Properties

Label 405.2.j.c.379.1
Level $405$
Weight $2$
Character 405.379
Analytic conductor $3.234$
Analytic rank $0$
Dimension $4$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,2,Mod(109,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.j (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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 379.1
Root \(-1.93649 - 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 405.379
Dual form 405.2.j.c.109.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.93649 - 1.11803i) q^{2} +(1.50000 + 2.59808i) q^{4} +(-1.93649 + 1.11803i) q^{5} -2.23607i q^{8} +O(q^{10})\) \(q+(-1.93649 - 1.11803i) q^{2} +(1.50000 + 2.59808i) q^{4} +(-1.93649 + 1.11803i) q^{5} -2.23607i q^{8} +5.00000 q^{10} +(0.500000 - 0.866025i) q^{16} -4.47214i q^{17} -4.00000 q^{19} +(-5.80948 - 3.35410i) q^{20} +(7.74597 - 4.47214i) q^{23} +(2.50000 - 4.33013i) q^{25} +(-4.00000 - 6.92820i) q^{31} +(-5.80948 + 3.35410i) q^{32} +(-5.00000 + 8.66025i) q^{34} +(7.74597 + 4.47214i) q^{38} +(2.50000 + 4.33013i) q^{40} -20.0000 q^{46} +(-7.74597 - 4.47214i) q^{47} +(-3.50000 - 6.06218i) q^{49} +(-9.68246 + 5.59017i) q^{50} -4.47214i q^{53} +(-1.00000 + 1.73205i) q^{61} +17.8885i q^{62} +13.0000 q^{64} +(11.6190 - 6.70820i) q^{68} +(-6.00000 - 10.3923i) q^{76} +(8.00000 - 13.8564i) q^{79} +2.23607i q^{80} +(15.4919 + 8.94427i) q^{83} +(5.00000 + 8.66025i) q^{85} +(23.2379 + 13.4164i) q^{92} +(10.0000 + 17.3205i) q^{94} +(7.74597 - 4.47214i) q^{95} +15.6525i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{4} + 20 q^{10} + 2 q^{16} - 16 q^{19} + 10 q^{25} - 16 q^{31} - 20 q^{34} + 10 q^{40} - 80 q^{46} - 14 q^{49} - 4 q^{61} + 52 q^{64} - 24 q^{76} + 32 q^{79} + 20 q^{85} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93649 1.11803i −1.36931 0.790569i −0.378467 0.925615i \(-0.623549\pi\)
−0.990839 + 0.135045i \(0.956882\pi\)
\(3\) 0 0
\(4\) 1.50000 + 2.59808i 0.750000 + 1.29904i
\(5\) −1.93649 + 1.11803i −0.866025 + 0.500000i
\(6\) 0 0
\(7\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(8\) 2.23607i 0.790569i
\(9\) 0 0
\(10\) 5.00000 1.58114
\(11\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 0.866025i 0.125000 0.216506i
\(17\) 4.47214i 1.08465i −0.840168 0.542326i \(-0.817544\pi\)
0.840168 0.542326i \(-0.182456\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) −5.80948 3.35410i −1.29904 0.750000i
\(21\) 0 0
\(22\) 0 0
\(23\) 7.74597 4.47214i 1.61515 0.932505i 0.626994 0.779024i \(-0.284285\pi\)
0.988152 0.153481i \(-0.0490483\pi\)
\(24\) 0 0
\(25\) 2.50000 4.33013i 0.500000 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(30\) 0 0
\(31\) −4.00000 6.92820i −0.718421 1.24434i −0.961625 0.274367i \(-0.911532\pi\)
0.243204 0.969975i \(-0.421802\pi\)
\(32\) −5.80948 + 3.35410i −1.02698 + 0.592927i
\(33\) 0 0
\(34\) −5.00000 + 8.66025i −0.857493 + 1.48522i
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 7.74597 + 4.47214i 1.25656 + 0.725476i
\(39\) 0 0
\(40\) 2.50000 + 4.33013i 0.395285 + 0.684653i
\(41\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(42\) 0 0
\(43\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −20.0000 −2.94884
\(47\) −7.74597 4.47214i −1.12987 0.652328i −0.185964 0.982556i \(-0.559541\pi\)
−0.943901 + 0.330228i \(0.892874\pi\)
\(48\) 0 0
\(49\) −3.50000 6.06218i −0.500000 0.866025i
\(50\) −9.68246 + 5.59017i −1.36931 + 0.790569i
\(51\) 0 0
\(52\) 0 0
