Properties

Label 2268.2.i.h.2053.1
Level $2268$
Weight $2$
Character 2268.2053
Analytic conductor $18.110$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(865,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.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: \(18.1100711784\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 2053.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2268.2053
Dual form 2268.2.i.h.865.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 + 2.59808i) q^{5} +(-0.500000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(1.50000 + 2.59808i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-1.50000 + 2.59808i) q^{11} +(-1.00000 + 1.73205i) q^{13} +(1.50000 + 2.59808i) q^{17} +(0.500000 - 0.866025i) q^{19} +(1.50000 + 2.59808i) q^{23} +(-2.00000 + 3.46410i) q^{25} +(-3.00000 - 5.19615i) q^{29} -7.00000 q^{31} +(-7.50000 + 2.59808i) q^{35} +(0.500000 - 0.866025i) q^{37} +(3.00000 - 5.19615i) q^{41} +(2.00000 + 3.46410i) q^{43} +9.00000 q^{47} +(-6.50000 - 2.59808i) q^{49} +(1.50000 + 2.59808i) q^{53} -9.00000 q^{55} -9.00000 q^{59} -1.00000 q^{61} -6.00000 q^{65} -7.00000 q^{67} +(0.500000 + 0.866025i) q^{73} +(-6.00000 - 5.19615i) q^{77} -13.0000 q^{79} +(6.00000 + 10.3923i) q^{83} +(-4.50000 + 7.79423i) q^{85} +(7.50000 - 12.9904i) q^{89} +(-4.00000 - 3.46410i) q^{91} +3.00000 q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{5} - q^{7} - 3 q^{11} - 2 q^{13} + 3 q^{17} + q^{19} + 3 q^{23} - 4 q^{25} - 6 q^{29} - 14 q^{31} - 15 q^{35} + q^{37} + 6 q^{41} + 4 q^{43} + 18 q^{47} - 13 q^{49} + 3 q^{53} - 18 q^{55} - 18 q^{59} - 2 q^{61} - 12 q^{65} - 14 q^{67} + q^{73} - 12 q^{77} - 26 q^{79} + 12 q^{83} - 9 q^{85} + 15 q^{89} - 8 q^{91} + 6 q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 1.50000 + 2.59808i 0.670820 + 1.16190i 0.977672 + 0.210138i \(0.0673912\pi\)
−0.306851 + 0.951757i \(0.599275\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.50000 + 2.59808i −0.452267 + 0.783349i −0.998526 0.0542666i \(-0.982718\pi\)
0.546259 + 0.837616i \(0.316051\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 0 0
\(19\) 0.500000 0.866025i 0.114708 0.198680i −0.802955 0.596040i \(-0.796740\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.50000 + 2.59808i 0.312772 + 0.541736i 0.978961 0.204046i \(-0.0654092\pi\)
−0.666190 + 0.745782i \(0.732076\pi\)
\(24\) 0 0
\(25\) −2.00000 + 3.46410i −0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 5.19615i −0.557086 0.964901i −0.997738 0.0672232i \(-0.978586\pi\)
0.440652 0.897678i \(-0.354747\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.50000 + 2.59808i −1.26773 + 0.439155i
\(36\) 0 0
\(37\) 0.500000 0.866025i 0.0821995 0.142374i −0.821995 0.569495i \(-0.807139\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 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
\(43\) 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
\(47\) 9.00000 1.31278 0.656392 0.754420i \(-0.272082\pi\)
0.656392 + 0.754420i \(0.272082\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.50000 + 2.59808i 0.206041 + 0.356873i 0.950464 0.310835i \(-0.100609\pi\)
−0.744423 + 0.667708i \(0.767275\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.00000 −1.17170 −0.585850 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) −7.00000 −0.855186 −0.427593 0.903971i \(-0.640638\pi\)
−0.427593 + 0.903971i \(0.640638\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0.500000 + 0.866025i 0.0585206 + 0.101361i 0.893801 0.448463i \(-0.148028\pi\)
−0.835281 + 0.549823i \(0.814695\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.00000 5.19615i −0.683763 0.592157i
\(78\) 0 0
\(79\) −13.0000 −1.46261 −0.731307 0.682048i \(-0.761089\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.00000 + 10.3923i 0.658586 + 1.14070i 0.980982 + 0.194099i \(0.0621783\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(84\) 0 0
