Properties

Label 3240.2.q.s.1081.1
Level $3240$
Weight $2$
Character 3240.1081
Analytic conductor $25.872$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3240,2,Mod(1081,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3240.1081");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
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

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: \(25.8715302549\)
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: \(\Z[a_1, \ldots, a_{5}]\)
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.s.2161.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{11} -3.00000 q^{17} -1.00000 q^{19} +(1.50000 - 2.59808i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(2.00000 + 3.46410i) q^{29} +(2.50000 - 4.33013i) q^{31} +10.0000 q^{37} +(3.00000 - 5.19615i) q^{41} +(3.00000 + 5.19615i) q^{43} +(4.00000 + 6.92820i) q^{47} +(3.50000 - 6.06218i) q^{49} -3.00000 q^{53} +2.00000 q^{55} +(-2.50000 - 4.33013i) q^{61} +(1.00000 - 1.73205i) q^{67} -2.00000 q^{71} +6.00000 q^{73} +(5.50000 + 9.52628i) q^{79} +(4.50000 + 7.79423i) q^{83} +(-1.50000 + 2.59808i) q^{85} -10.0000 q^{89} +(-0.500000 + 0.866025i) q^{95} +(-4.00000 - 6.92820i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + 2 q^{11} - 6 q^{17} - 2 q^{19} + 3 q^{23} - q^{25} + 4 q^{29} + 5 q^{31} + 20 q^{37} + 6 q^{41} + 6 q^{43} + 8 q^{47} + 7 q^{49} - 6 q^{53} + 4 q^{55} - 5 q^{61} + 2 q^{67} - 4 q^{71} + 12 q^{73} + 11 q^{79} + 9 q^{83} - 3 q^{85} - 20 q^{89} - q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.50000 2.59808i 0.312772 0.541736i −0.666190 0.745782i \(-0.732076\pi\)
0.978961 + 0.204046i \(0.0654092\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 + 3.46410i 0.371391 + 0.643268i 0.989780 0.142605i \(-0.0455477\pi\)
−0.618389 + 0.785872i \(0.712214\pi\)
\(30\) 0 0
\(31\) 2.50000 4.33013i 0.449013 0.777714i −0.549309 0.835619i \(-0.685109\pi\)
0.998322 + 0.0579057i \(0.0184423\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 10.0000 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 5.19615i 0.468521 0.811503i −0.530831 0.847477i \(-0.678120\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) 0 0
\(43\) 3.00000 + 5.19615i 0.457496 + 0.792406i 0.998828 0.0484030i \(-0.0154132\pi\)
−0.541332 + 0.840809i \(0.682080\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.00000 + 6.92820i 0.583460 + 1.01058i 0.995066 + 0.0992202i \(0.0316348\pi\)
−0.411606 + 0.911362i \(0.635032\pi\)
\(48\) 0 0
\(49\) 3.50000 6.06218i 0.500000 0.866025i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) −2.50000 4.33013i −0.320092 0.554416i 0.660415 0.750901i \(-0.270381\pi\)
−0.980507 + 0.196485i \(0.937047\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.00000 1.73205i 0.122169 0.211604i −0.798454 0.602056i \(-0.794348\pi\)
0.920623 + 0.390453i \(0.127682\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 5.50000 + 9.52628i 0.618798 + 1.07179i 0.989705 + 0.143120i \(0.0457135\pi\)
−0.370907 + 0.928670i \(0.620953\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.50000 + 7.79423i 0.493939 + 0.855528i 0.999976 0.00698436i \(-0.00222321\pi\)
−0.506036 + 0.862512i \(0.668890\pi\)
\(84\) 0 0
\(85\) −1.50000 + 2.59808i −0.162698 + 0.281801i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.500000 + 0.866025i −0.0512989 + 0.0888523i
\(96\) 0 0
\(97\) −4.00000 6.92820i −0.406138 0.703452i 0.588315 0.808632i \(-0.299792\pi\)
−0.994453 + 0.105180i \(0.966458\pi\)
\(98\) 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
