Properties

Label 1620.2.i.j.1081.1
Level $1620$
Weight $2$
Character 1620.1081
Analytic conductor $12.936$
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] = [1620,2,Mod(541,1620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1620, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1620.541");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.i (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: \(12.9357651274\)
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 540)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1081.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1620.1081
Dual form 1620.2.i.j.541.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{7} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{7} +(3.00000 + 5.19615i) q^{11} +(0.500000 - 0.866025i) q^{13} -1.00000 q^{19} +(3.00000 - 5.19615i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(3.00000 + 5.19615i) q^{29} +(-4.00000 + 6.92820i) q^{31} +1.00000 q^{35} -7.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +(6.00000 + 10.3923i) q^{47} +(3.00000 - 5.19615i) q^{49} +6.00000 q^{53} +6.00000 q^{55} +(-5.50000 - 9.52628i) q^{61} +(-0.500000 - 0.866025i) q^{65} +(3.50000 - 6.06218i) q^{67} +6.00000 q^{71} +11.0000 q^{73} +(-3.00000 + 5.19615i) q^{77} +(0.500000 + 0.866025i) q^{79} +(3.00000 + 5.19615i) q^{83} +12.0000 q^{89} +1.00000 q^{91} +(-0.500000 + 0.866025i) q^{95} +(6.50000 + 11.2583i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} + q^{7} + 6 q^{11} + q^{13} - 2 q^{19} + 6 q^{23} - q^{25} + 6 q^{29} - 8 q^{31} + 2 q^{35} - 14 q^{37} - 6 q^{41} + 4 q^{43} + 12 q^{47} + 6 q^{49} + 12 q^{53} + 12 q^{55} - 11 q^{61} - q^{65} + 7 q^{67} + 12 q^{71} + 22 q^{73} - 6 q^{77} + q^{79} + 6 q^{83} + 24 q^{89} + 2 q^{91} - q^{95} + 13 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) 0.500000 + 0.866025i 0.188982 + 0.327327i 0.944911 0.327327i \(-0.106148\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 + 5.19615i 0.904534 + 1.56670i 0.821541 + 0.570149i \(0.193114\pi\)
0.0829925 + 0.996550i \(0.473552\pi\)
\(12\) 0 0
\(13\) 0.500000 0.866025i 0.138675 0.240192i −0.788320 0.615265i \(-0.789049\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 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
\(23\) 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i \(-0.618211\pi\)
0.988436 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i \(0.421802\pi\)
−0.961625 + 0.274367i \(0.911532\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 + 5.19615i −0.468521 + 0.811503i −0.999353 0.0359748i \(-0.988546\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) 2.00000 + 3.46410i 0.304997 + 0.528271i 0.977261 0.212041i \(-0.0680112\pi\)
−0.672264 + 0.740312i \(0.734678\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 + 10.3923i 0.875190 + 1.51587i 0.856560 + 0.516047i \(0.172597\pi\)
0.0186297 + 0.999826i \(0.494070\pi\)
\(48\) 0 0
\(49\) 3.00000 5.19615i 0.428571 0.742307i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 6.00000 0.809040
\(56\) 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
\(61\) −5.50000 9.52628i −0.704203 1.21972i −0.966978 0.254858i \(-0.917971\pi\)
0.262776 0.964857i \(-0.415362\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.500000 0.866025i −0.0620174 0.107417i
\(66\) 0 0
\(67\) 3.50000 6.06218i 0.427593 0.740613i −0.569066 0.822292i \(-0.692695\pi\)
0.996659 + 0.0816792i \(0.0260283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.00000 + 5.19615i −0.341882 + 0.592157i
\(78\) 0 0
\(79\) 0.500000 + 0.866025i 0.0562544 + 0.0974355i 0.892781 0.450490i \(-0.148751\pi\)
−0.836527 + 0.547926i \(0.815418\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.00000 + 5.19615i 0.329293 + 0.570352i 0.982372 0.186938i \(-0.0598564\pi\)