\(85\) −4.50000 + 7.79423i −0.488094 + 0.845403i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.50000 12.9904i 0.794998 1.37698i −0.127842 0.991795i \(-0.540805\pi\)
0.922840 0.385183i \(-0.125862\pi\)
\(90\) 0 0
\(91\) −4.00000 3.46410i −0.419314 0.363137i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) 5.00000 + 8.66025i 0.507673 + 0.879316i 0.999961 + 0.00888289i \(0.00282755\pi\)
−0.492287 + 0.870433i \(0.663839\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.50000 12.9904i 0.746278 1.29259i −0.203317 0.979113i \(-0.565172\pi\)
0.949595 0.313478i \(-0.101494\pi\)
\(102\) 0 0
\(103\) −5.50000 9.52628i −0.541931 0.938652i −0.998793 0.0491146i \(-0.984360\pi\)
0.456862 0.889538i \(-0.348973\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.50000 12.9904i 0.725052 1.25583i −0.233900 0.972261i \(-0.575149\pi\)
0.958952 0.283567i \(-0.0915178\pi\)
\(108\) 0 0
\(109\) 0.500000 + 0.866025i 0.0478913 + 0.0829502i 0.888977 0.457951i \(-0.151417\pi\)
−0.841086 + 0.540901i \(0.818083\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.00000 5.19615i 0.282216 0.488813i −0.689714 0.724082i \(-0.742264\pi\)
0.971930 + 0.235269i \(0.0755971\pi\)
\(114\) 0 0
\(115\) −4.50000 + 7.79423i −0.419627 + 0.726816i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.50000 + 2.59808i −0.687524 + 0.238165i
\(120\) 0 0
\(121\) 1.00000 + 1.73205i 0.0909091 + 0.157459i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.50000 + 2.59808i 0.131056 + 0.226995i 0.924084 0.382190i \(-0.124830\pi\)
−0.793028 + 0.609185i \(0.791497\pi\)
\(132\) 0 0
\(133\) 2.00000 + 1.73205i 0.173422 + 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.5000 + 18.1865i −0.897076 + 1.55378i −0.0658609 + 0.997829i \(0.520979\pi\)
−0.831215 + 0.555952i \(0.812354\pi\)
\(138\) 0 0
\(139\) −10.0000 + 17.3205i −0.848189 + 1.46911i 0.0346338 + 0.999400i \(0.488974\pi\)
−0.882823 + 0.469706i \(0.844360\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.00000 5.19615i −0.250873 0.434524i
\(144\) 0 0
\(145\) 9.00000 15.5885i 0.747409 1.29455i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.50000 + 2.59808i 0.122885 + 0.212843i 0.920904 0.389789i \(-0.127452\pi\)
−0.798019 + 0.602632i \(0.794119\pi\)
\(150\) 0 0
\(151\) −8.50000 + 14.7224i −0.691720 + 1.19809i 0.279554 + 0.960130i \(0.409814\pi\)
−0.971274 + 0.237964i \(0.923520\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −10.5000 18.1865i −0.843380 1.46078i
\(156\) 0 0
\(157\) −13.0000 −1.03751 −0.518756 0.854922i \(-0.673605\pi\)
−0.518756 + 0.854922i \(0.673605\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.50000 + 2.59808i −0.591083 + 0.204757i
\(162\) 0 0
\(163\) −5.50000 + 9.52628i −0.430793 + 0.746156i −0.996942 0.0781474i \(-0.975100\pi\)
0.566149 + 0.824303i \(0.308433\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.00000 + 10.3923i −0.464294 + 0.804181i −0.999169 0.0407502i \(-0.987025\pi\)
0.534875 + 0.844931i \(0.320359\pi\)
\(168\) 0 0
\(169\) 4.50000 + 7.79423i 0.346154 + 0.599556i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.00000 0.684257 0.342129 0.939653i \(-0.388852\pi\)
0.342129 + 0.939653i \(0.388852\pi\)
\(174\) 0 0
\(175\) −8.00000 6.92820i −0.604743 0.523723i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 10.5000 + 18.1865i 0.784807 + 1.35933i 0.929114 + 0.369792i \(0.120571\pi\)
−0.144308 + 0.989533i \(0.546095\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 0.220564
\(186\) 0 0
\(187\) −9.00000 −0.658145
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 9.00000 0.651217 0.325609 0.945505i \(-0.394431\pi\)
0.325609 + 0.945505i \(0.394431\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 3.50000 + 6.06218i 0.248108 + 0.429736i 0.963001 0.269498i \(-0.0868577\pi\)
