Properties

Label 3240.2.q.b.1081.1
Level $3240$
Weight $2$
Character 3240.1081
Analytic conductor $25.872$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.8715302549\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1080)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1081.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3240.1081
Dual form 3240.2.q.b.2161.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-2.00000 - 3.46410i) q^{11} +(1.00000 - 1.73205i) q^{13} +5.00000 q^{17} -5.00000 q^{19} +(-0.500000 + 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(1.00000 + 1.73205i) q^{29} +(-3.50000 + 6.06218i) q^{31} +2.00000 q^{35} -6.00000 q^{37} +(-2.00000 - 3.46410i) q^{43} +(-2.00000 - 3.46410i) q^{47} +(1.50000 - 2.59808i) q^{49} +9.00000 q^{53} +4.00000 q^{55} +(-7.00000 + 12.1244i) q^{59} +(5.50000 + 9.52628i) q^{61} +(1.00000 + 1.73205i) q^{65} +(-7.00000 + 12.1244i) q^{67} -12.0000 q^{73} +(-4.00000 + 6.92820i) q^{77} +(1.50000 + 2.59808i) q^{79} +(0.500000 + 0.866025i) q^{83} +(-2.50000 + 4.33013i) q^{85} -4.00000 q^{91} +(2.50000 - 4.33013i) q^{95} +(-8.00000 - 13.8564i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{5} - 2q^{7} + O(q^{10}) \) \( 2q - q^{5} - 2q^{7} - 4q^{11} + 2q^{13} + 10q^{17} - 10q^{19} - q^{23} - q^{25} + 2q^{29} - 7q^{31} + 4q^{35} - 12q^{37} - 4q^{43} - 4q^{47} + 3q^{49} + 18q^{53} + 8q^{55} - 14q^{59} + 11q^{61} + 2q^{65} - 14q^{67} - 24q^{73} - 8q^{77} + 3q^{79} + q^{83} - 5q^{85} - 8q^{91} + 5q^{95} - 16q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) −1.00000 1.73205i −0.377964 0.654654i 0.612801 0.790237i \(-0.290043\pi\)
−0.990766 + 0.135583i \(0.956709\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 3.46410i −0.603023 1.04447i −0.992361 0.123371i \(-0.960630\pi\)
0.389338 0.921095i \(-0.372704\pi\)
\(12\) 0 0
\(13\) 1.00000 1.73205i 0.277350 0.480384i −0.693375 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.00000 1.21268 0.606339 0.795206i \(-0.292637\pi\)
0.606339 + 0.795206i \(0.292637\pi\)
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.500000 + 0.866025i −0.104257 + 0.180579i −0.913434 0.406986i \(-0.866580\pi\)
0.809177 + 0.587565i \(0.199913\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.00000 + 1.73205i 0.185695 + 0.321634i 0.943811 0.330487i \(-0.107213\pi\)
−0.758115 + 0.652121i \(0.773880\pi\)
\(30\) 0 0
\(31\) −3.50000 + 6.06218i −0.628619 + 1.08880i 0.359211 + 0.933257i \(0.383046\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) −2.00000 3.46410i −0.304997 0.528271i 0.672264 0.740312i \(-0.265322\pi\)
−0.977261 + 0.212041i \(0.931989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.00000 3.46410i −0.291730 0.505291i 0.682489 0.730896i \(-0.260898\pi\)
−0.974219 + 0.225605i \(0.927564\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.00000 + 12.1244i −0.911322 + 1.57846i −0.0991242 + 0.995075i \(0.531604\pi\)
−0.812198 + 0.583382i \(0.801729\pi\)
\(60\) 0 0
\(61\) 5.50000 + 9.52628i 0.704203 + 1.21972i 0.966978 + 0.254858i \(0.0820288\pi\)
−0.262776 + 0.964857i \(0.584638\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.00000 + 1.73205i 0.124035 + 0.214834i
\(66\) 0 0
\(67\) −7.00000 + 12.1244i −0.855186 + 1.48123i 0.0212861 + 0.999773i \(0.493224\pi\)
−0.876472 + 0.481452i \(0.840109\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 + 6.92820i −0.455842 + 0.789542i
\(78\) 0 0
