Properties

Label 1764.3.bk.g.1745.7
Level $1764$
Weight $3$
Character 1764.1745
Analytic conductor $48.066$
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] = [1764,3,Mod(557,1764)]
 
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, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.557");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.bk (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: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: 16.0.329365073333488765586374656.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 199x^{12} + 24960x^{8} - 2913559x^{4} + 214358881 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1745.7
Root \(-3.03759 + 1.33156i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1745
Dual form 1764.3.bk.g.557.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(8.27698 + 4.77872i) q^{5} +O(q^{10})\) \(q+(8.27698 + 4.77872i) q^{5} +(-8.03119 + 4.63681i) q^{11} -4.24264 q^{13} +(-13.4731 + 7.77872i) q^{17} +(17.4460 - 30.2173i) q^{19} +(-15.3797 - 8.87945i) q^{23} +(33.1723 + 57.4561i) q^{25} +26.2442i q^{29} +(16.0317 + 27.7678i) q^{31} +(-27.6723 + 47.9299i) q^{37} +38.4426i q^{41} +29.3446 q^{43} +(59.0887 + 34.1149i) q^{47} +(0.943355 - 0.544646i) q^{53} -88.6320 q^{55} +(-59.0887 + 34.1149i) q^{59} +(-23.8099 + 41.2400i) q^{61} +(-35.1163 - 20.2744i) q^{65} +(4.32768 + 7.49577i) q^{67} +95.7031i q^{71} +(-45.0231 - 77.9823i) q^{73} +(74.3446 - 128.769i) q^{79} +88.8851i q^{83} -148.689 q^{85} +(-66.0316 - 38.1234i) q^{89} +(288.800 - 166.739i) q^{95} +12.7519 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 216 q^{25} - 128 q^{37} - 160 q^{43} + 384 q^{67} + 560 q^{79} - 1120 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 8.27698 + 4.77872i 1.65540 + 0.955744i 0.974798 + 0.223087i \(0.0716135\pi\)
0.680598 + 0.732657i \(0.261720\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −8.03119 + 4.63681i −0.730108 + 0.421528i −0.818462 0.574561i \(-0.805173\pi\)
0.0883536 + 0.996089i \(0.471839\pi\)
\(12\) 0 0
\(13\) −4.24264 −0.326357 −0.163178 0.986597i \(-0.552175\pi\)
−0.163178 + 0.986597i \(0.552175\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −13.4731 + 7.77872i −0.792537 + 0.457572i −0.840855 0.541260i \(-0.817947\pi\)
0.0483176 + 0.998832i \(0.484614\pi\)
\(18\) 0 0
\(19\) 17.4460 30.2173i 0.918209 1.59038i 0.116074 0.993241i \(-0.462969\pi\)
0.802135 0.597143i \(-0.203698\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −15.3797 8.87945i −0.668681 0.386063i 0.126896 0.991916i \(-0.459499\pi\)
−0.795577 + 0.605853i \(0.792832\pi\)
\(24\) 0 0
\(25\) 33.1723 + 57.4561i 1.32689 + 2.29825i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 26.2442i 0.904972i 0.891772 + 0.452486i \(0.149463\pi\)
−0.891772 + 0.452486i \(0.850537\pi\)
\(30\) 0 0
\(31\) 16.0317 + 27.7678i 0.517153 + 0.895736i 0.999802 + 0.0199213i \(0.00634158\pi\)
−0.482648 + 0.875814i \(0.660325\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −27.6723 + 47.9299i −0.747900 + 1.29540i 0.200927 + 0.979606i \(0.435605\pi\)
−0.948827 + 0.315795i \(0.897729\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 38.4426i 0.937623i 0.883298 + 0.468812i \(0.155318\pi\)
−0.883298 + 0.468812i \(0.844682\pi\)
\(42\) 0 0
\(43\) 29.3446 0.682433 0.341217 0.939985i \(-0.389161\pi\)
0.341217 + 0.939985i \(0.389161\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 59.0887 + 34.1149i 1.25721 + 0.725848i 0.972531 0.232775i \(-0.0747807\pi\)
0.284676 + 0.958624i \(0.408114\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.943355 0.544646i 0.0177991 0.0102763i −0.491074 0.871118i \(-0.663396\pi\)
0.508873 + 0.860842i \(0.330062\pi\)
\(54\) 0 0
\(55\) −88.6320 −1.61149
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −59.0887 + 34.1149i −1.00150 + 0.578218i −0.908693 0.417466i \(-0.862918\pi\)
−0.0928106 + 0.995684i \(0.529585\pi\)
\(60\) 0 0
\(61\) −23.8099 + 41.2400i −0.390327 + 0.676066i −0.992493 0.122305i \(-0.960971\pi\)
0.602166 + 0.798371i \(0.294305\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −35.1163 20.2744i −0.540250 0.311914i
\(66\) 0 0
\(67\) 4.32768 + 7.49577i 0.0645923 + 0.111877i 0.896513 0.443017i \(-0.146092\pi\)
−0.831921 + 0.554894i \(0.812759\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 95.7031i 1.34793i 0.738763 + 0.673966i \(0.235410\pi\)
−0.738763 + 0.673966i \(0.764590\pi\)
