.533159i −0.670066 0.742302i \(-0.733734\pi\)
0.977885 + 0.209143i \(0.0670674\pi\)
\(60\) 0 0
\(61\) −123.000 213.042i −0.258173 0.447168i 0.707580 0.706634i \(-0.249787\pi\)
−0.965752 + 0.259465i \(0.916454\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −182.000 315.233i −0.347297 0.601536i
\(66\) 0 0
\(67\) 365.000 632.199i 0.665550 1.15277i −0.313586 0.949560i \(-0.601530\pi\)
0.979136 0.203207i \(-0.0651363\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −338.000 −0.564975 −0.282487 0.959271i \(-0.591160\pi\)
−0.282487 + 0.959271i \(0.591160\pi\)
\(72\) 0 0
\(73\) 271.000 469.386i 0.434495 0.752568i −0.562759 0.826621i \(-0.690260\pi\)
0.997254 + 0.0740532i \(0.0235935\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −24.5000 + 127.306i −0.0362602 + 0.188413i
\(78\) 0 0
\(79\) 152.500 + 264.138i 0.217185 + 0.376175i 0.953946 0.299978i \(-0.0969793\pi\)
−0.736761 + 0.676153i \(0.763646\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1123.00 −1.48512 −0.742562 0.669778i \(-0.766389\pi\)
−0.742562 + 0.669778i \(0.766389\pi\)
\(84\) 0 0
\(85\) 504.000 0.643135
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −213.000 368.927i −0.253685 0.439395i 0.710853 0.703341i \(-0.248309\pi\)
−0.964537 + 0.263946i \(0.914976\pi\)
\(90\) 0 0
\(91\) 910.000 315.233i 1.04828 0.363137i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 70.0000 121.244i 0.0755984 0.130940i
\(96\) 0 0
\(97\) −369.000 −0.386250 −0.193125 0.981174i \(-0.561862\pi\)
−0.193125 + 0.981174i \(0.561862\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 635.000 1099.85i 0.625593 1.08356i −0.362833 0.931854i \(-0.618191\pi\)
0.988426 0.151704i \(-0.0484761\pi\)
\(102\) 0 0
\(103\) −916.000 1586.56i −0.876273 1.51775i −0.855400 0.517968i \(-0.826689\pi\)
−0.0208734 0.999782i \(-0.506645\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −499.500 865.159i −0.451294 0.781665i 0.547172 0.837020i \(-0.315704\pi\)
−0.998467 + 0.0553553i \(0.982371\pi\)
\(108\) 0 0
\(109\) 833.000 1442.80i 0.731990 1.26784i −0.224041 0.974580i \(-0.571925\pi\)
0.956031 0.293264i \(-0.0947417\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1832.00 1.52513 0.762567 0.646910i \(-0.223939\pi\)
0.762567 + 0.646910i \(0.223939\pi\)
\(114\) 0 0
\(115\) 168.000 290.985i 0.136227 0.235952i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −252.000 + 1309.43i −0.194124 + 1.00870i
\(120\) 0 0
\(121\) 641.000 + 1110.24i 0.481593 + 0.834143i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1407.00 1.00677
\(126\) 0 0
\(127\) −1931.00 −1.34920 −0.674601 0.738183i \(-0.735684\pi\)
−0.674601 + 0.738183i \(0.735684\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 489.500 + 847.839i 0.326472 + 0.565466i 0.981809 0.189871i \(-0.0608068\pi\)
−0.655337 + 0.755336i \(0.727473\pi\)
\(132\) 0 0
\(133\) 280.000 + 242.487i 0.182549 + 0.158092i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −243.000 + 420.888i −0.151539 + 0.262474i −0.931794 0.362989i \(-0.881756\pi\)
0.780254 + 0.625463i \(0.215090\pi\)
\(138\) 0 0
\(139\) −2630.00 −1.60485 −0.802423 0.596755i \(-0.796456\pi\)
−0.802423 + 0.596755i \(0.796456\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −182.000 + 315.233i −0.106431 + 0.184344i
\(144\) 0 0
\(145\) 850.500 + 1473.11i 0.487105 + 0.843690i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −939.000 1626.40i −0.516281 0.894225i −0.999821 0.0189029i \(-0.993983\pi\)
0.483540 0.875322i \(-0.339351\pi\)
\(150\) 0 0
\(151\) −1051.50 + 1821.25i −0.566688 + 0.981532i 0.430203 + 0.902732i \(0.358442\pi\)
−0.996890 + 0.0787997i \(0.974891\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −665.000 −0.344607
\(156\) 0 0
\(157\) −702.000 + 1215.90i −0.356852 + 0.618085i −0.987433 0.158038i \(-0.949483\pi\)
0.630581 + 0.776123i \(0.282816\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 672.000 + 581.969i 0.328950 + 0.284879i
\(162\) 0 0
\(163\) 768.000 + 1330.22i 0.369045 + 0.639205i 0.989417 0.145103i \(-0.0463513\pi\)
−0.620371 + 0.784308i \(0.713018\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3634.00 1.68388 0.841938 0.539574i \(-0.181415\pi\)
0.841938 + 0.539574i \(0.181415\pi\)
\(168\) 0 0