\(103\) 6.00000 10.3923i 0.591198 1.02398i −0.402874 0.915255i \(-0.631989\pi\)
0.994071 0.108729i \(-0.0346780\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) 9.00000 0.862044 0.431022 0.902342i \(-0.358153\pi\)
0.431022 + 0.902342i \(0.358153\pi\)
\(110\) 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
\(115\) −1.50000 2.59808i −0.139876 0.242272i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 22.0000 1.95218 0.976092 0.217357i \(-0.0697436\pi\)
0.976092 + 0.217357i \(0.0697436\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.00000 8.66025i 0.436852 0.756650i −0.560593 0.828092i \(-0.689427\pi\)
0.997445 + 0.0714417i \(0.0227600\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.50000 6.06218i −0.299025 0.517927i 0.676888 0.736086i \(-0.263328\pi\)
−0.975913 + 0.218159i \(0.929995\pi\)
\(138\) 0 0
\(139\) 10.0000 17.3205i 0.848189 1.46911i −0.0346338 0.999400i \(-0.511026\pi\)
0.882823 0.469706i \(-0.155640\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 4.00000 0.332182
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.00000 1.73205i 0.0819232 0.141895i −0.822153 0.569267i \(-0.807227\pi\)
0.904076 + 0.427372i \(0.140560\pi\)
\(150\) 0 0
\(151\) 4.00000 + 6.92820i 0.325515 + 0.563809i 0.981617 0.190864i \(-0.0611289\pi\)
−0.656101 + 0.754673i \(0.727796\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.50000 4.33013i −0.200805 0.347804i
\(156\) 0 0
\(157\) −9.00000 + 15.5885i −0.718278 + 1.24409i 0.243403 + 0.969925i \(0.421736\pi\)
−0.961681 + 0.274169i \(0.911597\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.00000 −0.156652 −0.0783260 0.996928i \(-0.524958\pi\)
−0.0783260 + 0.996928i \(0.524958\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.50000 + 14.7224i −0.657750 + 1.13926i 0.323447 + 0.946246i \(0.395158\pi\)
−0.981197 + 0.193010i \(0.938175\pi\)
\(168\) 0 0
\(169\) 6.50000 + 11.2583i 0.500000 + 0.866025i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.50000 16.4545i −0.722272 1.25101i −0.960087 0.279701i \(-0.909765\pi\)
0.237816 0.971310i \(-0.423569\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0000 1.79384 0.896922 0.442189i \(-0.145798\pi\)
0.896922 + 0.442189i \(0.145798\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.00000 8.66025i 0.367607 0.636715i
\(186\) 0 0
\(187\) −3.00000 5.19615i −0.219382 0.379980i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3.00000 5.19615i −0.217072 0.375980i 0.736839 0.676068i \(-0.236317\pi\)
−0.953912 + 0.300088i \(0.902984\pi\)
\(192\) 0 0
\(193\) 10.0000 17.3205i 0.719816 1.24676i −0.241257 0.970461i \(-0.577560\pi\)
0.961073 0.276296i \(-0.0891071\pi\)
\(194\) 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
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −3.00000 5.19615i −0.209529 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.00000 1.73205i −0.0691714 0.119808i
\(210\) 0 0
\(211\) 1.50000 2.59808i 0.103264 0.178859i −0.809763 0.586756i \(-0.800405\pi\)
0.913028 + 0.407898i \(0.133738\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 1.73205i −0.0669650 0.115987i 0.830599 0.556871i \(-0.187998\pi\)
−0.897564 + 0.440884i \(0.854665\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.50000 + 14.7224i 0.564165 + 0.977162i 0.997127 + 0.0757500i \(0.0241351\pi\)
−0.432962 + 0.901412i \(0.642532\pi\)
\(228\) 0 0
\(229\) 6.50000 11.2583i 0.429532 0.743971i −0.567300 0.823511i \(-0.692012\pi\)
0.996832 + 0.0795401i \(0.0253452\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.0000 + 25.9808i −0.970269 + 1.68056i −0.275533 + 0.961292i \(0.588854\pi\)