\(79\) 1.50000 + 2.59808i 0.168763 + 0.292306i 0.937985 0.346675i \(-0.112689\pi\)
−0.769222 + 0.638982i \(0.779356\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.500000 + 0.866025i 0.0548821 + 0.0950586i 0.892161 0.451717i \(-0.149188\pi\)
−0.837279 + 0.546776i \(0.815855\pi\)
\(84\) 0 0
\(85\) −2.50000 + 4.33013i −0.271163 + 0.469668i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.50000 4.33013i 0.256495 0.444262i
\(96\) 0 0
\(97\) −8.00000 13.8564i −0.812277 1.40690i −0.911267 0.411816i \(-0.864894\pi\)
0.0989899 0.995088i \(-0.468439\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 10.3923i −0.597022 1.03407i −0.993258 0.115924i \(-0.963017\pi\)
0.396236 0.918149i \(-0.370316\pi\)
\(102\) 0 0
\(103\) −2.00000 + 3.46410i −0.197066 + 0.341328i −0.947576 0.319531i \(-0.896475\pi\)
0.750510 + 0.660859i \(0.229808\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −19.0000 −1.81987 −0.909935 0.414751i \(-0.863869\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.00000 + 5.19615i −0.282216 + 0.488813i −0.971930 0.235269i \(-0.924403\pi\)
0.689714 + 0.724082i \(0.257736\pi\)
\(114\) 0 0
\(115\) −0.500000 0.866025i −0.0466252 0.0807573i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.00000 8.66025i −0.458349 0.793884i
\(120\) 0 0
\(121\) −2.50000 + 4.33013i −0.227273 + 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 6.00000 0.532414 0.266207 0.963916i \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 9.00000 15.5885i 0.786334 1.36197i −0.141865 0.989886i \(-0.545310\pi\)
0.928199 0.372084i \(-0.121357\pi\)
\(132\) 0 0
\(133\) 5.00000 + 8.66025i 0.433555 + 0.750939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.50000 + 14.7224i 0.726204 + 1.25782i 0.958477 + 0.285171i \(0.0920506\pi\)
−0.232273 + 0.972651i \(0.574616\pi\)
\(138\) 0 0
\(139\) −6.00000 + 10.3923i −0.508913 + 0.881464i 0.491033 + 0.871141i \(0.336619\pi\)
−0.999947 + 0.0103230i \(0.996714\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.00000 13.8564i 0.655386 1.13516i −0.326411 0.945228i \(-0.605840\pi\)
0.981797 0.189933i \(-0.0608272\pi\)
\(150\) 0 0
\(151\) −4.00000 6.92820i −0.325515 0.563809i 0.656101 0.754673i \(-0.272204\pi\)
−0.981617 + 0.190864i \(0.938871\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.50000 6.06218i −0.281127 0.486926i
\(156\) 0 0
\(157\) −8.00000 + 13.8564i −0.638470 + 1.10586i 0.347299 + 0.937754i \(0.387099\pi\)
−0.985769 + 0.168107i \(0.946235\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.50000 2.59808i 0.116073 0.201045i −0.802135 0.597143i \(-0.796303\pi\)
0.918208 + 0.396098i \(0.129636\pi\)
\(168\) 0 0
\(169\) 4.50000 + 7.79423i 0.346154 + 0.599556i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.50000 + 11.2583i 0.494186 + 0.855955i 0.999978 0.00670064i \(-0.00213290\pi\)
−0.505792 + 0.862656i \(0.668800\pi\)
\(174\) 0 0
\(175\) −1.00000 + 1.73205i −0.0755929 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −19.0000 −1.41226 −0.706129 0.708083i \(-0.749560\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 5.19615i 0.220564 0.382029i
\(186\) 0 0
\(187\) −10.0000 17.3205i −0.731272 1.26660i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −7.00000 12.1244i −0.506502 0.877288i −0.999972 0.00752447i \(-0.997605\pi\)
0.493469 0.869763i \(-0.335728\pi\)
\(192\) 0 0
\(193\) −5.00000 + 8.66025i −0.359908 + 0.623379i −0.987945 0.154805i \(-0.950525\pi\)