\(169\) 507.000 0.230769
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 429.000 + 743.050i 0.188533 + 0.326549i 0.944761 0.327759i \(-0.106293\pi\)
−0.756228 + 0.654308i \(0.772960\pi\)
\(174\) 0 0
\(175\) −266.000 + 1382.18i −0.114901 + 0.597044i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1210.00 + 2095.78i −0.505249 + 0.875118i 0.494732 + 0.869046i \(0.335266\pi\)
−0.999982 + 0.00607215i \(0.998067\pi\)
\(180\) 0 0
\(181\) −2672.00 −1.09728 −0.548641 0.836058i \(-0.684855\pi\)
−0.548641 + 0.836058i \(0.684855\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1232.00 2133.89i 0.489613 0.848035i
\(186\) 0 0
\(187\) −252.000 436.477i −0.0985458 0.170686i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −856.000 1482.64i −0.324283 0.561674i 0.657084 0.753817i \(-0.271790\pi\)
−0.981367 + 0.192143i \(0.938456\pi\)
\(192\) 0 0
\(193\) 1697.50 2940.16i 0.633102 1.09657i −0.353812 0.935317i \(-0.615115\pi\)
0.986914 0.161248i \(-0.0515521\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −510.000 −0.184447 −0.0922233 0.995738i \(-0.529397\pi\)
−0.0922233 + 0.995738i \(0.529397\pi\)
\(198\) 0 0
\(199\) −138.000 + 239.023i −0.0491586 + 0.0851452i −0.889558 0.456823i \(-0.848987\pi\)
0.840399 + 0.541968i \(0.182321\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4252.50 + 1473.11i −1.47028 + 0.509320i
\(204\) 0 0
\(205\) 1036.00 + 1794.40i 0.352963 + 0.611350i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −140.000 −0.0463349
\(210\) 0 0
\(211\) 3198.00 1.04341 0.521705 0.853126i \(-0.325296\pi\)
0.521705 + 0.853126i \(0.325296\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 553.000 + 957.824i 0.175415 + 0.303828i
\(216\) 0 0
\(217\) 332.500 1727.72i 0.104016 0.540485i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1872.00 + 3242.40i −0.569793 + 0.986911i
\(222\) 0 0
\(223\) −5091.00 −1.52878 −0.764391 0.644752i \(-0.776960\pi\)
−0.764391 + 0.644752i \(0.776960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1947.50 + 3373.17i −0.569428 + 0.986278i 0.427195 + 0.904160i \(0.359502\pi\)
−0.996623 + 0.0821183i \(0.973831\pi\)
\(228\) 0 0
\(229\) 3214.00 + 5566.81i 0.927455 + 1.60640i 0.787565 + 0.616231i \(0.211341\pi\)
0.139889 + 0.990167i \(0.455325\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1912.00 3311.68i −0.537593 0.931139i −0.999033 0.0439676i \(-0.986000\pi\)
0.461439 0.887172i \(-0.347333\pi\)
\(234\) 0 0
\(235\) 497.000 860.829i 0.137960 0.238955i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −568.000 −0.153727 −0.0768637 0.997042i \(-0.524491\pi\)
−0.0768637 + 0.997042i \(0.524491\pi\)
\(240\) 0 0
\(241\) −1767.50 + 3061.40i −0.472426 + 0.818266i −0.999502 0.0315522i \(-0.989955\pi\)
0.527076 + 0.849818i \(0.323288\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2229.50 + 891.140i 0.581378 + 0.232379i
\(246\) 0 0
\(247\) 520.000 + 900.666i 0.133955 + 0.232016i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4335.00 −1.09013 −0.545065 0.838394i \(-0.683495\pi\)
−0.545065 + 0.838394i \(0.683495\pi\)
\(252\) 0 0
\(253\) −336.000 −0.0834946
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1787.00 + 3095.17i 0.433735 + 0.751252i 0.997191 0.0748943i \(-0.0238619\pi\)
−0.563456 + 0.826146i \(0.690529\pi\)
\(258\) 0 0
\(259\) 4928.00 + 4267.77i 1.18228 + 1.02389i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2733.00 4733.69i 0.640776 1.10986i −0.344484 0.938792i \(-0.611946\pi\)
0.985260 0.171064i \(-0.0547205\pi\)
\(264\) 0 0
\(265\) −2625.00 −0.608500
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2907.50 + 5035.94i −0.659009 + 1.14144i 0.321864 + 0.946786i \(0.395691\pi\)
−0.980873 + 0.194651i \(0.937643\pi\)
\(270\) 0 0
\(271\) 1618.50 + 2803.32i 0.362793 + 0.628376i 0.988419 0.151747i \(-0.0484899\pi\)
−0.625626 + 0.780123i \(0.715157\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −266.000 460.726i −0.0583287 0.101028i
\(276\) 0 0
\(277\) 1588.00 2750.50i 0.344454 0.596611i −0.640801 0.767707i \(-0.721397\pi\)
0.985254 + 0.171096i \(0.0547308\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3282.00 0.696753 0.348377 0.937355i \(-0.386733\pi\)
0.348377 + 0.937355i \(0.386733\pi\)
\(282\) 0 0