\(53\) 4.47214i 0.614295i −0.951662 0.307148i \(-0.900625\pi\)
0.951662 0.307148i \(-0.0993745\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −1.00000 + 1.73205i −0.128037 + 0.221766i −0.922916 0.385002i \(-0.874201\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(62\) 17.8885i 2.27185i
\(63\) 0 0
\(64\) 13.0000 1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 11.6190 6.70820i 1.40900 0.813489i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −6.00000 10.3923i −0.688247 1.19208i
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 13.8564i 0.900070 1.55897i 0.0726692 0.997356i \(-0.476848\pi\)
0.827401 0.561611i \(-0.189818\pi\)
\(80\) 2.23607i 0.250000i
\(81\) 0 0
\(82\) 0 0
\(83\) 15.4919 + 8.94427i 1.70046 + 0.981761i 0.945289 + 0.326234i \(0.105780\pi\)
0.755172 + 0.655527i \(0.227553\pi\)
\(84\) 0 0
\(85\) 5.00000 + 8.66025i 0.542326 + 0.939336i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 23.2379 + 13.4164i 2.42272 + 1.39876i
\(93\) 0 0
\(94\) 10.0000 + 17.3205i 1.03142 + 1.78647i
\(95\) 7.74597 4.47214i 0.794719 0.458831i
\(96\) 0 0
\(97\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(98\) 15.6525i 1.58114i
\(99\) 0 0
\(100\) 15.0000 1.50000
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −5.00000 + 8.66025i −0.485643 + 0.841158i
\(107\) 17.8885i 1.72935i −0.502331 0.864675i \(-0.667524\pi\)
0.502331 0.864675i \(-0.332476\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.87298 + 2.23607i −0.364340 + 0.210352i −0.670983 0.741473i \(-0.734128\pi\)
0.306643 + 0.951825i \(0.400794\pi\)
\(114\) 0 0
\(115\) −10.0000 + 17.3205i −0.932505 + 1.61515i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 + 9.52628i 0.500000 + 0.866025i
\(122\) 3.87298 2.23607i 0.350643 0.202444i
\(123\) 0 0
\(124\) 12.0000 20.7846i 1.07763 1.86651i
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −13.5554 7.82624i −1.19814 0.691748i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −10.0000 −0.857493
\(137\) −19.3649 11.1803i −1.65446 0.955201i −0.975207 0.221293i \(-0.928972\pi\)
−0.679249 0.733908i \(-0.737694\pi\)
\(138\) 0 0
\(139\) 2.00000 + 3.46410i 0.169638 + 0.293821i 0.938293 0.345843i \(-0.112407\pi\)
−0.768655 + 0.639664i \(0.779074\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) −4.00000 + 6.92820i −0.325515 + 0.563809i −0.981617 0.190864i \(-0.938871\pi\)
0.656101 + 0.754673i \(0.272204\pi\)
\(152\) 8.94427i 0.725476i
\(153\) 0 0
\(154\) 0 0
\(155\) 15.4919 + 8.94427i 1.24434 + 0.718421i
\(156\) 0 0
\(157\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(158\) −30.9839 + 17.8885i −2.46494 + 1.42314i
\(159\) 0 0
\(160\) 7.50000 12.9904i 0.592927 1.02698i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −20.0000 34.6410i −1.55230 2.68866i
\(167\) 7.74597 4.47214i 0.599401 0.346064i −0.169405 0.985547i \(-0.554185\pi\)
0.768806 + 0.639482i \(0.220851\pi\)
\(168\) 0 0
\(169\) −6.50000 + 11.2583i −0.500000 + 0.866025i
\(170\) 22.3607i 1.71499i
\(171\) 0 0
\(172\) 0 0
\(173\) −19.3649 11.1803i −1.47229 0.850026i −0.472773 0.881184i \(-0.656747\pi\)
−0.999514 + 0.0311588i \(0.990080\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −10.0000 17.3205i −0.737210 1.27688i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 26.8328i 1.95698i
\(189\) 0 0
\(190\) −20.0000 −1.45095
\(191\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(192\) 0 0
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 10.5000 18.1865i 0.750000 1.29904i