0.628037 + 0.778183i \(0.283859\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.00000 −0.356235 −0.178118 0.984009i \(-0.557001\pi\)
−0.178118 + 0.984009i \(0.557001\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.00000 3.46410i 0.140372 0.243132i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0000 + 17.3205i 0.691714 + 1.19808i
\(210\) 0 0
\(211\) 9.50000 16.4545i 0.654007 1.13277i −0.328135 0.944631i \(-0.606420\pi\)
0.982142 0.188142i \(-0.0602466\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 14.0000 0.950382
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.00000 8.66025i 0.336336 0.582552i
\(222\) 0 0
\(223\) 5.00000 + 8.66025i 0.334825 + 0.579934i 0.983451 0.181173i \(-0.0579895\pi\)
−0.648626 + 0.761107i \(0.724656\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.50000 2.59808i −0.0995585 0.172440i 0.811943 0.583736i \(-0.198410\pi\)
−0.911502 + 0.411296i \(0.865076\pi\)
\(228\) 0 0
\(229\) 14.5000 25.1147i 0.958187 1.65963i 0.231287 0.972886i \(-0.425707\pi\)
0.726900 0.686743i \(-0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.00000 5.19615i 0.194054 0.336111i −0.752536 0.658551i \(-0.771170\pi\)
0.946590 + 0.322440i \(0.104503\pi\)
\(240\) 0 0
\(241\) −5.50000 9.52628i −0.354286 0.613642i 0.632709 0.774389i \(-0.281943\pi\)
−0.986996 + 0.160748i \(0.948609\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.50000 + 2.59808i 0.0958315 + 0.165985i
\(246\) 0 0
\(247\) −5.00000 + 8.66025i −0.318142 + 0.551039i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −13.5000 + 23.3827i −0.842107 + 1.45857i 0.0460033 + 0.998941i \(0.485352\pi\)
−0.888110 + 0.459631i \(0.847982\pi\)
\(258\) 0 0
\(259\) 6.00000 + 10.3923i 0.372822 + 0.645746i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 + 20.7846i 0.739952 + 1.28163i 0.952517 + 0.304487i \(0.0984850\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(264\) 0 0
\(265\) −4.50000 + 7.79423i −0.276433 + 0.478796i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.0000 −0.609711 −0.304855 0.952399i \(-0.598608\pi\)
−0.304855 + 0.952399i \(0.598608\pi\)
\(270\) 0 0
\(271\) −9.00000 −0.546711 −0.273356 0.961913i \(-0.588134\pi\)
−0.273356 + 0.961913i \(0.588134\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.00000 + 3.46410i −0.120605 + 0.208893i
\(276\) 0 0
\(277\) 2.00000 + 3.46410i 0.120168 + 0.208138i 0.919834 0.392308i \(-0.128323\pi\)
−0.799666 + 0.600446i \(0.794990\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.00000 + 12.1244i 0.417585 + 0.723278i 0.995696 0.0926797i \(-0.0295433\pi\)
−0.578111 + 0.815958i \(0.696210\pi\)
\(282\) 0 0
\(283\) −4.00000 + 6.92820i −0.237775 + 0.411839i −0.960076 0.279741i \(-0.909752\pi\)
0.722300 + 0.691580i \(0.243085\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.50000 + 6.06218i −0.204472 + 0.354156i −0.949964 0.312358i \(-0.898881\pi\)
0.745492 + 0.666514i \(0.232214\pi\)
\(294\) 0 0
\(295\) −7.00000 12.1244i −0.407556 0.705907i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.00000 + 1.73205i 0.0578315 + 0.100167i
\(300\) 0 0
\(301\) −4.00000 + 6.92820i −0.230556 + 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −11.0000 −0.629858
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.0000 19.0526i 0.623753 1.08037i −0.365028 0.930997i \(-0.618941\pi\)
0.988781 0.149375i \(-0.0477261\pi\)
\(312\) 0 0
\(313\) −4.00000 6.92820i −0.226093 0.391605i 0.730554 0.682855i \(-0.239262\pi\)