−0.653079 + 0.757290i \(0.726523\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.500000 + 0.866025i −0.0512989 + 0.0888523i
\(96\) 0 0
\(97\) 6.50000 + 11.2583i 0.659975 + 1.14311i 0.980622 + 0.195911i \(0.0627665\pi\)
−0.320647 + 0.947199i \(0.603900\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 6.50000 11.2583i 0.640464 1.10932i −0.344865 0.938652i \(-0.612075\pi\)
0.985329 0.170664i \(-0.0545913\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.00000 + 15.5885i −0.846649 + 1.46644i 0.0375328 + 0.999295i \(0.488050\pi\)
−0.884182 + 0.467143i \(0.845283\pi\)
\(114\) 0 0
\(115\) −3.00000 5.19615i −0.279751 0.484544i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −12.5000 + 21.6506i −1.13636 + 1.96824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i \(-0.657689\pi\)
0.999602 0.0281993i \(-0.00897729\pi\)
\(132\) 0 0
\(133\) −0.500000 0.866025i −0.0433555 0.0750939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 15.5885i −0.768922 1.33181i −0.938148 0.346235i \(-0.887460\pi\)
0.169226 0.985577i \(-0.445873\pi\)
\(138\) 0 0
\(139\) −2.50000 + 4.33013i −0.212047 + 0.367277i −0.952355 0.304991i \(-0.901346\pi\)
0.740308 + 0.672268i \(0.234680\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −6.00000 + 10.3923i −0.491539 + 0.851371i −0.999953 0.00974235i \(-0.996899\pi\)
0.508413 + 0.861113i \(0.330232\pi\)
\(150\) 0 0
\(151\) −5.50000 9.52628i −0.447584 0.775238i 0.550645 0.834740i \(-0.314382\pi\)
−0.998228 + 0.0595022i \(0.981049\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 + 6.92820i 0.321288 + 0.556487i
\(156\) 0 0
\(157\) −1.00000 + 1.73205i −0.0798087 + 0.138233i −0.903167 0.429289i \(-0.858764\pi\)
0.823359 + 0.567521i \(0.192098\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) 0 0
\(163\) 11.0000 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.00000 10.3923i 0.464294 0.804181i −0.534875 0.844931i \(-0.679641\pi\)
0.999169 + 0.0407502i \(0.0129748\pi\)
\(168\) 0 0
\(169\) 6.00000 + 10.3923i 0.461538 + 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.00000 + 10.3923i 0.456172 + 0.790112i 0.998755 0.0498898i \(-0.0158870\pi\)
−0.542583 + 0.840002i \(0.682554\pi\)
\(174\) 0 0
\(175\) 0.500000 0.866025i 0.0377964 0.0654654i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.50000 + 6.06218i −0.257325 + 0.445700i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 10.3923i −0.434145 0.751961i 0.563081 0.826402i \(-0.309616\pi\)
−0.997225 + 0.0744412i \(0.976283\pi\)
\(192\) 0 0
\(193\) 9.50000 16.4545i 0.683825 1.18442i −0.289980 0.957033i \(-0.593649\pi\)
0.973805 0.227387i \(-0.0730182\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) 5.00000 0.354441 0.177220 0.984171i \(-0.443289\pi\)
0.177220 + 0.984171i \(0.443289\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00000 + 5.19615i −0.210559 + 0.364698i
\(204\) 0 0
\(205\) 3.00000 + 5.19615i 0.209529 + 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.00000 5.19615i −0.207514 0.359425i
\(210\) 0 0
\(211\) −8.50000 + 14.7224i −0.585164 + 1.01353i 0.409691 + 0.912224i \(0.365637\pi\)
−0.994855 + 0.101310i \(0.967697\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 8.00000 + 13.8564i 0.535720 + 0.927894i 0.999128 + 0.0417488i \(0.0132929\pi\)
−0.463409 + 0.886145i \(0.653374\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.00000 + 5.19615i 0.199117 + 0.344881i 0.948242 0.317547i \(-0.102859\pi\)
−0.749125 + 0.662428i \(0.769526\pi\)
\(228\) 0 0
\(229\) 5.00000 8.66025i 0.330409 0.572286i −0.652183 0.758062i \(-0.726147\pi\)
0.982592 + 0.185776i \(0.0594799\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −24.0000 −1.57229 −0.786146 0.618041i \(-0.787927\pi\)