\(197\) 4.47214i 0.318626i −0.987228 0.159313i \(-0.949072\pi\)
0.987228 0.159313i \(-0.0509280\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −9.68246 5.59017i −0.684653 0.395285i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 14.0000 + 24.2487i 0.963800 + 1.66935i 0.712806 + 0.701361i \(0.247424\pi\)
0.250994 + 0.967989i \(0.419243\pi\)
\(212\) 11.6190 6.70820i 0.797993 0.460721i
\(213\) 0 0
\(214\) −20.0000 + 34.6410i −1.36717 + 2.36801i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −27.1109 15.6525i −1.83618 1.06012i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) 15.4919 + 8.94427i 1.02824 + 0.593652i 0.916479 0.400083i \(-0.131019\pi\)
0.111757 + 0.993736i \(0.464352\pi\)
\(228\) 0 0
\(229\) −13.0000 22.5167i −0.859064 1.48794i −0.872823 0.488037i \(-0.837713\pi\)
0.0137585 0.999905i \(-0.495620\pi\)
\(230\) 38.7298 22.3607i 2.55377 1.47442i
\(231\) 0 0
\(232\) 0 0
\(233\) 22.3607i 1.46490i 0.680823 + 0.732448i \(0.261622\pi\)
−0.680823 + 0.732448i \(0.738378\pi\)
\(234\) 0 0
\(235\) 20.0000 1.30466
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(240\) 0 0
\(241\) −1.00000 + 1.73205i −0.0644157 + 0.111571i −0.896435 0.443176i \(-0.853852\pi\)
0.832019 + 0.554747i \(0.187185\pi\)
\(242\) 24.5967i 1.58114i
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) 13.5554 + 7.82624i 0.866025 + 0.500000i
\(246\) 0 0
\(247\) 0 0
\(248\) −15.4919 + 8.94427i −0.983739 + 0.567962i
\(249\) 0 0
\(250\) 12.5000 21.6506i 0.790569 1.36931i
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.50000 + 7.79423i 0.281250 + 0.487139i
\(257\) −27.1109 + 15.6525i −1.69113 + 0.976375i −0.737523 + 0.675322i \(0.764005\pi\)
−0.953608 + 0.301052i \(0.902662\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.74597 4.47214i −0.477637 0.275764i 0.241794 0.970328i \(-0.422264\pi\)
−0.719431 + 0.694564i \(0.755597\pi\)
\(264\) 0 0
\(265\) 5.00000 + 8.66025i 0.307148 + 0.531995i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 32.0000 1.94386 0.971931 0.235267i \(-0.0755965\pi\)
0.971931 + 0.235267i \(0.0755965\pi\)
\(272\) −3.87298 2.23607i −0.234834 0.135582i
\(273\) 0 0
\(274\) 25.0000 + 43.3013i 1.51031 + 2.61593i
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(278\) 8.94427i 0.536442i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(282\) 0 0
\(283\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3.00000 −0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −27.1109 + 15.6525i −1.58383 + 0.914427i −0.589542 + 0.807737i \(0.700692\pi\)
−0.994292 + 0.106690i \(0.965975\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 15.4919 8.94427i 0.891461 0.514685i
\(303\) 0 0
\(304\) −2.00000 + 3.46410i −0.114708 + 0.198680i
\(305\) 4.47214i 0.256074i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −20.0000 34.6410i −1.13592 1.96748i
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 48.0000 2.70021
\(317\) −19.3649 11.1803i −1.08764 0.627950i −0.154694 0.987962i \(-0.549439\pi\)
−0.932948 + 0.360012i \(0.882773\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −25.1744 + 14.5344i −1.40729 + 0.812500i
\(321\) 0 0
\(322\) 0 0
\(323\) 17.8885i 0.995345i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 14.0000 24.2487i 0.769510 1.33283i −0.168320 0.985732i \(-0.553834\pi\)
0.937829 0.347097i \(-0.112833\pi\)
\(332\) 53.6656i 2.94528i
\(333\) 0 0
\(334\) −20.0000 −1.09435
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(338\) 25.1744 14.5344i 1.36931 0.790569i