−0.956647 + 0.291250i \(0.905929\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 13.5000 + 23.3827i 0.758236 + 1.31330i 0.943750 + 0.330661i \(0.107272\pi\)
−0.185514 + 0.982642i \(0.559395\pi\)
\(318\) 0 0
\(319\) 4.00000 6.92820i 0.223957 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −25.0000 −1.39104
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.00000 + 6.92820i −0.220527 + 0.381964i
\(330\) 0 0
\(331\) −6.00000 10.3923i −0.329790 0.571213i 0.652680 0.757634i \(-0.273645\pi\)
−0.982470 + 0.186421i \(0.940311\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.00000 12.1244i −0.382451 0.662424i
\(336\) 0 0
\(337\) −11.0000 + 19.0526i −0.599208 + 1.03786i 0.393730 + 0.919226i \(0.371184\pi\)
−0.992938 + 0.118633i \(0.962149\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 28.0000 1.51629
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10.0000 + 17.3205i −0.536828 + 0.929814i 0.462244 + 0.886753i \(0.347044\pi\)
−0.999072 + 0.0430610i \(0.986289\pi\)
\(348\) 0 0
\(349\) 1.50000 + 2.59808i 0.0802932 + 0.139072i 0.903376 0.428850i \(-0.141081\pi\)
−0.823083 + 0.567922i \(0.807748\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.0000 22.5167i −0.691920 1.19844i −0.971208 0.238233i \(-0.923432\pi\)
0.279288 0.960207i \(-0.409902\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.00000 10.3923i 0.314054 0.543958i
\(366\) 0 0
\(367\) 11.0000 + 19.0526i 0.574195 + 0.994535i 0.996129 + 0.0879086i \(0.0280183\pi\)
−0.421933 + 0.906627i \(0.638648\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −9.00000 15.5885i −0.467257 0.809312i
\(372\) 0 0
\(373\) 16.0000 27.7128i 0.828449 1.43492i −0.0708063 0.997490i \(-0.522557\pi\)
0.899255 0.437425i \(-0.144109\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) −23.0000 −1.18143 −0.590715 0.806880i \(-0.701154\pi\)
−0.590715 + 0.806880i \(0.701154\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.5000 23.3827i 0.689818 1.19480i −0.282079 0.959391i \(-0.591024\pi\)
0.971897 0.235408i \(-0.0756427\pi\)
\(384\) 0 0
\(385\) −4.00000 6.92820i −0.203859 0.353094i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 18.0000 + 31.1769i 0.912636 + 1.58073i 0.810326 + 0.585980i \(0.199290\pi\)
0.102311 + 0.994753i \(0.467376\pi\)
\(390\) 0 0
\(391\) −2.50000 + 4.33013i −0.126430 + 0.218984i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −3.00000 −0.150946
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.0000 25.9808i 0.749064 1.29742i −0.199207 0.979957i \(-0.563837\pi\)
0.948272 0.317460i \(-0.102830\pi\)
\(402\) 0 0
\(403\) 7.00000 + 12.1244i 0.348695 + 0.603957i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.0000 + 20.7846i 0.594818 + 1.03025i
\(408\) 0 0
\(409\) −12.5000 + 21.6506i −0.618085 + 1.07056i 0.371750 + 0.928333i \(0.378758\pi\)
−0.989835 + 0.142222i \(0.954575\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 28.0000 1.37779
\(414\) 0 0
\(415\) −1.00000 −0.0490881
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −18.0000 + 31.1769i −0.879358 + 1.52309i −0.0273103 + 0.999627i \(0.508694\pi\)
−0.852047 + 0.523465i \(0.824639\pi\)
\(420\) 0 0
\(421\) −1.50000 2.59808i −0.0731055 0.126622i 0.827155 0.561973i \(-0.189958\pi\)
−0.900261 + 0.435351i \(0.856624\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.50000 4.33013i −0.121268 0.210042i
\(426\) 0 0
\(427\) 11.0000 19.0526i 0.532327 0.922018i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.50000 4.33013i 0.119591 0.207138i