−0.786146 + 0.618041i \(0.787927\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.0000 25.9808i 0.970269 1.68056i 0.275533 0.961292i \(-0.411146\pi\)
0.694737 0.719264i \(-0.255521\pi\)
\(240\) 0 0
\(241\) −8.50000 14.7224i −0.547533 0.948355i −0.998443 0.0557856i \(-0.982234\pi\)
0.450910 0.892570i \(-0.351100\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.00000 5.19615i −0.191663 0.331970i
\(246\) 0 0
\(247\) −0.500000 + 0.866025i −0.0318142 + 0.0551039i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 0 0
\(253\) 36.0000 2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) −3.50000 6.06218i −0.217479 0.376685i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.00000 10.3923i −0.369976 0.640817i 0.619586 0.784929i \(-0.287301\pi\)
−0.989561 + 0.144112i \(0.953967\pi\)
\(264\) 0 0
\(265\) 3.00000 5.19615i 0.184289 0.319197i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −24.0000 −1.46331 −0.731653 0.681677i \(-0.761251\pi\)
−0.731653 + 0.681677i \(0.761251\pi\)
\(270\) 0 0
\(271\) −25.0000 −1.51864 −0.759321 0.650716i \(-0.774469\pi\)
−0.759321 + 0.650716i \(0.774469\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 5.19615i 0.180907 0.313340i
\(276\) 0 0
\(277\) −1.00000 1.73205i −0.0600842 0.104069i 0.834419 0.551131i \(-0.185804\pi\)
−0.894503 + 0.447062i \(0.852470\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(282\) 0 0
\(283\) 14.0000 24.2487i 0.832214 1.44144i −0.0640654 0.997946i \(-0.520407\pi\)
0.896279 0.443491i \(-0.146260\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 15.0000 25.9808i 0.876309 1.51781i 0.0209480 0.999781i \(-0.493332\pi\)
0.855361 0.518032i \(-0.173335\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.00000 5.19615i −0.173494 0.300501i
\(300\) 0 0
\(301\) −2.00000 + 3.46410i −0.115278 + 0.199667i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −11.0000 −0.629858
\(306\) 0 0
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) −8.50000 14.7224i −0.480448 0.832161i 0.519300 0.854592i \(-0.326193\pi\)
−0.999748 + 0.0224310i \(0.992859\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 + 10.3923i 0.336994 + 0.583690i 0.983866 0.178908i \(-0.0572566\pi\)
−0.646872 + 0.762598i \(0.723923\pi\)
\(318\) 0 0
\(319\) −18.0000 + 31.1769i −1.00781 + 1.74557i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −6.00000 + 10.3923i −0.330791 + 0.572946i
\(330\) 0 0
\(331\) 9.50000 + 16.4545i 0.522167 + 0.904420i 0.999667 + 0.0257885i \(0.00820965\pi\)
−0.477500 + 0.878632i \(0.658457\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.50000 6.06218i −0.191225 0.331212i
\(336\) 0 0
\(337\) −11.5000 + 19.9186i −0.626445 + 1.08503i 0.361815 + 0.932250i \(0.382157\pi\)
−0.988260 + 0.152784i \(0.951176\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −48.0000 −2.59935
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.00000 + 5.19615i −0.161048 + 0.278944i −0.935245 0.354001i \(-0.884821\pi\)
0.774197 + 0.632945i \(0.218154\pi\)
\(348\) 0 0
\(349\) 0.500000 + 0.866025i 0.0267644 + 0.0463573i 0.879097 0.476642i \(-0.158146\pi\)
−0.852333 + 0.523000i \(0.824813\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 31.1769i −0.958043 1.65938i −0.727245 0.686378i \(-0.759200\pi\)
−0.230799 0.973002i \(-0.574134\pi\)
\(354\) 0 0
\(355\) 3.00000 5.19615i 0.159223 0.275783i
\(356\) 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
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.50000 9.52628i 0.287883 0.498628i
\(366\) 0 0
\(367\) −11.5000 19.9186i −0.600295 1.03974i −0.992776 0.119982i \(-0.961716\pi\)
0.392481 0.919760i \(-0.371617\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.00000 + 5.19615i 0.155752 + 0.269771i
\(372\) 0 0
\(373\) 3.50000 6.06218i 0.181223 0.313888i −0.761074 0.648665i \(-0.775328\pi\)