\(72\) 0 0
\(73\) −45.0231 77.9823i −0.616755 1.06825i −0.990074 0.140548i \(-0.955114\pi\)
0.373319 0.927703i \(-0.378220\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 74.3446 128.769i 0.941071 1.62998i 0.177639 0.984096i \(-0.443154\pi\)
0.763433 0.645887i \(-0.223512\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 88.8851i 1.07091i 0.844565 + 0.535453i \(0.179859\pi\)
−0.844565 + 0.535453i \(0.820141\pi\)
\(84\) 0 0
\(85\) −148.689 −1.74929
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −66.0316 38.1234i −0.741928 0.428352i 0.0808420 0.996727i \(-0.474239\pi\)
−0.822770 + 0.568375i \(0.807572\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 288.800 166.739i 3.04000 1.75514i
\(96\) 0 0
\(97\) 12.7519 0.131463 0.0657314 0.997837i \(-0.479062\pi\)
0.0657314 + 0.997837i \(0.479062\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.64326 2.68079i 0.0459729 0.0265425i −0.476837 0.878992i \(-0.658217\pi\)
0.522810 + 0.852449i \(0.324884\pi\)
\(102\) 0 0
\(103\) −68.3576 + 118.399i −0.663667 + 1.14950i 0.315979 + 0.948766i \(0.397667\pi\)
−0.979645 + 0.200738i \(0.935666\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 129.182 + 74.5831i 1.20731 + 0.697038i 0.962170 0.272450i \(-0.0878340\pi\)
0.245136 + 0.969489i \(0.421167\pi\)
\(108\) 0 0
\(109\) −14.3446 24.8456i −0.131602 0.227942i 0.792692 0.609622i \(-0.208679\pi\)
−0.924294 + 0.381681i \(0.875345\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 145.712i 1.28949i −0.764399 0.644743i \(-0.776964\pi\)
0.764399 0.644743i \(-0.223036\pi\)
\(114\) 0 0
\(115\) −84.8648 146.990i −0.737955 1.27818i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −17.5000 + 30.3109i −0.144628 + 0.250503i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 395.149i 3.16119i
\(126\) 0 0
\(127\) −119.345 −0.939722 −0.469861 0.882741i \(-0.655696\pi\)
−0.469861 + 0.882741i \(0.655696\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 83.1384 + 48.0000i 0.634645 + 0.366412i 0.782549 0.622590i \(-0.213919\pi\)
−0.147904 + 0.989002i \(0.547253\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 149.018 86.0353i 1.08772 0.627995i 0.154751 0.987954i \(-0.450542\pi\)
0.932968 + 0.359958i \(0.117209\pi\)
\(138\) 0 0
\(139\) −211.205 −1.51946 −0.759731 0.650238i \(-0.774669\pi\)
−0.759731 + 0.650238i \(0.774669\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 34.0735 19.6723i 0.238276 0.137569i
\(144\) 0 0
\(145\) −125.414 + 217.223i −0.864921 + 1.49809i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −130.808 75.5219i −0.877905 0.506859i −0.00793782 0.999968i \(-0.502527\pi\)
−0.869967 + 0.493110i \(0.835860\pi\)
\(150\) 0 0
\(151\) −6.01695 10.4217i −0.0398473 0.0690176i 0.845414 0.534112i \(-0.179354\pi\)
−0.885261 + 0.465094i \(0.846020\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 306.445i 1.97706i
\(156\) 0 0
\(157\) −80.3785 139.220i −0.511965 0.886749i −0.999904 0.0138712i \(-0.995585\pi\)
0.487939 0.872878i \(-0.337749\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 129.017 223.464i 0.791515 1.37094i −0.133514 0.991047i \(-0.542626\pi\)
0.925029 0.379897i \(-0.124041\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 107.838i 0.645737i −0.946444 0.322868i \(-0.895353\pi\)
0.946444 0.322868i \(-0.104647\pi\)
\(168\) 0 0
\(169\) −151.000 −0.893491
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 21.9344 + 12.6638i 0.126789 + 0.0732014i 0.562053 0.827101i \(-0.310012\pi\)
−0.435264 + 0.900303i \(0.643345\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 48.5470 28.0286i 0.271212 0.156584i −0.358226 0.933635i \(-0.616618\pi\)
0.629438 + 0.777050i \(0.283285\pi\)
\(180\) 0 0
\(181\) −118.307 −0.653627 −0.326814 0.945089i \(-0.605975\pi\)
−0.326814 + 0.945089i \(0.605975\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −458.087 + 264.476i −2.47614 + 1.42960i
\(186\) 0 0
\(187\) 72.1369 124.945i 0.385759 0.668154i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −81.5159 47.0632i −0.426785 0.246404i 0.271191 0.962525i \(-0.412582\pi\)
−0.697976 + 0.716121i \(0.745916\pi\)
\(192\) 0 0
\(193\) 152.689 + 264.466i 0.791136 + 1.37029i 0.925264 + 0.379324i \(0.123843\pi\)
−0.134128 + 0.990964i \(0.542823\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 369.483i 1.87555i 0.347249 + 0.937773i \(0.387116\pi\)
−0.347249 + 0.937773i \(0.612884\pi\)