\(283\) −1091.00 + 1889.67i −0.229163 + 0.396923i −0.957560 0.288233i \(-0.906932\pi\)
0.728397 + 0.685155i \(0.240266\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −5180.00 + 1794.40i −1.06539 + 0.369060i
\(288\) 0 0
\(289\) −135.500 234.693i −0.0275799 0.0477698i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3021.00 −0.602351 −0.301175 0.953569i \(-0.597379\pi\)
−0.301175 + 0.953569i \(0.597379\pi\)
\(294\) 0 0
\(295\) 1953.00 0.385451
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1248.00 + 2161.60i 0.241384 + 0.418089i
\(300\) 0 0
\(301\) −2765.00 + 957.824i −0.529475 + 0.183415i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 861.000 1491.30i 0.161642 0.279972i
\(306\) 0 0
\(307\) −1304.00 −0.242421 −0.121210 0.992627i \(-0.538678\pi\)
−0.121210 + 0.992627i \(0.538678\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3104.00 5376.29i 0.565954 0.980261i −0.431006 0.902349i \(-0.641841\pi\)
0.996960 0.0779121i \(-0.0248254\pi\)
\(312\) 0 0
\(313\) −1668.50 2889.93i −0.301307 0.521880i 0.675125 0.737703i \(-0.264090\pi\)
−0.976432 + 0.215824i \(0.930756\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4318.50 + 7479.86i 0.765146 + 1.32527i 0.940170 + 0.340707i \(0.110666\pi\)
−0.175024 + 0.984564i \(0.556000\pi\)
\(318\) 0 0
\(319\) 850.500 1473.11i 0.149275 0.258553i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1440.00 −0.248061
\(324\) 0 0
\(325\) −1976.00 + 3422.53i −0.337258 + 0.584148i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1988.00 + 1721.66i 0.333137 + 0.288505i
\(330\) 0 0
\(331\) −970.000 1680.09i −0.161076 0.278991i 0.774179 0.632967i \(-0.218163\pi\)
−0.935255 + 0.353975i \(0.884830\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5110.00 0.833400
\(336\) 0 0
\(337\) −5527.00 −0.893397 −0.446699 0.894684i \(-0.647400\pi\)
−0.446699 + 0.894684i \(0.647400\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 332.500 + 575.907i 0.0528032 + 0.0914578i
\(342\) 0 0
\(343\) −3430.00 + 5346.84i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4782.00 8282.67i 0.739802 1.28137i −0.212783 0.977100i \(-0.568253\pi\)
0.952584 0.304275i \(-0.0984141\pi\)
\(348\) 0 0
\(349\) −918.000 −0.140801 −0.0704003 0.997519i \(-0.522428\pi\)
−0.0704003 + 0.997519i \(0.522428\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5740.00 9941.97i 0.865466 1.49903i −0.00111843 0.999999i \(-0.500356\pi\)
0.866584 0.499031i \(-0.166311\pi\)
\(354\) 0 0
\(355\) −1183.00 2049.02i −0.176865 0.306339i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1827.00 + 3164.46i 0.268594 + 0.465219i 0.968499 0.249017i \(-0.0801076\pi\)
−0.699905 + 0.714236i \(0.746774\pi\)
\(360\) 0 0
\(361\) 3229.50 5593.66i 0.470841 0.815521i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3794.00 0.544074
\(366\) 0 0
\(367\) 5033.50 8718.28i 0.715931 1.24003i −0.246669 0.969100i \(-0.579336\pi\)
0.962599 0.270928i \(-0.0873307\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1312.50 6819.95i 0.183670 0.954378i
\(372\) 0 0
\(373\) −4400.00 7621.02i −0.610786 1.05791i −0.991108 0.133059i \(-0.957520\pi\)
0.380322 0.924854i \(-0.375813\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12636.0 −1.72623
\(378\) 0 0
\(379\) 1136.00 0.153964 0.0769821 0.997032i \(-0.475472\pi\)
0.0769821 + 0.997032i \(0.475472\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2145.00 + 3715.25i 0.286173 + 0.495667i 0.972893 0.231256i \(-0.0742834\pi\)
−0.686720 + 0.726922i \(0.740950\pi\)
\(384\) 0 0
\(385\) −857.500 + 297.047i −0.113512 + 0.0393218i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −431.000 + 746.514i −0.0561763 + 0.0973001i −0.892746 0.450560i \(-0.851224\pi\)
0.836570 + 0.547861i \(0.184558\pi\)
\(390\) 0 0
\(391\) −3456.00 −0.447001
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1067.50 + 1848.96i −0.135979 + 0.235523i
\(396\) 0 0
\(397\) 4570.00 + 7915.47i 0.577737 + 1.00067i 0.995738 + 0.0922238i \(0.0293975\pi\)
−0.418001 + 0.908447i \(0.637269\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7116.00 12325.3i −0.886175 1.53490i −0.844361 0.535774i \(-0.820020\pi\)
−0.0418135 0.999125i \(-0.513314\pi\)
\(402\) 0 0