−0.714893 + 0.699234i \(0.753524\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 15.0000 5.19615i 1.05279 0.364698i
\(204\) 0 0
\(205\) 18.0000 1.25717
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.50000 + 2.59808i 0.103757 + 0.179713i
\(210\) 0 0
\(211\) 2.00000 3.46410i 0.137686 0.238479i −0.788935 0.614477i \(-0.789367\pi\)
0.926620 + 0.375999i \(0.122700\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −6.00000 + 10.3923i −0.409197 + 0.708749i
\(216\) 0 0
\(217\) 3.50000 18.1865i 0.237595 1.23458i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −4.00000 6.92820i −0.267860 0.463947i 0.700449 0.713702i \(-0.252983\pi\)
−0.968309 + 0.249756i \(0.919650\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.50000 + 2.59808i −0.0995585 + 0.172440i −0.911502 0.411296i \(-0.865076\pi\)
0.811943 + 0.583736i \(0.198410\pi\)
\(228\) 0 0
\(229\) −5.50000 9.52628i −0.363450 0.629514i 0.625076 0.780564i \(-0.285068\pi\)
−0.988526 + 0.151050i \(0.951735\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.5000 + 18.1865i −0.687878 + 1.19144i 0.284645 + 0.958633i \(0.408124\pi\)
−0.972523 + 0.232806i \(0.925209\pi\)
\(234\) 0 0
\(235\) 13.5000 + 23.3827i 0.880643 + 1.52532i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 + 10.3923i −0.388108 + 0.672222i −0.992195 0.124696i \(-0.960204\pi\)
0.604087 + 0.796918i \(0.293538\pi\)
\(240\) 0 0
\(241\) 0.500000 0.866025i 0.0322078 0.0557856i −0.849472 0.527633i \(-0.823079\pi\)
0.881680 + 0.471848i \(0.156413\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.00000 20.7846i −0.191663 1.32788i
\(246\) 0 0
\(247\) 1.00000 + 1.73205i 0.0636285 + 0.110208i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.50000 + 2.59808i 0.0935674 + 0.162064i 0.909010 0.416775i \(-0.136840\pi\)
−0.815442 + 0.578838i \(0.803506\pi\)
\(258\) 0 0
\(259\) 2.00000 + 1.73205i 0.124274 + 0.107624i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.50000 + 2.59808i −0.0924940 + 0.160204i −0.908560 0.417755i \(-0.862817\pi\)
0.816066 + 0.577959i \(0.196151\pi\)
\(264\) 0 0
\(265\) −4.50000 + 7.79423i −0.276433 + 0.478796i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.50000 + 2.59808i 0.0914566 + 0.158408i 0.908124 0.418701i \(-0.137514\pi\)
−0.816668 + 0.577108i \(0.804181\pi\)
\(270\) 0 0
\(271\) −5.50000 + 9.52628i −0.334101 + 0.578680i −0.983312 0.181928i \(-0.941766\pi\)
0.649211 + 0.760609i \(0.275099\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.00000 10.3923i −0.361814 0.626680i
\(276\) 0 0
\(277\) 6.50000 11.2583i 0.390547 0.676448i −0.601975 0.798515i \(-0.705619\pi\)
0.992522 + 0.122068i \(0.0389525\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.0000 + 25.9808i 0.894825 + 1.54988i 0.834021 + 0.551733i \(0.186033\pi\)
0.0608039 + 0.998150i \(0.480634\pi\)
\(282\) 0 0
\(283\) 29.0000 1.72387 0.861936 0.507018i \(-0.169252\pi\)
0.861936 + 0.507018i \(0.169252\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 + 10.3923i 0.708338 + 0.613438i
\(288\) 0 0
\(289\) 4.00000 6.92820i 0.235294 0.407541i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.00000 5.19615i 0.175262 0.303562i −0.764990 0.644042i \(-0.777256\pi\)
0.940252 + 0.340480i \(0.110589\pi\)
\(294\) 0 0
\(295\) −13.5000 23.3827i −0.786000 1.36139i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −10.0000 + 3.46410i −0.576390 + 0.199667i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.50000 2.59808i −0.0858898 0.148765i
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 27.0000 1.53103 0.765515 0.643418i \(-0.222484\pi\)
0.765515 + 0.643418i \(0.222484\pi\)
\(312\) 0 0
\(313\) 23.0000 1.30004 0.650018 0.759918i \(-0.274761\pi\)
0.650018 + 0.759918i \(0.274761\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 0.505490 0.252745 0.967533i \(-0.418667\pi\)