\(438\) 0 0
\(439\) 8.50000 + 14.7224i 0.405683 + 0.702663i 0.994401 0.105675i \(-0.0337004\pi\)
−0.588718 + 0.808339i \(0.700367\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −19.5000 33.7750i −0.926473 1.60470i −0.789175 0.614168i \(-0.789492\pi\)
−0.137298 0.990530i \(-0.543842\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.00000 3.46410i 0.0937614 0.162400i
\(456\) 0 0
\(457\) −14.0000 24.2487i −0.654892 1.13431i −0.981921 0.189292i \(-0.939381\pi\)
0.327028 0.945015i \(-0.393953\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −21.0000 36.3731i −0.978068 1.69406i −0.669417 0.742887i \(-0.733456\pi\)
−0.308651 0.951175i \(-0.599877\pi\)
\(462\) 0 0
\(463\) 9.00000 15.5885i 0.418265 0.724457i −0.577500 0.816391i \(-0.695972\pi\)
0.995765 + 0.0919339i \(0.0293048\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −13.0000 −0.601568 −0.300784 0.953692i \(-0.597248\pi\)
−0.300784 + 0.953692i \(0.597248\pi\)
\(468\) 0 0
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.00000 + 13.8564i −0.367840 + 0.637118i
\(474\) 0 0
\(475\) 2.50000 + 4.33013i 0.114708 + 0.198680i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.0000 24.2487i −0.639676 1.10795i −0.985504 0.169654i \(-0.945735\pi\)
0.345827 0.938298i \(-0.387598\pi\)
\(480\) 0 0
\(481\) −6.00000 + 10.3923i −0.273576 + 0.473848i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 16.0000 0.726523
\(486\) 0 0
\(487\) 40.0000 1.81257 0.906287 0.422664i \(-0.138905\pi\)
0.906287 + 0.422664i \(0.138905\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.00000 5.19615i 0.135388 0.234499i −0.790358 0.612646i \(-0.790105\pi\)
0.925746 + 0.378147i \(0.123439\pi\)
\(492\) 0 0
\(493\) 5.00000 + 8.66025i 0.225189 + 0.390038i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.5000 21.6506i 0.559577 0.969216i −0.437955 0.898997i \(-0.644297\pi\)
0.997532 0.0702185i \(-0.0223697\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15.0000 −0.668817 −0.334408 0.942428i \(-0.608537\pi\)
−0.334408 + 0.942428i \(0.608537\pi\)
\(504\) 0 0
\(505\) 12.0000 0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −14.0000 + 24.2487i −0.620539 + 1.07481i 0.368846 + 0.929490i \(0.379753\pi\)
−0.989385 + 0.145315i \(0.953580\pi\)
\(510\) 0 0
\(511\) 12.0000 + 20.7846i 0.530849 + 0.919457i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.00000 3.46410i −0.0881305 0.152647i
\(516\) 0 0
\(517\) −8.00000 + 13.8564i −0.351840 + 0.609404i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 40.0000 1.75243 0.876216 0.481919i \(-0.160060\pi\)
0.876216 + 0.481919i \(0.160060\pi\)
\(522\) 0 0
\(523\) 10.0000 0.437269 0.218635 0.975807i \(-0.429840\pi\)
0.218635 + 0.975807i \(0.429840\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −17.5000 + 30.3109i −0.762312 + 1.32036i
\(528\) 0 0
\(529\) 11.0000 + 19.0526i 0.478261 + 0.828372i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 6.00000 10.3923i 0.259403 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 22.0000 0.945854 0.472927 0.881102i \(-0.343197\pi\)
0.472927 + 0.881102i \(0.343197\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 9.50000 16.4545i 0.406935 0.704833i
\(546\) 0 0
\(547\) 14.0000 + 24.2487i 0.598597 + 1.03680i 0.993028 + 0.117875i \(0.0376081\pi\)
−0.394432 + 0.918925i \(0.629059\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.00000 8.66025i −0.213007 0.368939i