\(403\) 2470.00 4278.17i 0.305309 0.528810i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2464.00 −0.300088
\(408\) 0 0
\(409\) 5000.50 8661.12i 0.604545 1.04710i −0.387578 0.921837i \(-0.626688\pi\)
0.992123 0.125266i \(-0.0399783\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −976.500 + 5074.04i −0.116345 + 0.604546i
\(414\) 0 0
\(415\) −3930.50 6807.83i −0.464917 0.805260i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10256.0 1.19580 0.597898 0.801572i \(-0.296003\pi\)
0.597898 + 0.801572i \(0.296003\pi\)
\(420\) 0 0
\(421\) −4502.00 −0.521174 −0.260587 0.965450i \(-0.583916\pi\)
−0.260587 + 0.965450i \(0.583916\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2736.00 4738.89i −0.312272 0.540871i
\(426\) 0 0
\(427\) 3444.00 + 2982.59i 0.390320 + 0.338027i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6178.00 10700.6i 0.690450 1.19589i −0.281241 0.959637i \(-0.590746\pi\)
0.971691 0.236257i \(-0.0759206\pi\)
\(432\) 0 0
\(433\) 862.000 0.0956699 0.0478350 0.998855i \(-0.484768\pi\)
0.0478350 + 0.998855i \(0.484768\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −480.000 + 831.384i −0.0525435 + 0.0910080i
\(438\) 0 0
\(439\) 682.500 + 1182.12i 0.0742003 + 0.128519i 0.900738 0.434362i \(-0.143026\pi\)
−0.826538 + 0.562881i \(0.809693\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4465.50 + 7734.47i 0.478922 + 0.829517i 0.999708 0.0241705i \(-0.00769445\pi\)
−0.520786 + 0.853687i \(0.674361\pi\)
\(444\) 0 0
\(445\) 1491.00 2582.49i 0.158832 0.275105i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −11228.0 −1.18014 −0.590069 0.807353i \(-0.700900\pi\)
−0.590069 + 0.807353i \(0.700900\pi\)
\(450\) 0 0
\(451\) 1036.00 1794.40i 0.108167 0.187351i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5096.00 + 4413.27i 0.525064 + 0.454719i
\(456\) 0 0
\(457\) −6075.50 10523.1i −0.621882 1.07713i −0.989135 0.147010i \(-0.953035\pi\)
0.367253 0.930121i \(-0.380298\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −18534.0 −1.87248 −0.936241 0.351358i \(-0.885720\pi\)
−0.936241 + 0.351358i \(0.885720\pi\)
\(462\) 0 0
\(463\) 17096.0 1.71602 0.858011 0.513631i \(-0.171700\pi\)
0.858011 + 0.513631i \(0.171700\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1070.00 1853.29i −0.106025 0.183641i 0.808131 0.589002i \(-0.200479\pi\)
−0.914157 + 0.405361i \(0.867146\pi\)
\(468\) 0 0
\(469\) −2555.00 + 13276.2i −0.251554 + 1.30711i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 553.000 957.824i 0.0537568 0.0931095i
\(474\) 0 0
\(475\) −1520.00 −0.146826
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6633.00 + 11488.7i −0.632713 + 1.09589i 0.354282 + 0.935139i \(0.384725\pi\)
−0.986995 + 0.160752i \(0.948608\pi\)
\(480\) 0 0
\(481\) 9152.00 + 15851.7i 0.867558 + 1.50265i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1291.50 2236.94i −0.120915 0.209432i
\(486\) 0 0
\(487\) 9591.50 16613.0i 0.892469 1.54580i 0.0555627 0.998455i \(-0.482305\pi\)
0.836906 0.547346i \(-0.184362\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 21693.0 1.99387 0.996936 0.0782185i \(-0.0249232\pi\)
0.996936 + 0.0782185i \(0.0249232\pi\)
\(492\) 0 0
\(493\) 8748.00 15152.0i 0.799169 1.38420i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5915.00 2049.02i 0.533851 0.184931i
\(498\) 0 0
\(499\) −9829.00 17024.3i −0.881776 1.52728i −0.849364 0.527807i \(-0.823014\pi\)
−0.0324122 0.999475i \(-0.510319\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −19436.0 −1.72288 −0.861440 0.507860i \(-0.830437\pi\)
−0.861440 + 0.507860i \(0.830437\pi\)
\(504\) 0 0
\(505\) 8890.00 0.783366
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1544.50 + 2675.15i 0.134497 + 0.232955i 0.925405 0.378980i \(-0.123725\pi\)
−0.790908 + 0.611934i \(0.790392\pi\)
\(510\) 0 0
\(511\) −1897.00 + 9857.10i −0.164224 + 0.853332i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6412.00 11105.9i 0.548634 0.950262i
\(516\) 0 0
\(517\) −994.000 −0.0845572
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 975.000 1688.75i 0.0819876 0.142007i −0.822116 0.569320i \(-0.807207\pi\)
0.904104 + 0.427313i \(0.140540\pi\)
\(522\) 0 0
\(523\) −6566.00 11372.6i −0.548970 0.950843i −0.998345 0.0575005i \(-0.981687\pi\)