\(198\) 0 0
\(199\) 134.374 + 232.743i 0.675248 + 1.16956i 0.976396 + 0.215986i \(0.0692965\pi\)
−0.301149 + 0.953577i \(0.597370\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −183.706 + 318.188i −0.896128 + 1.55214i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 323.574i 1.54820i
\(210\) 0 0
\(211\) 296.068 1.40316 0.701582 0.712588i \(-0.252477\pi\)
0.701582 + 0.712588i \(0.252477\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 242.885 + 140.230i 1.12970 + 0.652231i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 57.1617 33.0023i 0.258650 0.149332i
\(222\) 0 0
\(223\) −298.934 −1.34051 −0.670256 0.742130i \(-0.733816\pi\)
−0.670256 + 0.742130i \(0.733816\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 114.912 66.3446i 0.506221 0.292267i −0.225058 0.974345i \(-0.572257\pi\)
0.731279 + 0.682078i \(0.238924\pi\)
\(228\) 0 0
\(229\) 117.647 203.771i 0.513744 0.889831i −0.486129 0.873887i \(-0.661591\pi\)
0.999873 0.0159436i \(-0.00507523\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 82.8813 + 47.8516i 0.355714 + 0.205371i 0.667199 0.744880i \(-0.267493\pi\)
−0.311485 + 0.950251i \(0.600826\pi\)
\(234\) 0 0
\(235\) 326.051 + 564.737i 1.38745 + 2.40313i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 382.397i 1.59999i −0.600009 0.799994i \(-0.704836\pi\)
0.600009 0.799994i \(-0.295164\pi\)
\(240\) 0 0
\(241\) 4.03490 + 6.98865i 0.0167423 + 0.0289985i 0.874275 0.485431i \(-0.161337\pi\)
−0.857533 + 0.514429i \(0.828004\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −74.0169 + 128.201i −0.299664 + 0.519033i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 216.264i 0.861608i 0.902445 + 0.430804i \(0.141770\pi\)
−0.902445 + 0.430804i \(0.858230\pi\)
\(252\) 0 0
\(253\) 164.689 0.650946
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 250.056 + 144.370i 0.972982 + 0.561751i 0.900144 0.435593i \(-0.143461\pi\)
0.0728377 + 0.997344i \(0.476795\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −58.4279 + 33.7334i −0.222159 + 0.128264i −0.606950 0.794740i \(-0.707607\pi\)
0.384790 + 0.923004i \(0.374274\pi\)
\(264\) 0 0
\(265\) 10.4108 0.0392862
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −50.5834 + 29.2043i −0.188042 + 0.108566i −0.591066 0.806623i \(-0.701293\pi\)
0.403023 + 0.915190i \(0.367959\pi\)
\(270\) 0 0
\(271\) 69.3324 120.087i 0.255839 0.443126i −0.709284 0.704923i \(-0.750982\pi\)
0.965123 + 0.261796i \(0.0843149\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −532.826 307.627i −1.93755 1.11865i
\(276\) 0 0
\(277\) 64.3446 + 111.448i 0.232291 + 0.402340i 0.958482 0.285153i \(-0.0920445\pi\)
−0.726191 + 0.687493i \(0.758711\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 225.534i 0.802611i 0.915944 + 0.401306i \(0.131443\pi\)
−0.915944 + 0.401306i \(0.868557\pi\)
\(282\) 0 0
\(283\) 11.7651 + 20.3778i 0.0415729 + 0.0720064i 0.886063 0.463565i \(-0.153430\pi\)
−0.844490 + 0.535571i \(0.820096\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −23.4831 + 40.6738i −0.0812562 + 0.140740i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 301.855i 1.03022i −0.857124 0.515111i \(-0.827751\pi\)
0.857124 0.515111i \(-0.172249\pi\)
\(294\) 0 0
\(295\) −652.102 −2.21051
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 65.2504 + 37.6723i 0.218229 + 0.125994i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −394.149 + 227.562i −1.29229 + 0.746104i
\(306\) 0 0
\(307\) 320.611 1.04434 0.522168 0.852843i \(-0.325123\pi\)
0.522168 + 0.852843i \(0.325123\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −339.400 + 195.953i −1.09132 + 0.630074i −0.933927 0.357463i \(-0.883642\pi\)
−0.157392 + 0.987536i \(0.550309\pi\)
\(312\) 0 0
\(313\) 199.648 345.800i 0.637852 1.10479i −0.348051 0.937476i \(-0.613156\pi\)
0.985903 0.167317i \(-0.0535104\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 496.345 + 286.565i 1.56576 + 0.903990i 0.996655 + 0.0817216i \(0.0260418\pi\)
0.569101 + 0.822268i \(0.307292\pi\)
\(318\) 0 0
\(319\) −121.689 210.772i −0.381471 0.660727i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 542.829i 1.68058i
\(324\) 0 0
\(325\) −140.738 243.766i −0.433041 0.750048i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 211.345 366.060i 0.638503 1.10592i −0.347258 0.937770i \(-0.612887\pi\)
0.985761 0.168151i \(-0.0537795\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 82.7232i 0.246935i