0.252745 + 0.967533i \(0.418667\pi\)
\(318\) 0 0
\(319\) 18.0000 1.00781
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) −4.00000 6.92820i −0.221880 0.384308i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −4.50000 + 23.3827i −0.248093 + 1.28913i
\(330\) 0 0
\(331\) −13.0000 −0.714545 −0.357272 0.934000i \(-0.616293\pi\)
−0.357272 + 0.934000i \(0.616293\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −10.5000 18.1865i −0.573676 0.993636i
\(336\) 0 0
\(337\) 17.0000 29.4449i 0.926049 1.60396i 0.136184 0.990684i \(-0.456516\pi\)
0.789865 0.613280i \(-0.210150\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 10.5000 18.1865i 0.568607 0.984856i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.00000 −0.483145 −0.241573 0.970383i \(-0.577663\pi\)
−0.241573 + 0.970383i \(0.577663\pi\)
\(348\) 0 0
\(349\) −13.0000 22.5167i −0.695874 1.20529i −0.969885 0.243563i \(-0.921684\pi\)
0.274011 0.961727i \(-0.411649\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.5000 + 18.1865i −0.558859 + 0.967972i 0.438733 + 0.898617i \(0.355427\pi\)
−0.997592 + 0.0693543i \(0.977906\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.50000 12.9904i 0.395835 0.685606i −0.597372 0.801964i \(-0.703789\pi\)
0.993207 + 0.116358i \(0.0371219\pi\)
\(360\) 0 0
\(361\) 9.00000 + 15.5885i 0.473684 + 0.820445i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.50000 + 2.59808i −0.0785136 + 0.135990i
\(366\) 0 0
\(367\) −2.50000 + 4.33013i −0.130499 + 0.226031i −0.923869 0.382709i \(-0.874991\pi\)
0.793370 + 0.608740i \(0.208325\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.50000 + 2.59808i −0.389381 + 0.134885i
\(372\) 0 0
\(373\) 12.5000 + 21.6506i 0.647225 + 1.12103i 0.983783 + 0.179364i \(0.0574041\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −16.5000 28.5788i −0.843111 1.46031i −0.887252 0.461285i \(-0.847389\pi\)
0.0441413 0.999025i \(-0.485945\pi\)
\(384\) 0 0
\(385\) 4.50000 23.3827i 0.229341 1.19169i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.50000 12.9904i 0.380265 0.658638i −0.610835 0.791758i \(-0.709166\pi\)
0.991100 + 0.133120i \(0.0424994\pi\)
\(390\) 0 0
\(391\) −4.50000 + 7.79423i −0.227575 + 0.394171i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −19.5000 33.7750i −0.981151 1.69940i
\(396\) 0 0
\(397\) 18.5000 32.0429i 0.928488 1.60819i 0.142636 0.989775i \(-0.454442\pi\)
0.785853 0.618414i \(-0.212224\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.50000 + 2.59808i 0.0749064 + 0.129742i 0.901046 0.433724i \(-0.142801\pi\)
−0.826139 + 0.563466i \(0.809468\pi\)
\(402\) 0 0
\(403\) 7.00000 12.1244i 0.348695 0.603957i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.50000 + 2.59808i 0.0743522 + 0.128782i
\(408\) 0 0
\(409\) 11.0000 0.543915 0.271957 0.962309i \(-0.412329\pi\)
0.271957 + 0.962309i \(0.412329\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.50000 23.3827i 0.221431 1.15059i
\(414\) 0 0
\(415\) −18.0000 + 31.1769i −0.883585 + 1.53041i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 + 10.3923i −0.293119 + 0.507697i −0.974546 0.224189i \(-0.928027\pi\)
0.681426 + 0.731887i \(0.261360\pi\)
\(420\) 0 0
\(421\) 11.0000 + 19.0526i 0.536107 + 0.928565i 0.999109 + 0.0422075i \(0.0134391\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) 0.500000 2.59808i 0.0241967 0.125730i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.50000 12.9904i −0.361262 0.625725i 0.626907 0.779094i \(-0.284321\pi\)
−0.988169 + 0.153370i \(0.950987\pi\)
\(432\) 0 0
\(433\) −10.0000 −0.480569 −0.240285 0.970702i \(-0.577241\pi\)
−0.240285 + 0.970702i \(0.577241\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) −1.00000 −0.0477274 −0.0238637 0.999715i \(-0.507597\pi\)
−0.0238637 + 0.999715i \(0.507597\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.00000 0.427603 0.213801 0.976877i \(-0.431415\pi\)