0.449376 0.893343i \(-0.351646\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3420.00 + 5923.61i 0.282690 + 0.489633i
\(528\) 0 0
\(529\) 4931.50 8541.61i 0.405318 0.702031i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15392.0 −1.25085
\(534\) 0 0
\(535\) 3496.50 6056.12i 0.282555 0.489399i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −343.000 2376.37i −0.0274101 0.189903i
\(540\) 0 0
\(541\) −1465.00 2537.45i −0.116424 0.201652i 0.801924 0.597426i \(-0.203810\pi\)
−0.918348 + 0.395774i \(0.870476\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 11662.0 0.916597
\(546\) 0 0
\(547\) 19824.0 1.54957 0.774783 0.632227i \(-0.217859\pi\)
0.774783 + 0.632227i \(0.217859\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2430.00 4208.88i −0.187879 0.325416i
\(552\) 0 0
\(553\) −4270.00 3697.93i −0.328352 0.284362i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9584.50 + 16600.8i −0.729099 + 1.26284i 0.228165 + 0.973622i \(0.426727\pi\)
−0.957264 + 0.289215i \(0.906606\pi\)
\(558\) 0 0
\(559\) −8216.00 −0.621645
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3633.50 6293.41i 0.271996 0.471111i −0.697377 0.716705i \(-0.745650\pi\)
0.969373 + 0.245594i \(0.0789829\pi\)
\(564\) 0 0
\(565\) 6412.00 + 11105.9i 0.477442 + 0.826954i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2446.00 4236.60i −0.180214 0.312139i 0.761739 0.647883i \(-0.224346\pi\)
−0.941953 + 0.335744i \(0.891012\pi\)
\(570\) 0 0
\(571\) −11063.0 + 19161.7i −0.810809 + 1.40436i 0.101489 + 0.994837i \(0.467639\pi\)
−0.912298 + 0.409526i \(0.865694\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3648.00 −0.264578
\(576\) 0 0
\(577\) 548.500 950.030i 0.0395743 0.0685446i −0.845560 0.533881i \(-0.820733\pi\)
0.885134 + 0.465336i \(0.154066\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 19652.5 6807.83i 1.40331 0.486121i
\(582\) 0 0
\(583\) 1312.50 + 2273.32i 0.0932388 + 0.161494i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9969.00 −0.700962 −0.350481 0.936570i \(-0.613982\pi\)
−0.350481 + 0.936570i \(0.613982\pi\)
\(588\) 0 0
\(589\) 1900.00 0.132917
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3212.00 5563.35i −0.222430 0.385260i 0.733115 0.680104i \(-0.238066\pi\)
−0.955545 + 0.294844i \(0.904732\pi\)
\(594\) 0 0
\(595\) −8820.00 + 3055.34i −0.607705 + 0.210515i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 8523.00 14762.3i 0.581370 1.00696i −0.413948 0.910301i \(-0.635850\pi\)
0.995317 0.0966609i \(-0.0308162\pi\)
\(600\) 0 0
\(601\) −18877.0 −1.28121 −0.640606 0.767869i \(-0.721317\pi\)
−0.640606 + 0.767869i \(0.721317\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4487.00 + 7771.71i −0.301525 + 0.522256i
\(606\) 0 0
\(607\) 6891.50 + 11936.4i 0.460819 + 0.798163i 0.999002 0.0446657i \(-0.0142223\pi\)
−0.538183 + 0.842828i \(0.680889\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3692.00 + 6394.73i 0.244456 + 0.423409i
\(612\) 0 0
\(613\) −9096.00 + 15754.7i −0.599321 + 1.03806i 0.393600 + 0.919282i \(0.371229\pi\)
−0.992921 + 0.118773i \(0.962104\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13054.0 0.851757 0.425879 0.904780i \(-0.359965\pi\)
0.425879 + 0.904780i \(0.359965\pi\)
\(618\) 0 0
\(619\) −3623.00 + 6275.22i −0.235251 + 0.407468i −0.959346 0.282234i \(-0.908925\pi\)
0.724094 + 0.689701i \(0.242258\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5964.00 + 5164.98i 0.383535 + 0.332151i
\(624\) 0 0
\(625\) 174.500 + 302.243i 0.0111680 + 0.0193435i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −25344.0 −1.60657
\(630\) 0 0
\(631\) 2817.00 0.177723 0.0888613 0.996044i \(-0.471677\pi\)
0.0888613 + 0.996044i \(0.471677\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6758.50 11706.1i −0.422367 0.731561i
\(636\) 0 0
\(637\) −14014.0 + 11033.2i −0.871672 + 0.686264i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7893.00 13671.1i 0.486357 0.842395i −0.513520 0.858078i \(-0.671659\pi\)
0.999877 + 0.0156827i \(0.00499217\pi\)
\(642\) 0 0
\(643\) −17426.0 −1.06876 −0.534381 0.845244i \(-0.679455\pi\)
−0.534381 + 0.845244i \(0.679455\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12917.0 + 22372.9i −0.784884 + 1.35946i 0.144185 + 0.989551i \(0.453944\pi\)