\(336\) 0 0
\(337\) 299.345 0.888263 0.444132 0.895962i \(-0.353512\pi\)
0.444132 + 0.895962i \(0.353512\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −257.508 148.672i −0.755155 0.435989i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 367.908 212.412i 1.06025 0.612137i 0.134751 0.990880i \(-0.456977\pi\)
0.925502 + 0.378742i \(0.123643\pi\)
\(348\) 0 0
\(349\) −182.505 −0.522938 −0.261469 0.965212i \(-0.584207\pi\)
−0.261469 + 0.965212i \(0.584207\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 182.875 105.583i 0.518059 0.299102i −0.218081 0.975931i \(-0.569980\pi\)
0.736140 + 0.676829i \(0.236646\pi\)
\(354\) 0 0
\(355\) −457.338 + 792.133i −1.28828 + 2.23136i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −80.1504 46.2749i −0.223260 0.128899i 0.384199 0.923250i \(-0.374478\pi\)
−0.607459 + 0.794351i \(0.707811\pi\)
\(360\) 0 0
\(361\) −428.223 741.704i −1.18621 2.05458i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 860.612i 2.35784i
\(366\) 0 0
\(367\) 151.808 + 262.940i 0.413646 + 0.716457i 0.995285 0.0969905i \(-0.0309216\pi\)
−0.581639 + 0.813447i \(0.697588\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 173.723 300.897i 0.465746 0.806695i −0.533489 0.845807i \(-0.679119\pi\)
0.999235 + 0.0391117i \(0.0124528\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 111.345i 0.295344i
\(378\) 0 0
\(379\) 233.379 0.615774 0.307887 0.951423i \(-0.400378\pi\)
0.307887 + 0.951423i \(0.400378\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −191.380 110.493i −0.499687 0.288495i 0.228897 0.973451i \(-0.426488\pi\)
−0.728584 + 0.684956i \(0.759821\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 113.442 65.4959i 0.291625 0.168370i −0.347049 0.937847i \(-0.612816\pi\)
0.638675 + 0.769477i \(0.279483\pi\)
\(390\) 0 0
\(391\) 276.283 0.706606
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1230.70 710.544i 3.11569 1.79885i
\(396\) 0 0
\(397\) −98.3238 + 170.302i −0.247667 + 0.428972i −0.962878 0.269937i \(-0.912997\pi\)
0.715211 + 0.698909i \(0.246331\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 550.775 + 317.990i 1.37350 + 0.792993i 0.991368 0.131112i \(-0.0418549\pi\)
0.382137 + 0.924106i \(0.375188\pi\)
\(402\) 0 0
\(403\) −68.0169 117.809i −0.168777 0.292330i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 513.245i 1.26104i
\(408\) 0 0
\(409\) −35.1236 60.8359i −0.0858768 0.148743i 0.819888 0.572524i \(-0.194036\pi\)
−0.905765 + 0.423781i \(0.860702\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −424.757 + 735.701i −1.02351 + 1.77277i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 436.196i 1.04104i −0.853849 0.520520i \(-0.825738\pi\)
0.853849 0.520520i \(-0.174262\pi\)
\(420\) 0 0
\(421\) 7.37852 0.0175262 0.00876309 0.999962i \(-0.497211\pi\)
0.00876309 + 0.999962i \(0.497211\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −893.870 516.076i −2.10322 1.21430i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 173.595 100.225i 0.402774 0.232542i −0.284906 0.958555i \(-0.591963\pi\)
0.687680 + 0.726014i \(0.258629\pi\)
\(432\) 0 0
\(433\) −454.106 −1.04874 −0.524372 0.851489i \(-0.675700\pi\)
−0.524372 + 0.851489i \(0.675700\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −536.626 + 309.821i −1.22798 + 0.708973i
\(438\) 0 0
\(439\) −131.106 + 227.083i −0.298648 + 0.517273i −0.975827 0.218545i \(-0.929869\pi\)
0.677179 + 0.735818i \(0.263202\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −201.499 116.336i −0.454852 0.262609i 0.255025 0.966934i \(-0.417916\pi\)
−0.709877 + 0.704326i \(0.751250\pi\)
\(444\) 0 0
\(445\) −364.362 631.093i −0.818790 1.41819i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 92.0197i 0.204944i 0.994736 + 0.102472i \(0.0326752\pi\)
−0.994736 + 0.102472i \(0.967325\pi\)
\(450\) 0 0
\(451\) −178.251 308.739i −0.395235 0.684566i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 347.412 601.736i 0.760202 1.31671i −0.182544 0.983198i \(-0.558433\pi\)
0.942746 0.333511i \(-0.108233\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 432.476i 0.938127i −0.883164 0.469063i \(-0.844592\pi\)
0.883164 0.469063i \(-0.155408\pi\)
\(462\) 0 0
\(463\) −310.169 −0.669912 −0.334956 0.942234i \(-0.608722\pi\)
−0.334956 + 0.942234i \(0.608722\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −360.097 207.902i −0.771086 0.445186i 0.0621762 0.998065i \(-0.480196\pi\)