\(339\) 0 0
\(340\) −15.0000 + 25.9808i −0.813489 + 1.40900i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 25.0000 + 43.3013i 1.34401 + 2.32789i
\(347\) 30.9839 17.8885i 1.66330 0.960307i 0.692179 0.721726i \(-0.256651\pi\)
0.971123 0.238581i \(-0.0766823\pi\)
\(348\) 0 0
\(349\) 17.0000 29.4449i 0.909989 1.57615i 0.0959126 0.995390i \(-0.469423\pi\)
0.814076 0.580758i \(-0.197244\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 27.1109 + 15.6525i 1.44297 + 0.833097i 0.998047 0.0624731i \(-0.0198987\pi\)
0.444920 + 0.895570i \(0.353232\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 42.6028 + 24.5967i 2.23915 + 1.29278i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(368\) 8.94427i 0.466252i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −10.0000 + 17.3205i −0.515711 + 0.893237i
\(377\) 0 0
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 23.2379 + 13.4164i 1.19208 + 0.688247i
\(381\) 0 0
\(382\) 0 0
\(383\) 7.74597 4.47214i 0.395800 0.228515i −0.288870 0.957368i \(-0.593280\pi\)
0.684670 + 0.728853i \(0.259946\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) −20.0000 34.6410i −1.01144 1.75187i
\(392\) −13.5554 + 7.82624i −0.684653 + 0.395285i
\(393\) 0 0
\(394\) −5.00000 + 8.66025i −0.251896 + 0.436297i
\(395\) 35.7771i 1.80014i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 30.9839 + 17.8885i 1.55308 + 0.896672i
\(399\) 0 0
\(400\) −2.50000 4.33013i −0.125000 0.216506i
\(401\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −13.0000 22.5167i −0.642809 1.11338i −0.984803 0.173675i \(-0.944436\pi\)
0.341994 0.939702i \(-0.388898\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −40.0000 −1.96352
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(420\) 0 0
\(421\) −19.0000 + 32.9090i −0.926003 + 1.60388i −0.136064 + 0.990700i \(0.543445\pi\)
−0.789940 + 0.613185i \(0.789888\pi\)
\(422\) 62.6099i 3.04780i
\(423\) 0 0
\(424\) −10.0000 −0.485643
\(425\) −19.3649 11.1803i −0.939336 0.542326i
\(426\) 0 0
\(427\) 0 0
\(428\) 46.4758 26.8328i 2.24649 1.29701i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 21.0000 + 36.3731i 1.00572 + 1.74195i
\(437\) −30.9839 + 17.8885i −1.48216 + 0.855725i
\(438\) 0 0
\(439\) 8.00000 13.8564i 0.381819 0.661330i −0.609503 0.792784i \(-0.708631\pi\)
0.991322 + 0.131453i \(0.0419644\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 15.4919 + 8.94427i 0.736044 + 0.424955i 0.820629 0.571461i \(-0.193623\pi\)
−0.0845852 + 0.996416i \(0.526957\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −11.6190 6.70820i −0.546509 0.315527i
\(453\) 0 0
\(454\) −20.0000 34.6410i −0.938647 1.62578i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 58.1378i 2.71660i
\(459\) 0 0
\(460\) −60.0000 −2.79751
\(461\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(462\) 0 0
\(463\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 25.0000 43.3013i 1.15810 2.00589i
\(467\) 35.7771i 1.65557i 0.561048 + 0.827783i \(0.310398\pi\)
−0.561048 + 0.827783i \(0.689602\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −38.7298 22.3607i −1.78647 1.03142i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −10.0000 + 17.3205i −0.458831 + 0.794719i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 3.87298 2.23607i 0.176410 0.101850i
\(483\) 0 0
\(484\) −16.5000 + 28.5788i −0.750000 + 1.29904i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 3.87298 + 2.23607i 0.175322 + 0.101222i