0.942297 + 0.334777i \(0.108661\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 12.0000 20.7846i 0.613171 1.06204i −0.377531 0.925997i \(-0.623227\pi\)
0.990702 0.136047i \(-0.0434398\pi\)
\(384\) 0 0
\(385\) 3.00000 + 5.19615i 0.152894 + 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.00000 0.0503155
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 + 5.19615i −0.149813 + 0.259483i −0.931158 0.364615i \(-0.881200\pi\)
0.781345 + 0.624099i \(0.214534\pi\)
\(402\) 0 0
\(403\) 4.00000 + 6.92820i 0.199254 + 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21.0000 36.3731i −1.04093 1.80295i
\(408\) 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
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −12.0000 + 20.7846i −0.586238 + 1.01539i 0.408481 + 0.912767i \(0.366058\pi\)
−0.994720 + 0.102628i \(0.967275\pi\)
\(420\) 0 0
\(421\) 9.50000 + 16.4545i 0.463002 + 0.801942i 0.999109 0.0422075i \(-0.0134391\pi\)
−0.536107 + 0.844150i \(0.680106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.50000 9.52628i 0.266164 0.461009i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 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
\(437\) −3.00000 + 5.19615i −0.143509 + 0.248566i
\(438\) 0 0
\(439\) −4.00000 6.92820i −0.190910 0.330665i 0.754642 0.656136i \(-0.227810\pi\)
−0.945552 + 0.325471i \(0.894477\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.00000 10.3923i −0.285069 0.493753i 0.687557 0.726130i \(-0.258683\pi\)
−0.972626 + 0.232377i \(0.925350\pi\)
\(444\) 0 0
\(445\) 6.00000 10.3923i 0.284427 0.492642i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −36.0000 −1.69895 −0.849473 0.527633i \(-0.823080\pi\)
−0.849473 + 0.527633i \(0.823080\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.500000 0.866025i 0.0234404 0.0405999i
\(456\) 0 0
\(457\) 5.00000 + 8.66025i 0.233890 + 0.405110i 0.958950 0.283577i \(-0.0915211\pi\)
−0.725059 + 0.688686i \(0.758188\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −6.00000 10.3923i −0.279448 0.484018i 0.691800 0.722089i \(-0.256818\pi\)
−0.971248 + 0.238071i \(0.923485\pi\)
\(462\) 0 0
\(463\) 6.50000 11.2583i 0.302081 0.523219i −0.674526 0.738251i \(-0.735652\pi\)
0.976607 + 0.215032i \(0.0689855\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 7.00000 0.323230
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12.0000 + 20.7846i −0.551761 + 0.955677i
\(474\) 0 0
\(475\) 0.500000 + 0.866025i 0.0229416 + 0.0397360i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.00000 15.5885i −0.411220 0.712255i 0.583803 0.811895i \(-0.301564\pi\)
−0.995023 + 0.0996406i \(0.968231\pi\)
\(480\) 0 0
\(481\) −3.50000 + 6.06218i −0.159586 + 0.276412i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 13.0000 0.590300
\(486\) 0 0
\(487\) 23.0000 1.04223 0.521115 0.853487i \(-0.325516\pi\)
0.521115 + 0.853487i \(0.325516\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.0000 + 25.9808i −0.676941 + 1.17250i 0.298957 + 0.954267i \(0.403361\pi\)
−0.975898 + 0.218229i \(0.929972\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.00000 + 5.19615i 0.134568 + 0.233079i
\(498\) 0 0
\(499\) −16.0000 + 27.7128i −0.716258 + 1.24060i 0.246214 + 0.969216i \(0.420813\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.0000 25.9808i 0.664863 1.15158i −0.314459 0.949271i \(-0.601823\pi\)
0.979322 0.202306i \(-0.0648436\pi\)
\(510\) 0 0
\(511\) 5.50000 + 9.52628i 0.243306 + 0.421418i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −6.50000 11.2583i −0.286424 0.496101i
\(516\) 0 0
\(517\) −36.0000 + 62.3538i −1.58328 + 2.74232i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 0 0
\(523\) 5.00000 0.218635 0.109317 0.994007i \(-0.465134\pi\)