−0.694737 + 0.719264i \(0.744479\pi\)
\(240\) 0 0
\(241\) 8.50000 + 14.7224i 0.547533 + 0.948355i 0.998443 + 0.0557856i \(0.0177663\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.50000 6.06218i −0.223607 0.387298i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −11.5000 + 19.9186i −0.717350 + 1.24249i 0.244696 + 0.969600i \(0.421312\pi\)
−0.962046 + 0.272887i \(0.912021\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000 + 10.3923i 0.369976 + 0.640817i 0.989561 0.144112i \(-0.0460326\pi\)
−0.619586 + 0.784929i \(0.712699\pi\)
\(264\) 0 0
\(265\) −1.50000 + 2.59808i −0.0921443 + 0.159599i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) 3.00000 0.182237 0.0911185 0.995840i \(-0.470956\pi\)
0.0911185 + 0.995840i \(0.470956\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.00000 1.73205i 0.0603023 0.104447i
\(276\) 0 0
\(277\) −11.0000 19.0526i −0.660926 1.14476i −0.980373 0.197153i \(-0.936830\pi\)
0.319447 0.947604i \(-0.396503\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −14.0000 24.2487i −0.835170 1.44656i −0.893892 0.448282i \(-0.852036\pi\)
0.0587220 0.998274i \(-0.481297\pi\)
\(282\) 0 0
\(283\) −1.00000 + 1.73205i −0.0594438 + 0.102960i −0.894216 0.447636i \(-0.852266\pi\)
0.834772 + 0.550596i \(0.185599\pi\)
\(284\) 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
\(293\) −5.50000 + 9.52628i −0.321313 + 0.556531i −0.980759 0.195221i \(-0.937458\pi\)
0.659446 + 0.751752i \(0.270791\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −5.00000 −0.286299
\(306\) 0 0
\(307\) 26.0000 1.48390 0.741949 0.670456i \(-0.233902\pi\)
0.741949 + 0.670456i \(0.233902\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.0000 17.3205i 0.567048 0.982156i −0.429808 0.902920i \(-0.641419\pi\)
0.996856 0.0792356i \(-0.0252479\pi\)
\(312\) 0 0
\(313\) 10.0000 + 17.3205i 0.565233 + 0.979013i 0.997028 + 0.0770410i \(0.0245472\pi\)
−0.431795 + 0.901972i \(0.642119\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.50000 + 9.52628i 0.308911 + 0.535049i 0.978124 0.208021i \(-0.0667022\pi\)
−0.669214 + 0.743070i \(0.733369\pi\)
\(318\) 0 0
\(319\) −4.00000 + 6.92820i −0.223957 + 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 6.92820i −0.219860 0.380808i 0.734905 0.678170i \(-0.237227\pi\)
−0.954765 + 0.297361i \(0.903893\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.00000 1.73205i −0.0546358 0.0946320i
\(336\) 0 0
\(337\) −4.00000 + 6.92820i −0.217894 + 0.377403i −0.954164 0.299285i \(-0.903252\pi\)
0.736270 + 0.676688i \(0.236585\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.00000 + 10.3923i −0.322097 + 0.557888i −0.980921 0.194409i \(-0.937721\pi\)
0.658824 + 0.752297i \(0.271054\pi\)
\(348\) 0 0
\(349\) 3.50000 + 6.06218i 0.187351 + 0.324501i 0.944366 0.328896i \(-0.106677\pi\)
−0.757015 + 0.653397i \(0.773343\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −5.00000 8.66025i −0.266123 0.460939i 0.701734 0.712439i \(-0.252409\pi\)
−0.967857 + 0.251500i \(0.919076\pi\)
\(354\) 0 0
\(355\) −1.00000 + 1.73205i −0.0530745 + 0.0919277i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6.00000 0.316668 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.00000 5.19615i 0.157027 0.271979i
\(366\) 0 0
\(367\) −8.00000 13.8564i −0.417597 0.723299i 0.578101 0.815966i \(-0.303794\pi\)
−0.995697 + 0.0926670i \(0.970461\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.00000 + 3.46410i −0.103556 + 0.179364i −0.913147 0.407630i \(-0.866355\pi\)