\(552\) 0 0
\(553\) 3.00000 5.19615i 0.127573 0.220963i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.0000 −0.423714 −0.211857 0.977301i \(-0.567951\pi\)
−0.211857 + 0.977301i \(0.567951\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.00000 + 3.46410i −0.0842900 + 0.145994i −0.905088 0.425223i \(-0.860196\pi\)
0.820798 + 0.571218i \(0.193529\pi\)
\(564\) 0 0
\(565\) −3.00000 5.19615i −0.126211 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.00000 + 8.66025i 0.209611 + 0.363057i 0.951592 0.307364i \(-0.0994469\pi\)
−0.741981 + 0.670421i \(0.766114\pi\)
\(570\) 0 0
\(571\) 2.50000 4.33013i 0.104622 0.181210i −0.808962 0.587861i \(-0.799970\pi\)
0.913584 + 0.406651i \(0.133303\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) −16.0000 −0.666089 −0.333044 0.942911i \(-0.608076\pi\)
−0.333044 + 0.942911i \(0.608076\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.00000 1.73205i 0.0414870 0.0718576i
\(582\) 0 0
\(583\) −18.0000 31.1769i −0.745484 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.50000 + 12.9904i 0.309558 + 0.536170i 0.978266 0.207355i \(-0.0664855\pi\)
−0.668708 + 0.743525i \(0.733152\pi\)
\(588\) 0 0
\(589\) 17.5000 30.3109i 0.721075 1.24894i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −9.00000 −0.369586 −0.184793 0.982777i \(-0.559161\pi\)
−0.184793 + 0.982777i \(0.559161\pi\)
\(594\) 0 0
\(595\) 10.0000 0.409960
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.00000 + 15.5885i −0.367730 + 0.636927i −0.989210 0.146503i \(-0.953198\pi\)
0.621480 + 0.783430i \(0.286532\pi\)
\(600\) 0 0
\(601\) 5.50000 + 9.52628i 0.224350 + 0.388585i 0.956124 0.292962i \(-0.0946409\pi\)
−0.731774 + 0.681547i \(0.761308\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.50000 4.33013i −0.101639 0.176045i
\(606\) 0 0
\(607\) 18.0000 31.1769i 0.730597 1.26543i −0.226031 0.974120i \(-0.572575\pi\)
0.956628 0.291312i \(-0.0940917\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −16.5000 + 28.5788i −0.664265 + 1.15054i 0.315219 + 0.949019i \(0.397922\pi\)
−0.979484 + 0.201522i \(0.935411\pi\)
\(618\) 0 0
\(619\) −16.0000 27.7128i −0.643094 1.11387i −0.984738 0.174042i \(-0.944317\pi\)
0.341644 0.939829i \(-0.389016\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −30.0000 −1.19618
\(630\) 0 0
\(631\) 13.0000 0.517522 0.258761 0.965941i \(-0.416686\pi\)
0.258761 + 0.965941i \(0.416686\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.00000 + 5.19615i −0.119051 + 0.206203i
\(636\) 0 0
\(637\) −3.00000 5.19615i −0.118864 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) −3.00000 + 5.19615i −0.118308 + 0.204916i −0.919097 0.394030i \(-0.871080\pi\)
0.800789 + 0.598947i \(0.204414\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.0000 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(648\) 0 0
\(649\) 56.0000 2.19819
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −20.5000 + 35.5070i −0.802227 + 1.38950i 0.115920 + 0.993259i \(0.463018\pi\)
−0.918147 + 0.396239i \(0.870315\pi\)
\(654\) 0 0
\(655\) 9.00000 + 15.5885i 0.351659 + 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −9.00000 15.5885i −0.350590 0.607240i 0.635763 0.771885i \(-0.280686\pi\)
−0.986353 + 0.164644i \(0.947352\pi\)
\(660\) 0 0
\(661\) 21.0000 36.3731i 0.816805 1.41475i −0.0912190 0.995831i \(-0.529076\pi\)
0.908024 0.418917i \(-0.137590\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.0000 −0.387783