0.213801 + 0.976877i \(0.431415\pi\)
\(444\) 0 0
\(445\) 45.0000 2.13320
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 9.00000 + 15.5885i 0.423793 + 0.734032i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.00000 15.5885i 0.140642 0.730798i
\(456\) 0 0
\(457\) 23.0000 1.07589 0.537947 0.842978i \(-0.319200\pi\)
0.537947 + 0.842978i \(0.319200\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.00000 5.19615i −0.139724 0.242009i 0.787668 0.616100i \(-0.211288\pi\)
−0.927392 + 0.374091i \(0.877955\pi\)
\(462\) 0 0
\(463\) 8.00000 13.8564i 0.371792 0.643962i −0.618050 0.786139i \(-0.712077\pi\)
0.989841 + 0.142177i \(0.0454103\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −10.5000 + 18.1865i −0.485882 + 0.841572i −0.999868 0.0162260i \(-0.994835\pi\)
0.513986 + 0.857798i \(0.328168\pi\)
\(468\) 0 0
\(469\) 3.50000 18.1865i 0.161615 0.839776i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −12.0000 −0.551761
\(474\) 0 0
\(475\) 2.00000 + 3.46410i 0.0917663 + 0.158944i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.50000 + 2.59808i −0.0685367 + 0.118709i −0.898257 0.439470i \(-0.855166\pi\)
0.829721 + 0.558179i \(0.188500\pi\)
\(480\) 0 0
\(481\) 1.00000 + 1.73205i 0.0455961 + 0.0789747i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.0000 + 25.9808i −0.681115 + 1.17973i
\(486\) 0 0
\(487\) 9.50000 + 16.4545i 0.430486 + 0.745624i 0.996915 0.0784867i \(-0.0250088\pi\)
−0.566429 + 0.824110i \(0.691675\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 12.0000 20.7846i 0.541552 0.937996i −0.457263 0.889332i \(-0.651170\pi\)
0.998815 0.0486647i \(-0.0154966\pi\)
\(492\) 0 0
\(493\) 9.00000 15.5885i 0.405340 0.702069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −5.50000 9.52628i −0.246214 0.426455i 0.716258 0.697835i \(-0.245853\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 45.0000 2.00247
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.50000 + 2.59808i 0.0664863 + 0.115158i 0.897352 0.441315i \(-0.145488\pi\)
−0.830866 + 0.556473i \(0.812154\pi\)
\(510\) 0 0
\(511\) −2.50000 + 0.866025i −0.110593 + 0.0383107i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 16.5000 28.5788i 0.727077 1.25933i
\(516\) 0 0
\(517\) −13.5000 + 23.3827i −0.593729 + 1.02837i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 19.5000 + 33.7750i 0.854311 + 1.47971i 0.877283 + 0.479973i \(0.159354\pi\)
−0.0229727 + 0.999736i \(0.507313\pi\)
\(522\) 0 0
\(523\) 0.500000 0.866025i 0.0218635 0.0378686i −0.854887 0.518815i \(-0.826373\pi\)
0.876750 + 0.480946i \(0.159707\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.5000 18.1865i −0.457387 0.792218i
\(528\) 0 0
\(529\) 7.00000 12.1244i 0.304348 0.527146i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 + 10.3923i 0.259889 + 0.450141i
\(534\) 0 0
\(535\) 45.0000 1.94552
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 16.5000 12.9904i 0.710705 0.559535i
\(540\) 0 0
\(541\) −17.5000 + 30.3109i −0.752384 + 1.30317i 0.194281 + 0.980946i \(0.437763\pi\)
−0.946664 + 0.322221i \(0.895571\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.50000 + 2.59808i −0.0642529 + 0.111289i
\(546\) 0 0
\(547\) −4.00000 6.92820i −0.171028 0.296229i 0.767752 0.640747i \(-0.221375\pi\)
−0.938779 + 0.344519i \(0.888042\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 6.50000 33.7750i 0.276408 1.43626i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.5000 28.5788i −0.699127 1.21092i −0.968769 0.247964i \(-0.920239\pi\)
0.269642 0.962961i \(-0.413095\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −9.00000 −0.379305 −0.189652 0.981851i \(-0.560736\pi\)
−0.189652 + 0.981851i \(0.560736\pi\)
\(564\) 0 0
\(565\) 18.0000 0.757266
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.00000 0.377300 0.188650 0.982044i \(-0.439589\pi\)