0.809591 + 0.586994i \(0.199689\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 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
\(383\) 1.50000 2.59808i 0.0766464 0.132755i −0.825155 0.564907i \(-0.808912\pi\)
0.901801 + 0.432151i \(0.142245\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 9.00000 + 15.5885i 0.456318 + 0.790366i 0.998763 0.0497253i \(-0.0158346\pi\)
−0.542445 + 0.840091i \(0.682501\pi\)
\(390\) 0 0
\(391\) −4.50000 + 7.79423i −0.227575 + 0.394171i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.0000 29.4449i 0.848939 1.47041i −0.0332161 0.999448i \(-0.510575\pi\)
0.882156 0.470958i \(-0.156092\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.0000 + 17.3205i 0.495682 + 0.858546i
\(408\) 0 0
\(409\) 15.5000 26.8468i 0.766426 1.32749i −0.173064 0.984911i \(-0.555367\pi\)
0.939490 0.342578i \(-0.111300\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 9.00000 0.441793
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3.00000 + 5.19615i −0.146560 + 0.253849i −0.929954 0.367677i \(-0.880153\pi\)
0.783394 + 0.621525i \(0.213487\pi\)
\(420\) 0 0
\(421\) 0.500000 + 0.866025i 0.0243685 + 0.0422075i 0.877952 0.478748i \(-0.158909\pi\)
−0.853584 + 0.520955i \(0.825576\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.50000 + 2.59808i 0.0727607 + 0.126025i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) −24.0000 −1.15337 −0.576683 0.816968i \(-0.695653\pi\)
−0.576683 + 0.816968i \(0.695653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.50000 + 2.59808i −0.0717547 + 0.124283i
\(438\) 0 0
\(439\) −3.50000 6.06218i −0.167046 0.289332i 0.770334 0.637641i \(-0.220089\pi\)
−0.937380 + 0.348309i \(0.886756\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14.5000 + 25.1147i 0.688916 + 1.19324i 0.972189 + 0.234198i \(0.0752464\pi\)
−0.283273 + 0.959039i \(0.591420\pi\)
\(444\) 0 0
\(445\) −5.00000 + 8.66025i −0.237023 + 0.410535i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 + 13.8564i 0.374224 + 0.648175i 0.990211 0.139581i \(-0.0445757\pi\)
−0.615986 + 0.787757i \(0.711242\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −17.0000 29.4449i −0.791769 1.37138i −0.924871 0.380282i \(-0.875827\pi\)
0.133102 0.991102i \(-0.457506\pi\)
\(462\) 0 0
\(463\) −7.00000 + 12.1244i −0.325318 + 0.563467i −0.981577 0.191069i \(-0.938805\pi\)
0.656259 + 0.754536i \(0.272138\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.00000 0.323921 0.161961 0.986797i \(-0.448218\pi\)
0.161961 + 0.986797i \(0.448218\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.00000 + 10.3923i −0.275880 + 0.477839i
\(474\) 0 0
\(475\) 0.500000 + 0.866025i 0.0229416 + 0.0397360i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.00000 + 8.66025i 0.228456 + 0.395697i 0.957351 0.288929i \(-0.0932990\pi\)
−0.728895 + 0.684626i \(0.759966\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 21.0000 36.3731i 0.947717 1.64149i 0.197499 0.980303i \(-0.436718\pi\)
0.750218 0.661190i \(-0.229948\pi\)
\(492\) 0 0
\(493\) −6.00000 10.3923i −0.270226 0.468046i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −21.5000 + 37.2391i −0.962472 + 1.66705i −0.246214 + 0.969216i \(0.579187\pi\)
−0.716258 + 0.697835i \(0.754147\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −35.0000 −1.56057 −0.780286 0.625422i \(-0.784927\pi\)
−0.780286 + 0.625422i \(0.784927\pi\)
\(504\) 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 \(-0.620247\pi\)
0.989385 0.145315i \(-0.0464195\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.00000 10.3923i −0.264392 0.457940i