−0.929069 + 0.369907i \(0.879389\pi\)
\(648\) 0 0
\(649\) −976.500 1691.35i −0.0590616 0.102298i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13991.5 + 24234.0i 0.838483 + 1.45230i 0.891162 + 0.453684i \(0.149891\pi\)
−0.0526789 + 0.998612i \(0.516776\pi\)
\(654\) 0 0
\(655\) −3426.50 + 5934.87i −0.204404 + 0.354038i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 28296.0 1.67262 0.836309 0.548258i \(-0.184709\pi\)
0.836309 + 0.548258i \(0.184709\pi\)
\(660\) 0 0
\(661\) −10627.0 + 18406.5i −0.625329 + 1.08310i 0.363148 + 0.931731i \(0.381702\pi\)
−0.988477 + 0.151370i \(0.951631\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −490.000 + 2546.11i −0.0285735 + 0.148472i
\(666\) 0 0
\(667\) −5832.00 10101.3i −0.338555 0.586394i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1722.00 −0.0990716
\(672\) 0 0
\(673\) −28259.0 −1.61858 −0.809290 0.587409i \(-0.800148\pi\)
−0.809290 + 0.587409i \(0.800148\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15889.5 + 27521.4i 0.902043 + 1.56238i 0.824838 + 0.565369i \(0.191266\pi\)
0.0772049 + 0.997015i \(0.475400\pi\)
\(678\) 0 0
\(679\) 6457.50 2236.94i 0.364972 0.126430i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4873.50 + 8441.15i −0.273030 + 0.472901i −0.969636 0.244552i \(-0.921359\pi\)
0.696606 + 0.717453i \(0.254692\pi\)
\(684\) 0 0
\(685\) −3402.00 −0.189757
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9750.00 16887.5i 0.539108 0.933762i
\(690\) 0 0
\(691\) 12256.0 + 21228.0i 0.674733 + 1.16867i 0.976547 + 0.215304i \(0.0690744\pi\)
−0.301814 + 0.953367i \(0.597592\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9205.00 15943.5i −0.502396 0.870176i
\(696\) 0 0
\(697\) 10656.0 18456.7i 0.579089 1.00301i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −14325.0 −0.771823 −0.385911 0.922536i \(-0.626113\pi\)
−0.385911 + 0.922536i \(0.626113\pi\)
\(702\) 0 0
\(703\) −3520.00 + 6096.82i −0.188847 + 0.327092i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4445.00 + 23096.9i −0.236452 + 1.22864i
\(708\) 0 0
\(709\) −6039.00 10459.9i −0.319886 0.554059i 0.660578 0.750758i \(-0.270311\pi\)
−0.980464 + 0.196698i \(0.936978\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4560.00 0.239514
\(714\) 0 0
\(715\) −2548.00 −0.133272
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −5329.00 9230.10i −0.276409 0.478755i 0.694081 0.719897i \(-0.255811\pi\)
−0.970490 + 0.241143i \(0.922478\pi\)
\(720\) 0 0
\(721\) 25648.0 + 22211.8i 1.32480 + 1.14731i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9234.00 15993.8i 0.473024 0.819301i
\(726\) 0 0
\(727\) 19099.0 0.974337 0.487168 0.873308i \(-0.338030\pi\)
0.487168 + 0.873308i \(0.338030\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5688.00 9851.90i 0.287795 0.498476i
\(732\) 0 0
\(733\) −703.000 1217.63i −0.0354241 0.0613564i 0.847770 0.530364i \(-0.177945\pi\)
−0.883194 + 0.469008i \(0.844612\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2555.00 4425.39i −0.127700 0.221182i
\(738\) 0 0
\(739\) −3365.00 + 5828.35i −0.167501 + 0.290121i −0.937541 0.347875i \(-0.886903\pi\)
0.770039 + 0.637996i \(0.220237\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1166.00 0.0575725 0.0287863 0.999586i \(-0.490836\pi\)
0.0287863 + 0.999586i \(0.490836\pi\)
\(744\) 0 0
\(745\) 6573.00 11384.8i 0.323243 0.559873i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13986.0 + 12112.2i 0.682293 + 0.590883i
\(750\) 0 0
\(751\) 8483.50 + 14693.9i 0.412207 + 0.713963i 0.995131 0.0985636i \(-0.0314248\pi\)
−0.582924 + 0.812527i \(0.698091\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −14721.0 −0.709605
\(756\) 0 0
\(757\) 11878.0 0.570295 0.285147 0.958484i \(-0.407957\pi\)
0.285147 + 0.958484i \(0.407957\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13662.0 + 23663.3i 0.650785 + 1.12719i 0.982933 + 0.183965i \(0.0588933\pi\)
−0.332148 + 0.943227i \(0.607773\pi\)
\(762\) 0 0
\(763\) −5831.00 + 30298.8i −0.276666 + 1.43760i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7254.00 + 12564.3i −0.341495 + 0.591487i
\(768\) 0 0
\(769\) −11971.0 −0.561359 −0.280680 0.959802i \(-0.590560\pi\)