0.109317 + 0.994007i \(0.465134\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.00000 + 5.19615i 0.129944 + 0.225070i
\(534\) 0 0
\(535\) 9.00000 15.5885i 0.389104 0.673948i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −43.0000 −1.84871 −0.924357 0.381528i \(-0.875398\pi\)
−0.924357 + 0.381528i \(0.875398\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.00000 + 8.66025i −0.214176 + 0.370965i
\(546\) 0 0
\(547\) −17.5000 30.3109i −0.748246 1.29600i −0.948663 0.316289i \(-0.897563\pi\)
0.200417 0.979711i \(-0.435770\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.00000 5.19615i −0.127804 0.221364i
\(552\) 0 0
\(553\) −0.500000 + 0.866025i −0.0212622 + 0.0368271i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 −0.508456 −0.254228 0.967144i \(-0.581821\pi\)
−0.254228 + 0.967144i \(0.581821\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −6.00000 + 10.3923i −0.252870 + 0.437983i −0.964315 0.264758i \(-0.914708\pi\)
0.711445 + 0.702742i \(0.248041\pi\)
\(564\) 0 0
\(565\) 9.00000 + 15.5885i 0.378633 + 0.655811i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.0000 20.7846i −0.503066 0.871336i −0.999994 0.00354413i \(-0.998872\pi\)
0.496928 0.867792i \(-0.334461\pi\)
\(570\) 0 0
\(571\) 6.50000 11.2583i 0.272017 0.471146i −0.697362 0.716720i \(-0.745643\pi\)
0.969378 + 0.245573i \(0.0789761\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −6.00000 −0.250217
\(576\) 0 0
\(577\) −31.0000 −1.29055 −0.645273 0.763952i \(-0.723257\pi\)
−0.645273 + 0.763952i \(0.723257\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.00000 + 5.19615i −0.124461 + 0.215573i
\(582\) 0 0
\(583\) 18.0000 + 31.1769i 0.745484 + 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.00000 + 5.19615i 0.123823 + 0.214468i 0.921272 0.388918i \(-0.127151\pi\)
−0.797449 + 0.603386i \(0.793818\pi\)
\(588\) 0 0
\(589\) 4.00000 6.92820i 0.164817 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 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
\(601\) −7.00000 12.1244i −0.285536 0.494563i 0.687203 0.726465i \(-0.258838\pi\)
−0.972739 + 0.231903i \(0.925505\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 12.5000 + 21.6506i 0.508197 + 0.880223i
\(606\) 0 0
\(607\) 6.50000 11.2583i 0.263827 0.456962i −0.703429 0.710766i \(-0.748349\pi\)
0.967256 + 0.253804i \(0.0816819\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.00000 5.19615i 0.120775 0.209189i −0.799298 0.600935i \(-0.794795\pi\)
0.920074 + 0.391745i \(0.128129\pi\)
\(618\) 0 0
\(619\) −5.50000 9.52628i −0.221064 0.382893i 0.734068 0.679076i \(-0.237620\pi\)
−0.955131 + 0.296183i \(0.904286\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000 + 10.3923i 0.240385 + 0.416359i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −25.0000 −0.995234 −0.497617 0.867397i \(-0.665792\pi\)
−0.497617 + 0.867397i \(0.665792\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.00000 + 13.8564i −0.317470 + 0.549875i
\(636\) 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
\(643\) 2.00000 3.46410i 0.0788723 0.136611i −0.823891 0.566748i \(-0.808201\pi\)
0.902764 + 0.430137i \(0.141535\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.0000 1.17942 0.589711 0.807614i \(-0.299242\pi\)
0.589711 + 0.807614i \(0.299242\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 + 10.3923i −0.234798 + 0.406682i −0.959214 0.282681i \(-0.908776\pi\)
0.724416 + 0.689363i \(0.242110\pi\)
\(654\) 0 0
\(655\) −6.00000 10.3923i −0.234439 0.406061i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 24.0000 + 41.5692i 0.934907 + 1.61931i 0.774799 + 0.632207i \(0.217851\pi\)
0.160108 + 0.987099i \(0.448816\pi\)
\(660\) 0 0
\(661\) 6.50000 11.2583i 0.252821 0.437898i −0.711481 0.702706i \(-0.751975\pi\)