\(666\) 0 0
\(667\) −2.00000 −0.0774403
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 22.0000 38.1051i 0.849301 1.47103i
\(672\) 0 0
\(673\) −15.0000 25.9808i −0.578208 1.00148i −0.995685 0.0927975i \(-0.970419\pi\)
0.417477 0.908687i \(-0.362914\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.00000 + 1.73205i 0.0384331 + 0.0665681i 0.884602 0.466347i \(-0.154430\pi\)
−0.846169 + 0.532915i \(0.821097\pi\)
\(678\) 0 0
\(679\) −16.0000 + 27.7128i −0.614024 + 1.06352i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) −17.0000 −0.649537
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.00000 15.5885i 0.342873 0.593873i
\(690\) 0 0
\(691\) 9.50000 + 16.4545i 0.361397 + 0.625958i 0.988191 0.153227i \(-0.0489666\pi\)
−0.626794 + 0.779185i \(0.715633\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.00000 10.3923i −0.227593 0.394203i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.0000 + 20.7846i −0.451306 + 0.781686i
\(708\) 0 0
\(709\) −13.0000 22.5167i −0.488225 0.845631i 0.511683 0.859174i \(-0.329022\pi\)
−0.999908 + 0.0135434i \(0.995689\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.50000 6.06218i −0.131076 0.227030i
\(714\) 0 0
\(715\) 4.00000 6.92820i 0.149592 0.259100i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 46.0000 1.71551 0.857755 0.514058i \(-0.171858\pi\)
0.857755 + 0.514058i \(0.171858\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.00000 1.73205i 0.0371391 0.0643268i
\(726\) 0 0
\(727\) −16.0000 27.7128i −0.593407 1.02781i −0.993770 0.111454i \(-0.964449\pi\)
0.400362 0.916357i \(-0.368884\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10.0000 17.3205i −0.369863 0.640622i
\(732\) 0 0
\(733\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 56.0000 2.06279
\(738\) 0 0
\(739\) −15.0000 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(744\) 0 0
\(745\) 8.00000 + 13.8564i 0.293097 + 0.507659i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 12.0000 + 20.7846i 0.438470 + 0.759453i
\(750\) 0 0
\(751\) 1.50000 2.59808i 0.0547358 0.0948051i −0.837359 0.546653i \(-0.815902\pi\)
0.892095 + 0.451848i \(0.149235\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 25.9808i 0.543750 0.941802i −0.454935 0.890525i \(-0.650337\pi\)
0.998684 0.0512772i \(-0.0163292\pi\)
\(762\) 0 0
\(763\) 19.0000 + 32.9090i 0.687846 + 1.19138i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.0000 + 24.2487i 0.505511 + 0.875570i
\(768\) 0 0
\(769\) 17.5000 30.3109i 0.631066 1.09304i −0.356268 0.934384i \(-0.615951\pi\)
0.987334 0.158655i \(-0.0507157\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −15.0000 −0.539513 −0.269756 0.962929i \(-0.586943\pi\)
−0.269756 + 0.962929i \(0.586943\pi\)
\(774\) 0 0
\(775\) 7.00000 0.251447
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8.00000 13.8564i −0.285532 0.494556i
\(786\) 0 0
\(787\) 20.0000 34.6410i 0.712923 1.23482i −0.250832 0.968031i \(-0.580704\pi\)
0.963755 0.266788i \(-0.0859624\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) 22.0000 0.781243
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.50000 + 2.59808i −0.0531327 + 0.0920286i −0.891368 0.453279i \(-0.850254\pi\)
0.838236 + 0.545308i \(0.183587\pi\)
\(798\) 0 0
\(799\) −10.0000 17.3205i −0.353775 0.612756i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 24.0000 + 41.5692i 0.846942 + 1.46695i
\(804\) 0