\(516\) 0 0
\(517\) −8.00000 + 13.8564i −0.351840 + 0.609404i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.50000 + 12.9904i −0.326705 + 0.565870i
\(528\) 0 0
\(529\) 7.00000 + 12.1244i 0.304348 + 0.527146i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 2.00000 3.46410i 0.0864675 0.149766i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 14.0000 0.603023
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.50000 7.79423i 0.192759 0.333868i
\(546\) 0 0
\(547\) −9.00000 15.5885i −0.384812 0.666514i 0.606931 0.794755i \(-0.292400\pi\)
−0.991743 + 0.128240i \(0.959067\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.00000 3.46410i −0.0852029 0.147576i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14.0000 24.2487i 0.590030 1.02196i −0.404198 0.914671i \(-0.632449\pi\)
0.994228 0.107290i \(-0.0342173\pi\)
\(564\) 0 0
\(565\) 3.00000 + 5.19615i 0.126211 + 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.0000 + 20.7846i 0.503066 + 0.871336i 0.999994 + 0.00354413i \(0.00112814\pi\)
−0.496928 + 0.867792i \(0.665539\pi\)
\(570\) 0 0
\(571\) 4.50000 7.79423i 0.188319 0.326178i −0.756371 0.654143i \(-0.773029\pi\)
0.944690 + 0.327965i \(0.106363\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) −46.0000 −1.91501 −0.957503 0.288425i \(-0.906868\pi\)
−0.957503 + 0.288425i \(0.906868\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −3.00000 5.19615i −0.124247 0.215203i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.5000 + 19.9186i 0.474656 + 0.822128i 0.999579 0.0290218i \(-0.00923921\pi\)
−0.524923 + 0.851150i \(0.675906\pi\)
\(588\) 0 0
\(589\) −2.50000 + 4.33013i −0.103011 + 0.178420i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.00000 0.123195 0.0615976 0.998101i \(-0.480380\pi\)
0.0615976 + 0.998101i \(0.480380\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −20.0000 + 34.6410i −0.817178 + 1.41539i 0.0905757 + 0.995890i \(0.471129\pi\)
−0.907754 + 0.419504i \(0.862204\pi\)
\(600\) 0 0
\(601\) −8.50000 14.7224i −0.346722 0.600541i 0.638943 0.769254i \(-0.279372\pi\)
−0.985665 + 0.168714i \(0.946039\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.50000 6.06218i −0.142295 0.246463i
\(606\) 0 0
\(607\) −5.00000 + 8.66025i −0.202944 + 0.351509i −0.949476 0.313841i \(-0.898384\pi\)
0.746532 + 0.665350i \(0.231718\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.5000 + 21.6506i −0.503231 + 0.871622i 0.496762 + 0.867887i \(0.334522\pi\)
−0.999993 + 0.00373492i \(0.998811\pi\)
\(618\) 0 0
\(619\) 2.00000 + 3.46410i 0.0803868 + 0.139234i 0.903416 0.428765i \(-0.141051\pi\)
−0.823029 + 0.567999i \(0.807718\pi\)
\(620\) 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
\(631\) −19.0000 −0.756378 −0.378189 0.925728i \(-0.623453\pi\)
−0.378189 + 0.925728i \(0.623453\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 11.0000 19.0526i 0.436522 0.756078i
\(636\) 0 0
\(637\) 0 0
\(638\) 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
\(643\) −20.0000 + 34.6410i −0.788723 + 1.36611i 0.138027 + 0.990429i \(0.455924\pi\)
−0.926750 + 0.375680i \(0.877409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −39.0000 −1.53325 −0.766624 0.642096i \(-0.778065\pi\)
−0.766624 + 0.642096i \(0.778065\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.50000 9.52628i 0.215232 0.372792i −0.738113 0.674678i \(-0.764283\pi\)
0.953344 + 0.301885i \(0.0976160\pi\)
\(654\) 0 0
\(655\) −5.00000 8.66025i −0.195366 0.338384i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.00000 + 3.46410i 0.0779089 + 0.134942i 0.902348 0.431009i \(-0.141842\pi\)