−0.833262 + 0.552879i \(0.813529\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −235.672 + 136.065i −0.498250 + 0.287665i
\(474\) 0 0
\(475\) 2314.89 4.87346
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 247.993 143.179i 0.517731 0.298912i −0.218275 0.975887i \(-0.570043\pi\)
0.736006 + 0.676975i \(0.236710\pi\)
\(480\) 0 0
\(481\) 117.404 203.349i 0.244083 0.422763i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 105.547 + 60.9377i 0.217623 + 0.125645i
\(486\) 0 0
\(487\) −242.706 420.379i −0.498370 0.863202i 0.501628 0.865083i \(-0.332735\pi\)
−0.999998 + 0.00188110i \(0.999401\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 138.130i 0.281323i −0.990058 0.140661i \(-0.955077\pi\)
0.990058 0.140661i \(-0.0449229\pi\)
\(492\) 0 0
\(493\) −204.146 353.591i −0.414089 0.717224i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 72.6723 125.872i 0.145636 0.252249i −0.783974 0.620794i \(-0.786811\pi\)
0.929610 + 0.368545i \(0.120144\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 619.115i 1.23084i −0.788198 0.615422i \(-0.788985\pi\)
0.788198 0.615422i \(-0.211015\pi\)
\(504\) 0 0
\(505\) 51.2430 0.101471
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 200.851 + 115.961i 0.394599 + 0.227822i 0.684151 0.729340i \(-0.260173\pi\)
−0.289552 + 0.957162i \(0.593506\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1131.59 + 653.324i −2.19726 + 1.26859i
\(516\) 0 0
\(517\) −632.737 −1.22386
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −45.7558 + 26.4171i −0.0878231 + 0.0507047i −0.543268 0.839559i \(-0.682813\pi\)
0.455445 + 0.890264i \(0.349480\pi\)
\(522\) 0 0
\(523\) 28.3082 49.0313i 0.0541266 0.0937501i −0.837693 0.546142i \(-0.816096\pi\)
0.891819 + 0.452392i \(0.149429\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −431.996 249.413i −0.819727 0.473269i
\(528\) 0 0
\(529\) −106.811 185.002i −0.201911 0.349720i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 163.098i 0.306000i
\(534\) 0 0
\(535\) 712.824 + 1234.65i 1.33238 + 2.30775i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −45.6215 + 79.0187i −0.0843281 + 0.146060i −0.905105 0.425189i \(-0.860208\pi\)
0.820777 + 0.571249i \(0.193541\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 274.196i 0.503112i
\(546\) 0 0
\(547\) 356.825 0.652331 0.326165 0.945313i \(-0.394243\pi\)
0.326165 + 0.945313i \(0.394243\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 793.028 + 457.855i 1.43925 + 0.830953i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −812.850 + 469.299i −1.45934 + 0.842548i −0.998979 0.0451859i \(-0.985612\pi\)
−0.460357 + 0.887734i \(0.652279\pi\)
\(558\) 0 0
\(559\) −124.499 −0.222717
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 877.413 506.574i 1.55846 0.899777i 0.561055 0.827779i \(-0.310396\pi\)
0.997405 0.0719980i \(-0.0229375\pi\)
\(564\) 0 0
\(565\) 696.316 1206.06i 1.23242 2.13461i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 837.689 + 483.640i 1.47221 + 0.849982i 0.999512 0.0312415i \(-0.00994608\pi\)
0.472700 + 0.881223i \(0.343279\pi\)
\(570\) 0 0
\(571\) 315.757 + 546.907i 0.552990 + 0.957806i 0.998057 + 0.0623089i \(0.0198464\pi\)
−0.445067 + 0.895497i \(0.646820\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1178.21i 2.04906i
\(576\) 0 0
\(577\) 307.176 + 532.044i 0.532367 + 0.922087i 0.999286 + 0.0377869i \(0.0120308\pi\)
−0.466918 + 0.884300i \(0.654636\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −5.05084 + 8.74831i −0.00866353 + 0.0150057i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 411.243i 0.700584i −0.936641 0.350292i \(-0.886082\pi\)
0.936641 0.350292i \(-0.113918\pi\)
\(588\) 0 0
\(589\) 1118.76 1.89942
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 330.755 + 190.961i 0.557765 + 0.322026i 0.752248 0.658880i \(-0.228970\pi\)
−0.194483 + 0.980906i \(0.562303\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −212.423 + 122.642i −0.354629 + 0.204745i −0.666722 0.745306i \(-0.732303\pi\)
0.312093 + 0.950052i \(0.398970\pi\)
\(600\) 0 0
\(601\) 168.243 0.279939 0.139970 0.990156i \(-0.455300\pi\)
0.139970 + 0.990156i \(0.455300\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −289.694 + 167.255i −0.478834 + 0.276455i
\(606\) 0 0
\(607\) 279.183 483.560i 0.459940 0.796639i −0.539018 0.842294i \(-0.681204\pi\)