0.964301 + 0.264807i \(0.0853084\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.00000 −0.0387783
\(666\) 0 0
\(667\) 36.0000 1.39393
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 33.0000 57.1577i 1.27395 2.20655i
\(672\) 0 0
\(673\) −8.50000 14.7224i −0.327651 0.567508i 0.654394 0.756153i \(-0.272924\pi\)
−0.982045 + 0.188645i \(0.939590\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 + 10.3923i 0.230599 + 0.399409i 0.957984 0.286820i \(-0.0925982\pi\)
−0.727386 + 0.686229i \(0.759265\pi\)
\(678\) 0 0
\(679\) −6.50000 + 11.2583i −0.249447 + 0.432055i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.0000 0.688751 0.344375 0.938832i \(-0.388091\pi\)
0.344375 + 0.938832i \(0.388091\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 3.00000 5.19615i 0.114291 0.197958i
\(690\) 0 0
\(691\) 14.0000 + 24.2487i 0.532585 + 0.922464i 0.999276 + 0.0380440i \(0.0121127\pi\)
−0.466691 + 0.884420i \(0.654554\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.50000 + 4.33013i 0.0948304 + 0.164251i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 7.00000 0.264010
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.50000 4.33013i −0.0938895 0.162621i 0.815255 0.579102i \(-0.196597\pi\)
−0.909145 + 0.416481i \(0.863263\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.0000 + 41.5692i 0.898807 + 1.55678i
\(714\) 0 0
\(715\) 3.00000 5.19615i 0.112194 0.194325i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.00000 −0.223762 −0.111881 0.993722i \(-0.535688\pi\)
−0.111881 + 0.993722i \(0.535688\pi\)
\(720\) 0 0
\(721\) 13.0000 0.484145
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.00000 5.19615i 0.111417 0.192980i
\(726\) 0 0
\(727\) −4.00000 6.92820i −0.148352 0.256953i 0.782267 0.622944i \(-0.214063\pi\)
−0.930618 + 0.365991i \(0.880730\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 11.0000 19.0526i 0.406294 0.703722i −0.588177 0.808732i \(-0.700154\pi\)
0.994471 + 0.105010i \(0.0334875\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 42.0000 1.54709
\(738\) 0 0
\(739\) 20.0000 0.735712 0.367856 0.929883i \(-0.380092\pi\)
0.367856 + 0.929883i \(0.380092\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −12.0000 + 20.7846i −0.440237 + 0.762513i −0.997707 0.0676840i \(-0.978439\pi\)
0.557470 + 0.830197i \(0.311772\pi\)
\(744\) 0 0
\(745\) 6.00000 + 10.3923i 0.219823 + 0.380745i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.00000 + 15.5885i 0.328853 + 0.569590i
\(750\) 0 0
\(751\) 6.50000 11.2583i 0.237188 0.410822i −0.722718 0.691143i \(-0.757107\pi\)
0.959906 + 0.280321i \(0.0904408\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −11.0000 −0.400331
\(756\) 0 0
\(757\) 23.0000 0.835949 0.417975 0.908459i \(-0.362740\pi\)
0.417975 + 0.908459i \(0.362740\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 36.3731i 0.761249 1.31852i −0.180957 0.983491i \(-0.557920\pi\)
0.942207 0.335032i \(-0.108747\pi\)
\(762\) 0 0
\(763\) −5.00000 8.66025i −0.181012 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 18.5000 32.0429i 0.667127 1.15550i −0.311577 0.950221i \(-0.600857\pi\)
0.978704 0.205277i \(-0.0658095\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −6.00000 −0.215805 −0.107903 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) 8.00000 0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.00000 5.19615i 0.107486 0.186171i
\(780\) 0 0
\(781\) 18.0000 + 31.1769i 0.644091 + 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.00000 + 1.73205i 0.0356915 + 0.0618195i
\(786\) 0 0
\(787\) −8.50000 + 14.7224i −0.302992 + 0.524798i −0.976812 0.214097i \(-0.931319\pi\)
0.673820 + 0.738896i \(0.264652\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −11.0000 −0.390621
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(798\) 0 0