−0.824439 + 0.565951i \(0.808509\pi\)
\(660\) 0 0
\(661\) −3.00000 + 5.19615i −0.116686 + 0.202107i −0.918453 0.395531i \(-0.870561\pi\)
0.801766 + 0.597638i \(0.203894\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0000 0.464642
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.00000 8.66025i 0.193023 0.334325i
\(672\) 0 0
\(673\) 4.00000 + 6.92820i 0.154189 + 0.267063i 0.932763 0.360489i \(-0.117390\pi\)
−0.778575 + 0.627552i \(0.784057\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.0000 + 43.3013i 0.960828 + 1.66420i 0.720429 + 0.693529i \(0.243945\pi\)
0.240399 + 0.970674i \(0.422722\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23.0000 0.880071 0.440035 0.897980i \(-0.354966\pi\)
0.440035 + 0.897980i \(0.354966\pi\)
\(684\) 0 0
\(685\) −7.00000 −0.267456
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −6.50000 11.2583i −0.247272 0.428287i 0.715496 0.698617i \(-0.246201\pi\)
−0.962768 + 0.270330i \(0.912867\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −10.0000 17.3205i −0.379322 0.657004i
\(696\) 0 0
\(697\) −9.00000 + 15.5885i −0.340899 + 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 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
\(713\) −7.50000 12.9904i −0.280877 0.486494i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.00000 3.46410i 0.0742781 0.128654i
\(726\) 0 0
\(727\) 22.0000 + 38.1051i 0.815935 + 1.41324i 0.908655 + 0.417548i \(0.137111\pi\)
−0.0927199 + 0.995692i \(0.529556\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −9.00000 15.5885i −0.332877 0.576560i
\(732\) 0 0
\(733\) 4.00000 6.92820i 0.147743 0.255899i −0.782650 0.622462i \(-0.786132\pi\)
0.930393 + 0.366563i \(0.119466\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −19.0000 −0.698926 −0.349463 0.936950i \(-0.613636\pi\)
−0.349463 + 0.936950i \(0.613636\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −8.00000 + 13.8564i −0.293492 + 0.508342i −0.974633 0.223810i \(-0.928151\pi\)
0.681141 + 0.732152i \(0.261484\pi\)
\(744\) 0 0
\(745\) −1.00000 1.73205i −0.0366372 0.0634574i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −26.5000 + 45.8993i −0.966999 + 1.67489i −0.262852 + 0.964836i \(0.584663\pi\)
−0.704146 + 0.710055i \(0.748670\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −50.0000 −1.81728 −0.908640 0.417579i \(-0.862879\pi\)
−0.908640 + 0.417579i \(0.862879\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 38.1051i 0.797499 1.38131i −0.123741 0.992315i \(-0.539489\pi\)
0.921240 0.388995i \(-0.127178\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −10.5000 + 18.1865i −0.378640 + 0.655823i −0.990865 0.134860i \(-0.956941\pi\)
0.612225 + 0.790684i \(0.290275\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3.00000 −0.107903 −0.0539513 0.998544i \(-0.517182\pi\)
−0.0539513 + 0.998544i \(0.517182\pi\)
\(774\) 0 0
\(775\) −5.00000 −0.179605
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.00000 + 5.19615i −0.107486 + 0.186171i
\(780\) 0 0
\(781\) −2.00000 3.46410i −0.0715656 0.123955i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.00000 + 15.5885i 0.321224 + 0.556376i
\(786\) 0 0
\(787\) 19.0000 32.9090i 0.677277 1.17308i −0.298521 0.954403i \(-0.596493\pi\)
0.975798 0.218675i \(-0.0701734\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −11.5000 + 19.9186i −0.407351 + 0.705552i −0.994592 0.103860i \(-0.966881\pi\)
0.587241 + 0.809412i \(0.300214\pi\)
\(798\) 0 0
\(799\) −12.0000 20.7846i −0.424529 0.735307i
\(800\) 0