0.998957 + 0.0456558i \(0.0145377\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −250.692 144.737i −0.410298 0.236886i
\(612\) 0 0
\(613\) 130.085 + 225.313i 0.212210 + 0.367559i 0.952406 0.304833i \(-0.0986006\pi\)
−0.740196 + 0.672391i \(0.765267\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 351.007i 0.568893i 0.958692 + 0.284447i \(0.0918099\pi\)
−0.958692 + 0.284447i \(0.908190\pi\)
\(618\) 0 0
\(619\) −158.903 275.229i −0.256710 0.444634i 0.708649 0.705561i \(-0.249305\pi\)
−0.965358 + 0.260927i \(0.915972\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1059.00 + 1834.24i −1.69440 + 2.93478i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 861.021i 1.36887i
\(630\) 0 0
\(631\) 297.966 0.472213 0.236106 0.971727i \(-0.424129\pi\)
0.236106 + 0.971727i \(0.424129\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −987.814 570.314i −1.55561 0.898133i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −659.835 + 380.956i −1.02938 + 0.594315i −0.916808 0.399328i \(-0.869244\pi\)
−0.112576 + 0.993643i \(0.535910\pi\)
\(642\) 0 0
\(643\) 21.5808 0.0335626 0.0167813 0.999859i \(-0.494658\pi\)
0.0167813 + 0.999859i \(0.494658\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 90.5462 52.2768i 0.139948 0.0807988i −0.428391 0.903593i \(-0.640920\pi\)
0.568339 + 0.822794i \(0.307586\pi\)
\(648\) 0 0
\(649\) 316.368 547.966i 0.487471 0.844324i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 311.528 + 179.861i 0.477072 + 0.275438i 0.719195 0.694808i \(-0.244511\pi\)
−0.242123 + 0.970245i \(0.577844\pi\)
\(654\) 0 0
\(655\) 458.757 + 794.591i 0.700392 + 1.21312i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 983.046i 1.49172i 0.666100 + 0.745862i \(0.267962\pi\)
−0.666100 + 0.745862i \(0.732038\pi\)
\(660\) 0 0
\(661\) 255.813 + 443.081i 0.387009 + 0.670319i 0.992046 0.125879i \(-0.0401750\pi\)
−0.605037 + 0.796197i \(0.706842\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 233.034 403.627i 0.349376 0.605137i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 441.608i 0.658135i
\(672\) 0 0
\(673\) −49.8644 −0.0740928 −0.0370464 0.999314i \(-0.511795\pi\)
−0.0370464 + 0.999314i \(0.511795\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −488.482 282.025i −0.721540 0.416581i 0.0937794 0.995593i \(-0.470105\pi\)
−0.815319 + 0.579012i \(0.803438\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 114.683 66.2124i 0.167911 0.0969434i −0.413689 0.910418i \(-0.635760\pi\)
0.581600 + 0.813475i \(0.302427\pi\)
\(684\) 0 0
\(685\) 1644.55 2.40081
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.00232 + 2.31074i −0.00580888 + 0.00335376i
\(690\) 0 0
\(691\) 551.056 954.457i 0.797476 1.38127i −0.123779 0.992310i \(-0.539501\pi\)
0.921255 0.388959i \(-0.127165\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1748.14 1009.29i −2.51531 1.45222i
\(696\) 0 0
\(697\) −299.034 517.942i −0.429030 0.743102i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 302.130i 0.430999i −0.976504 0.215500i \(-0.930862\pi\)
0.976504 0.215500i \(-0.0691380\pi\)
\(702\) 0 0
\(703\) 965.540 + 1672.36i 1.37346 + 2.37890i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.31074 10.9305i 0.00890090 0.0154168i −0.861541 0.507689i \(-0.830500\pi\)
0.870442 + 0.492272i \(0.163833\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 569.412i 0.798615i
\(714\) 0 0
\(715\) 376.034 0.525922
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 657.840 + 379.804i 0.914938 + 0.528239i 0.882016 0.471219i \(-0.156186\pi\)
0.0329210 + 0.999458i \(0.489519\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1507.89 + 870.580i −2.07985 + 1.20080i
\(726\) 0 0
\(727\) −829.041 −1.14036 −0.570179 0.821520i \(-0.693126\pi\)
−0.570179 + 0.821520i \(0.693126\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −395.364 + 228.264i −0.540854 + 0.312262i
\(732\) 0 0
\(733\) −677.260 + 1173.05i −0.923957 + 1.60034i −0.130727 + 0.991418i \(0.541731\pi\)
−0.793230 + 0.608922i \(0.791602\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −69.5129 40.1333i −0.0943187 0.0544549i
\(738\) 0 0
\(739\) −17.0169 29.4742i −0.0230270 0.0398839i 0.854282 0.519809i \(-0.173997\pi\)
−0.877309 + 0.479925i \(0.840664\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 61.0163i 0.0821215i −0.999157 0.0410607i \(-0.986926\pi\)
0.999157 0.0410607i \(-0.0130737\pi\)
\(744\) 0 0