−0.280680 + 0.959802i \(0.590560\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −169.000 + 292.717i −0.00786353 + 0.0136200i −0.869930 0.493175i \(-0.835836\pi\)
0.862067 + 0.506795i \(0.169170\pi\)
\(774\) 0 0
\(775\) 3610.00 + 6252.70i 0.167323 + 0.289811i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2960.00 5126.87i −0.136140 0.235801i
\(780\) 0 0
\(781\) −1183.00 + 2049.02i −0.0542011 + 0.0938791i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −9828.00 −0.446849
\(786\) 0 0
\(787\) −7057.00 + 12223.1i −0.319638 + 0.553629i −0.980412 0.196956i \(-0.936895\pi\)
0.660775 + 0.750584i \(0.270228\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −32060.0 + 11105.9i −1.44112 + 0.499217i
\(792\) 0 0
\(793\) 6396.00 + 11078.2i 0.286417 + 0.496089i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1563.00 −0.0694659 −0.0347329 0.999397i \(-0.511058\pi\)
−0.0347329 + 0.999397i \(0.511058\pi\)
\(798\) 0 0
\(799\) −10224.0 −0.452690
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1897.00 3285.70i −0.0833670 0.144396i
\(804\) 0 0
\(805\) −1176.00 + 6110.68i −0.0514889 + 0.267544i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −15784.0 + 27338.7i −0.685953 + 1.18811i 0.287183 + 0.957876i \(0.407281\pi\)
−0.973136 + 0.230230i \(0.926052\pi\)
\(810\) 0 0
\(811\) 2626.00 0.113701 0.0568504 0.998383i \(-0.481894\pi\)
0.0568504 + 0.998383i \(0.481894\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5376.00 + 9311.51i −0.231059 + 0.400206i
\(816\) 0 0
\(817\) −1580.00 2736.64i −0.0676588 0.117188i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 180.500 + 312.635i 0.00767295 + 0.0132899i 0.869836 0.493340i \(-0.164224\pi\)
−0.862163 + 0.506630i \(0.830891\pi\)
\(822\) 0 0
\(823\) 14772.0 25585.9i 0.625662 1.08368i −0.362751 0.931886i \(-0.618162\pi\)
0.988413 0.151792i \(-0.0485043\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −24163.0 −1.01600 −0.507999 0.861358i \(-0.669615\pi\)
−0.507999 + 0.861358i \(0.669615\pi\)
\(828\) 0 0
\(829\) 19598.0 33944.7i 0.821070 1.42213i −0.0838169 0.996481i \(-0.526711\pi\)
0.904886 0.425653i \(-0.139956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3528.00 24442.7i −0.146744 1.01667i
\(834\) 0 0
\(835\) 12719.0 + 22030.0i 0.527137 + 0.913028i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −6528.00 −0.268619 −0.134310 0.990939i \(-0.542882\pi\)
−0.134310 + 0.990939i \(0.542882\pi\)
\(840\) 0 0
\(841\) 34660.0 1.42113
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1774.50 + 3073.52i 0.0722422 + 0.125127i
\(846\) 0 0
\(847\) −17948.0 15543.4i −0.728100 0.630553i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −8448.00 + 14632.4i −0.340298 + 0.589414i
\(852\) 0 0
\(853\) 1046.00 0.0419864 0.0209932 0.999780i \(-0.493317\pi\)
0.0209932 + 0.999780i \(0.493317\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16455.0 28500.9i 0.655883 1.13602i −0.325788 0.945443i \(-0.605630\pi\)
0.981672 0.190581i \(-0.0610371\pi\)
\(858\) 0 0
\(859\) −11341.0 19643.2i −0.450466 0.780229i 0.547949 0.836512i \(-0.315409\pi\)
−0.998415 + 0.0562823i \(0.982075\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −3725.00 6451.89i −0.146930 0.254490i 0.783161 0.621818i \(-0.213606\pi\)
−0.930091 + 0.367328i \(0.880272\pi\)
\(864\) 0 0
\(865\) −3003.00 + 5201.35i −0.118041 + 0.204452i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2135.00 0.0833428
\(870\) 0 0
\(871\) −18980.0 + 32874.3i −0.738361 + 1.27888i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −24622.5 + 8529.48i −0.951306 + 0.329542i
\(876\) 0 0
\(877\) −5040.00 8729.54i −0.194058 0.336118i 0.752533 0.658554i \(-0.228832\pi\)
−0.946591 + 0.322436i \(0.895498\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 32958.0 1.26037 0.630183 0.776446i \(-0.282980\pi\)
0.630183 + 0.776446i \(0.282980\pi\)
\(882\) 0 0
\(883\) −19784.0 −0.754003 −0.377001 0.926213i \(-0.623045\pi\)
−0.377001 + 0.926213i \(0.623045\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −5740.00 9941.97i −0.217283 0.376346i 0.736693 0.676227i \(-0.236386\pi\)
−0.953976 + 0.299881i \(0.903053\pi\)
\(888\) 0 0
\(889\) 33792.5 11706.1i 1.27488 0.441630i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1420.00 + 2459.51i −0.0532122 + 0.0921662i