\(489\) 0 0
\(490\) −17.5000 30.3109i −0.790569 1.36931i
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) 0 0
\(498\) 0 0
\(499\) −22.0000 38.1051i −0.984855 1.70582i −0.642578 0.766220i \(-0.722135\pi\)
−0.342277 0.939599i \(-0.611198\pi\)
\(500\) −29.0474 + 16.7705i −1.29904 + 0.750000i
\(501\) 0 0
\(502\) 0 0
\(503\) 44.7214i 1.99403i −0.0772283 0.997013i \(-0.524607\pi\)
0.0772283 0.997013i \(-0.475393\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 11.1803i 0.494106i
\(513\) 0 0
\(514\) 70.0000 3.08757
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 10.0000 + 17.3205i 0.436021 + 0.755210i
\(527\) −30.9839 + 17.8885i −1.34968 + 0.779237i
\(528\) 0 0
\(529\) 28.5000 49.3634i 1.23913 2.14624i
\(530\) 22.3607i 0.971286i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 20.0000 + 34.6410i 0.864675 + 1.49766i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −61.9677 35.7771i −2.66174 1.53676i
\(543\) 0 0
\(544\) 15.0000 + 25.9808i 0.643120 + 1.11392i
\(545\) −27.1109 + 15.6525i −1.16130 + 0.670478i
\(546\) 0 0
\(547\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 67.0820i 2.86560i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −6.00000 + 10.3923i −0.254457 + 0.440732i
\(557\) 22.3607i 0.947452i 0.880672 + 0.473726i \(0.157091\pi\)
−0.880672 + 0.473726i \(0.842909\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 30.9839 17.8885i 1.30581 0.753912i 0.324420 0.945913i \(-0.394831\pi\)
0.981395 + 0.192001i \(0.0614977\pi\)
\(564\) 0 0
\(565\) 5.00000 8.66025i 0.210352 0.364340i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 14.0000 + 24.2487i 0.585882 + 1.01478i 0.994765 + 0.102190i \(0.0325850\pi\)
−0.408883 + 0.912587i \(0.634082\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 44.7214i 1.86501i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 5.80948 + 3.35410i 0.241642 + 0.139512i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 70.0000 2.89167
\(587\) 15.4919 + 8.94427i 0.639421 + 0.369170i 0.784391 0.620266i \(-0.212975\pi\)
−0.144971 + 0.989436i \(0.546309\pi\)
\(588\) 0 0
\(589\) 16.0000 + 27.7128i 0.659269 + 1.14189i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.47214i 0.183649i −0.995775 0.0918243i \(-0.970730\pi\)
0.995775 0.0918243i \(-0.0292698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(600\) 0 0
\(601\) −19.0000 + 32.9090i −0.775026 + 1.34238i 0.159754 + 0.987157i \(0.448930\pi\)
−0.934780 + 0.355228i \(0.884403\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −24.0000 −0.976546
\(605\) −21.3014 12.2984i −0.866025 0.500000i
\(606\) 0 0
\(607\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(608\) 23.2379 13.4164i 0.942421 0.544107i
\(609\) 0 0
\(610\) −5.00000 + 8.66025i −0.202444 + 0.350643i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 42.6028 24.5967i 1.71512 0.990228i 0.787833 0.615889i \(-0.211203\pi\)
0.927292 0.374338i \(-0.122130\pi\)
\(618\) 0 0
\(619\) −22.0000 + 38.1051i −0.884255 + 1.53157i −0.0376891 + 0.999290i \(0.512000\pi\)
−0.846566 + 0.532284i \(0.821334\pi\)
\(620\) 53.6656i 2.15526i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 21.6506i −0.500000 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) −30.9839 17.8885i −1.23247 0.711568i
\(633\) 0 0
\(634\) 25.0000 + 43.3013i 0.992877 + 1.71971i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 35.0000 1.38350
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 20.0000 34.6410i 0.786889 1.36293i