\(745\) −721.796 1250.19i −0.968854 1.67810i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 70.0678 121.361i 0.0932993 0.161599i −0.815598 0.578619i \(-0.803592\pi\)
0.908898 + 0.417019i \(0.136925\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 115.013i 0.152335i
\(756\) 0 0
\(757\) 518.000 0.684280 0.342140 0.939649i \(-0.388848\pi\)
0.342140 + 0.939649i \(0.388848\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −117.905 68.0725i −0.154934 0.0894514i 0.420528 0.907279i \(-0.361845\pi\)
−0.575463 + 0.817828i \(0.695178\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 250.692 144.737i 0.326848 0.188706i
\(768\) 0 0
\(769\) 237.244 0.308510 0.154255 0.988031i \(-0.450702\pi\)
0.154255 + 0.988031i \(0.450702\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −747.641 + 431.651i −0.967194 + 0.558410i −0.898380 0.439220i \(-0.855255\pi\)
−0.0688142 + 0.997629i \(0.521922\pi\)
\(774\) 0 0
\(775\) −1063.62 + 1842.24i −1.37241 + 2.37709i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1161.63 + 670.667i 1.49118 + 0.860934i
\(780\) 0 0
\(781\) −443.757 768.610i −0.568191 0.984135i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1536.42i 1.95723i
\(786\) 0 0
\(787\) −382.421 662.372i −0.485922 0.841642i 0.513947 0.857822i \(-0.328183\pi\)
−0.999869 + 0.0161799i \(0.994850\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 101.017 174.966i 0.127386 0.220639i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1413.60i 1.77365i 0.462106 + 0.886825i \(0.347094\pi\)
−0.462106 + 0.886825i \(0.652906\pi\)
\(798\) 0 0
\(799\) −1061.48 −1.32851
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 723.178 + 417.527i 0.900596 + 0.519959i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 624.645 360.639i 0.772120 0.445784i −0.0615102 0.998106i \(-0.519592\pi\)
0.833630 + 0.552323i \(0.186258\pi\)
\(810\) 0 0
\(811\) −928.795 −1.14525 −0.572623 0.819819i \(-0.694074\pi\)
−0.572623 + 0.819819i \(0.694074\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2135.74 1233.07i 2.62054 1.51297i
\(816\) 0 0
\(817\) 511.945 886.715i 0.626616 1.08533i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 876.578 + 506.093i 1.06770 + 0.616434i 0.927551 0.373697i \(-0.121910\pi\)
0.140145 + 0.990131i \(0.455243\pi\)
\(822\) 0 0
\(823\) −665.791 1153.18i −0.808980 1.40120i −0.913571 0.406680i \(-0.866686\pi\)
0.104590 0.994515i \(-0.466647\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 96.8217i 0.117076i −0.998285 0.0585379i \(-0.981356\pi\)
0.998285 0.0585379i \(-0.0186438\pi\)
\(828\) 0 0
\(829\) −137.422 238.023i −0.165769 0.287120i 0.771159 0.636642i \(-0.219677\pi\)
−0.936928 + 0.349522i \(0.886344\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 515.328 892.574i 0.617159 1.06895i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1196.53i 1.42614i −0.701091 0.713072i \(-0.747303\pi\)
0.701091 0.713072i \(-0.252697\pi\)
\(840\) 0 0
\(841\) 152.243 0.181026
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1249.82 721.587i −1.47908 0.853949i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 851.182 491.430i 1.00021 0.577473i
\(852\) 0 0
\(853\) 761.454 0.892678 0.446339 0.894864i \(-0.352728\pi\)
0.446339 + 0.894864i \(0.352728\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 177.030 102.208i 0.206569 0.119263i −0.393147 0.919476i \(-0.628614\pi\)
0.599716 + 0.800213i \(0.295280\pi\)
\(858\) 0 0
\(859\) −28.2244 + 48.8860i −0.0328572 + 0.0569104i −0.881986 0.471275i \(-0.843794\pi\)
0.849129 + 0.528185i \(0.177127\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 768.748 + 443.837i 0.890785 + 0.514295i 0.874199 0.485567i \(-0.161387\pi\)
0.0165860 + 0.999862i \(0.494720\pi\)
\(864\) 0 0
\(865\) 121.034 + 209.637i 0.139924 + 0.242355i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1378.89i 1.58675i
\(870\) 0 0
\(871\) −18.3608 31.8019i −0.0210801 0.0365119i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −38.5706 + 66.8063i −0.0439802 + 0.0761759i −0.887178 0.461428i \(-0.847337\pi\)
0.843197 + 0.537604i \(0.180671\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 918.740i 1.04284i 0.853301 + 0.521419i \(0.174597\pi\)
−0.853301 + 0.521419i \(0.825403\pi\)
\(882\) 0 0
\(883\) 6.72316 0.00761399 0.00380700 0.999993i \(-0.498788\pi\)
0.00380700 + 0.999993i \(0.498788\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −136.610 78.8719i −0.154014 0.0889199i 0.421013 0.907055i \(-0.361675\pi\)