\(894\) 0 0
\(895\) −16940.0 −0.632672
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11542.5 + 19992.2i −0.428213 + 0.741688i
\(900\) 0 0
\(901\) 13500.0 + 23382.7i 0.499168 + 0.864584i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9352.00 16198.1i −0.343504 0.594966i
\(906\) 0 0
\(907\) 22676.0 39276.0i 0.830148 1.43786i −0.0677725 0.997701i \(-0.521589\pi\)
0.897920 0.440158i \(-0.145077\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 39906.0 1.45131 0.725656 0.688058i \(-0.241537\pi\)
0.725656 + 0.688058i \(0.241537\pi\)
\(912\) 0 0
\(913\) −3930.50 + 6807.83i −0.142476 + 0.246776i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −13706.0 11869.7i −0.493579 0.427452i
\(918\) 0 0
\(919\) −8764.00 15179.7i −0.314579 0.544866i 0.664769 0.747049i \(-0.268530\pi\)
−0.979348 + 0.202183i \(0.935197\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 17576.0 0.626783
\(924\) 0 0
\(925\) −26752.0 −0.950919
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8533.00 + 14779.6i 0.301355 + 0.521962i 0.976443 0.215775i \(-0.0692278\pi\)
−0.675088 + 0.737737i \(0.735894\pi\)
\(930\) 0 0
\(931\) −6370.00 2546.11i −0.224241 0.0896300i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1764.00 3055.34i 0.0616994 0.106867i
\(936\) 0 0
\(937\) −30821.0 −1.07458 −0.537288 0.843399i \(-0.680551\pi\)
−0.537288 + 0.843399i \(0.680551\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2089.50 3619.12i 0.0723866 0.125377i −0.827560 0.561377i \(-0.810272\pi\)
0.899947 + 0.436000i \(0.143605\pi\)
\(942\) 0 0
\(943\) −7104.00 12304.5i −0.245321 0.424909i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4888.00 + 8466.26i 0.167728 + 0.290514i 0.937621 0.347660i \(-0.113024\pi\)
−0.769893 + 0.638174i \(0.779690\pi\)
\(948\) 0 0
\(949\) −14092.0 + 24408.1i −0.482029 + 0.834899i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −14778.0 −0.502315 −0.251158 0.967946i \(-0.580811\pi\)
−0.251158 + 0.967946i \(0.580811\pi\)
\(954\) 0 0
\(955\) 5992.00 10378.4i 0.203033 0.351664i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1701.00 8838.66i 0.0572765 0.297617i
\(960\) 0 0
\(961\) 10383.0 + 17983.9i 0.348528 + 0.603668i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 23765.0 0.792769
\(966\) 0 0
\(967\) 8759.00 0.291283 0.145641 0.989337i \(-0.453475\pi\)
0.145641 + 0.989337i \(0.453475\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −16789.5 29080.3i −0.554893 0.961102i −0.997912 0.0645901i \(-0.979426\pi\)
0.443019 0.896512i \(-0.353907\pi\)
\(972\) 0 0
\(973\) 46025.0 15943.5i 1.51644 0.525309i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21317.0 36922.1i 0.698046 1.20905i −0.271097 0.962552i \(-0.587386\pi\)
0.969143 0.246500i \(-0.0792804\pi\)
\(978\) 0 0
\(979\) −2982.00 −0.0973495
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 14574.0 25242.9i 0.472877 0.819048i −0.526641 0.850088i \(-0.676549\pi\)
0.999518 + 0.0310404i \(0.00988204\pi\)
\(984\) 0 0
\(985\) −1785.00 3091.71i −0.0577409 0.100010i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −3792.00 6567.94i −0.121920 0.211171i
\(990\) 0 0
\(991\) 22398.5 38795.3i 0.717974 1.24357i −0.243828 0.969819i \(-0.578403\pi\)
0.961801 0.273748i \(-0.0882635\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1932.00 −0.0615563
\(996\) 0 0
\(997\) −5327.00 + 9226.63i −0.169215 + 0.293090i −0.938144 0.346245i \(-0.887457\pi\)
0.768929 + 0.639334i \(0.220790\pi\)
\(998\) 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 504.4.s.e.361.1 2
3.2 odd 2 168.4.q.a.25.1 2
7.2 even 3 inner 504.4.s.e.289.1 2
12.11 even 2 336.4.q.a.193.1 2
21.2 odd 6 168.4.q.a.121.1 yes 2
21.11 odd 6 1176.4.a.f.1.1 1
21.17 even 6 1176.4.a.i.1.1 1
84.11 even 6 2352.4.a.bg.1.1 1
84.23 even 6 336.4.q.a.289.1 2
84.59 odd 6 2352.4.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.a.25.1 2 3.2 odd 2
168.4.q.a.121.1 yes 2 21.2 odd 6
336.4.q.a.193.1 2 12.11 even 2
336.4.q.a.289.1 2 84.23 even 6
504.4.s.e.289.1 2 7.2 even 3 inner
504.4.s.e.361.1 2 1.1 even 1 trivial
1176.4.a.f.1.1 1 21.11 odd 6
1176.4.a.i.1.1 1 21.17 even 6
2352.4.a.c.1.1 1 84.59 odd 6
2352.4.a.bg.1.1 1 84.11 even 6