\(647\) 44.7214i 1.75818i −0.476658 0.879089i \(-0.658152\pi\)
0.476658 0.879089i \(-0.341848\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 42.6028 24.5967i 1.66718 0.962545i 0.698028 0.716071i \(-0.254061\pi\)
0.969149 0.246474i \(-0.0792721\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(660\) 0 0
\(661\) 11.0000 + 19.0526i 0.427850 + 0.741059i 0.996682 0.0813955i \(-0.0259377\pi\)
−0.568831 + 0.822454i \(0.692604\pi\)
\(662\) −54.2218 + 31.3050i −2.10739 + 1.21670i
\(663\) 0 0
\(664\) 20.0000 34.6410i 0.776151 1.34433i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 23.2379 + 13.4164i 0.899101 + 0.519096i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −39.0000 −1.50000
\(677\) 27.1109 + 15.6525i 1.04196 + 0.601574i 0.920387 0.391009i \(-0.127874\pi\)
0.121569 + 0.992583i \(0.461207\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 19.3649 11.1803i 0.742611 0.428746i
\(681\) 0 0
\(682\) 0 0
\(683\) 35.7771i 1.36897i 0.729026 + 0.684486i \(0.239973\pi\)
−0.729026 + 0.684486i \(0.760027\pi\)
\(684\) 0 0
\(685\) 50.0000 1.91040
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 26.0000 45.0333i 0.989087 1.71315i 0.366947 0.930242i \(-0.380403\pi\)
0.622139 0.782907i \(-0.286264\pi\)
\(692\) 67.0820i 2.55008i
\(693\) 0 0
\(694\) −80.0000 −3.03676
\(695\) −7.74597 4.47214i −0.293821 0.169638i
\(696\) 0 0
\(697\) 0 0
\(698\) −65.8407 + 38.0132i −2.49211 + 1.43882i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −35.0000 60.6218i −1.31724 2.28153i
\(707\) 0 0
\(708\) 0 0
\(709\) −13.0000 + 22.5167i −0.488225 + 0.845631i −0.999908 0.0135434i \(-0.995689\pi\)
0.511683 + 0.859174i \(0.329022\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −61.9677 35.7771i −2.32071 1.33986i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 5.80948 + 3.35410i 0.216206 + 0.124827i
\(723\) 0 0
\(724\) −33.0000 57.1577i −1.22644 2.12425i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −30.0000 + 51.9615i −1.10581 + 1.91533i
\(737\) 0 0
\(738\) 0 0
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −38.7298 + 22.3607i −1.42086 + 0.820334i −0.996372 0.0851001i \(-0.972879\pi\)
−0.424487 + 0.905434i \(0.639546\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −4.00000 6.92820i −0.145962 0.252814i 0.783769 0.621052i \(-0.213294\pi\)
−0.929731 + 0.368238i \(0.879961\pi\)
\(752\) −7.74597 + 4.47214i −0.282466 + 0.163082i
\(753\) 0 0
\(754\) 0 0
\(755\) 17.8885i 0.651031i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 7.74597 + 4.47214i 0.281346 + 0.162435i
\(759\) 0 0
\(760\) −10.0000 17.3205i −0.362738 0.628281i
\(761\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −20.0000 −0.722629
\(767\) 0 0
\(768\) 0 0
\(769\) 23.0000 + 39.8372i 0.829401 + 1.43657i 0.898509 + 0.438956i \(0.144652\pi\)
−0.0691074 + 0.997609i \(0.522015\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.47214i 0.160852i −0.996761 0.0804258i \(-0.974372\pi\)
0.996761 0.0804258i \(-0.0256280\pi\)
\(774\) 0 0
\(775\) −40.0000 −1.43684
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 89.4427i 3.19847i
\(783\) 0 0
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) 11.6190 6.70820i 0.413908 0.238970i
\(789\) 0 0
\(790\) 40.0000 69.2820i 1.42314 2.46494i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −24.0000 41.5692i −0.850657 1.47338i
\(797\) 42.6028 24.5967i 1.50907 0.871262i 0.509125 0.860693i \(-0.329969\pi\)