−0.575026 + 0.818135i \(0.695008\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2061.72 1190.33i 2.30876 1.33296i
\(894\) 0 0
\(895\) 535.763 0.598618
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −728.743 + 420.740i −0.810615 + 0.468009i
\(900\) 0 0
\(901\) −8.47330 + 14.6762i −0.00940433 + 0.0162888i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −979.222 565.354i −1.08201 0.624700i
\(906\) 0 0
\(907\) −306.085 530.154i −0.337469 0.584514i 0.646487 0.762925i \(-0.276238\pi\)
−0.983956 + 0.178411i \(0.942904\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 570.650i 0.626399i −0.949687 0.313200i \(-0.898599\pi\)
0.949687 0.313200i \(-0.101401\pi\)
\(912\) 0 0
\(913\) −412.143 713.853i −0.451417 0.781876i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 633.463 1097.19i 0.689296 1.19390i −0.282770 0.959188i \(-0.591253\pi\)
0.972066 0.234708i \(-0.0754135\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 406.034i 0.439907i
\(924\) 0 0
\(925\) −3671.82 −3.96953
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1192.81 688.668i −1.28397 0.741300i −0.306398 0.951904i \(-0.599124\pi\)
−0.977572 + 0.210604i \(0.932457\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1194.15 689.444i 1.27717 0.737373i
\(936\) 0 0
\(937\) −754.751 −0.805497 −0.402748 0.915311i \(-0.631945\pi\)
−0.402748 + 0.915311i \(0.631945\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 658.289 380.063i 0.699563 0.403893i −0.107622 0.994192i \(-0.534324\pi\)
0.807185 + 0.590299i \(0.200990\pi\)
\(942\) 0 0
\(943\) 341.349 591.233i 0.361982 0.626971i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −33.0058 19.0559i −0.0348531 0.0201224i 0.482472 0.875911i \(-0.339739\pi\)
−0.517325 + 0.855789i \(0.673072\pi\)
\(948\) 0 0
\(949\) 191.017 + 330.851i 0.201282 + 0.348631i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1140.96i 1.19723i −0.801039 0.598613i \(-0.795719\pi\)
0.801039 0.598613i \(-0.204281\pi\)
\(954\) 0 0
\(955\) −449.804 779.083i −0.470999 0.815794i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −33.5339 + 58.0824i −0.0348948 + 0.0604395i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2918.64i 3.02449i
\(966\) 0 0
\(967\) 1212.79 1.25418 0.627089 0.778947i \(-0.284246\pi\)
0.627089 + 0.778947i \(0.284246\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −411.462 237.557i −0.423750 0.244652i 0.272930 0.962034i \(-0.412007\pi\)
−0.696681 + 0.717381i \(0.745341\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1156.85 + 667.908i −1.18409 + 0.683632i −0.956956 0.290233i \(-0.906267\pi\)
−0.227129 + 0.973865i \(0.572934\pi\)
\(978\) 0 0
\(979\) 707.083 0.722250
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 94.1799 54.3748i 0.0958086 0.0553151i −0.451330 0.892357i \(-0.649050\pi\)
0.547139 + 0.837042i \(0.315717\pi\)
\(984\) 0 0
\(985\) −1765.65 + 3058.20i −1.79254 + 3.10477i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −451.310 260.564i −0.456330 0.263462i
\(990\) 0 0
\(991\) −275.429 477.058i −0.277931 0.481390i 0.692940 0.720996i \(-0.256315\pi\)
−0.970870 + 0.239605i \(0.922982\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2568.55i 2.58145i
\(996\) 0 0
\(997\) 35.7308 + 61.8876i 0.0358384 + 0.0620739i 0.883388 0.468642i \(-0.155257\pi\)
−0.847550 + 0.530716i \(0.821923\pi\)
\(998\) 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 1764.3.bk.g.1745.7 16
3.2 odd 2 inner 1764.3.bk.g.1745.2 16
7.2 even 3 1764.3.c.h.197.1 8
7.3 odd 6 inner 1764.3.bk.g.557.8 16
7.4 even 3 inner 1764.3.bk.g.557.2 16
7.5 odd 6 1764.3.c.h.197.7 yes 8
7.6 odd 2 inner 1764.3.bk.g.1745.1 16
21.2 odd 6 1764.3.c.h.197.8 yes 8
21.5 even 6 1764.3.c.h.197.2 yes 8
21.11 odd 6 inner 1764.3.bk.g.557.7 16
21.17 even 6 inner 1764.3.bk.g.557.1 16
21.20 even 2 inner 1764.3.bk.g.1745.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.3.c.h.197.1 8 7.2 even 3
1764.3.c.h.197.2 yes 8 21.5 even 6
1764.3.c.h.197.7 yes 8 7.5 odd 6
1764.3.c.h.197.8 yes 8 21.2 odd 6
1764.3.bk.g.557.1 16 21.17 even 6 inner
1764.3.bk.g.557.2 16 7.4 even 3 inner
1764.3.bk.g.557.7 16 21.11 odd 6 inner
1764.3.bk.g.557.8 16 7.3 odd 6 inner
1764.3.bk.g.1745.1 16 7.6 odd 2 inner
1764.3.bk.g.1745.2 16 3.2 odd 2 inner
1764.3.bk.g.1745.7 16 1.1 even 1 trivial
1764.3.bk.g.1745.8 16 21.20 even 2 inner