0.188650 + 0.982044i \(0.439589\pi\)
\(570\) 0 0
\(571\) 29.0000 1.21361 0.606806 0.794850i \(-0.292450\pi\)
0.606806 + 0.794850i \(0.292450\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 0.500000 + 0.866025i 0.0208153 + 0.0360531i 0.876245 0.481865i \(-0.160040\pi\)
−0.855430 + 0.517918i \(0.826707\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −30.0000 + 10.3923i −1.24461 + 0.431145i
\(582\) 0 0
\(583\) −9.00000 −0.372742
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000 + 10.3923i 0.247647 + 0.428936i 0.962872 0.269957i \(-0.0870095\pi\)
−0.715226 + 0.698893i \(0.753676\pi\)
\(588\) 0 0
\(589\) −3.50000 + 6.06218i −0.144215 + 0.249788i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −10.5000 + 18.1865i −0.431183 + 0.746831i −0.996976 0.0777165i \(-0.975237\pi\)
0.565792 + 0.824548i \(0.308570\pi\)
\(594\) 0 0
\(595\) −18.0000 15.5885i −0.737928 0.639064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 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
\(605\) −3.00000 + 5.19615i −0.121967 + 0.211254i
\(606\) 0 0
\(607\) −23.5000 40.7032i −0.953836 1.65209i −0.737011 0.675881i \(-0.763763\pi\)
−0.216825 0.976210i \(-0.569570\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.00000 + 15.5885i −0.364101 + 0.630641i
\(612\) 0 0
\(613\) 12.5000 + 21.6506i 0.504870 + 0.874461i 0.999984 + 0.00563283i \(0.00179300\pi\)
−0.495114 + 0.868828i \(0.664874\pi\)
\(614\) 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
\(619\) 15.5000 26.8468i 0.622998 1.07906i −0.365927 0.930644i \(-0.619248\pi\)
0.988924 0.148420i \(-0.0474187\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 30.0000 + 25.9808i 1.20192 + 1.04090i
\(624\) 0 0
\(625\) 14.5000 + 25.1147i 0.580000 + 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0000 + 20.7846i 0.476205 + 0.824812i
\(636\) 0 0
\(637\) 11.0000 8.66025i 0.435836 0.343132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 7.50000 12.9904i 0.296232 0.513089i −0.679039 0.734103i \(-0.737603\pi\)
0.975271 + 0.221013i \(0.0709364\pi\)
\(642\) 0 0
\(643\) −10.0000 + 17.3205i −0.394362 + 0.683054i −0.993019 0.117951i \(-0.962368\pi\)
0.598658 + 0.801005i \(0.295701\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.5000 + 18.1865i 0.412798 + 0.714986i 0.995194 0.0979182i \(-0.0312184\pi\)
−0.582397 + 0.812905i \(0.697885\pi\)
\(648\) 0 0
\(649\) 13.5000 23.3827i 0.529921 0.917851i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.5000 + 33.7750i 0.763094 + 1.32172i 0.941248 + 0.337715i \(0.109654\pi\)
−0.178154 + 0.984003i \(0.557013\pi\)
\(654\) 0 0
\(655\) −4.50000 + 7.79423i −0.175830 + 0.304546i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 6.00000 + 10.3923i 0.233727 + 0.404827i 0.958902 0.283738i \(-0.0915745\pi\)
−0.725175 + 0.688565i \(0.758241\pi\)
\(660\) 0 0
\(661\) 11.0000 0.427850 0.213925 0.976850i \(-0.431375\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.50000 + 7.79423i −0.0581675 + 0.302247i
\(666\) 0 0
\(667\) 9.00000 15.5885i 0.348481 0.603587i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.50000 2.59808i 0.0579069 0.100298i
\(672\) 0 0
\(673\) −7.00000 12.1244i −0.269830 0.467360i 0.698988 0.715134i \(-0.253634\pi\)
−0.968818 + 0.247774i \(0.920301\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −27.0000 −1.03769 −0.518847 0.854867i \(-0.673639\pi\)
−0.518847 + 0.854867i \(0.673639\pi\)
\(678\) 0 0
\(679\) −25.0000 + 8.66025i −0.959412 + 0.332350i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 10.5000 + 18.1865i 0.401771 + 0.695888i 0.993940 0.109926i \(-0.0350613\pi\)
−0.592168 + 0.805814i \(0.701728\pi\)
\(684\) 0 0
\(685\) −63.0000 −2.40711
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −13.0000 −0.494543 −0.247272 0.968946i \(-0.579534\pi\)
−0.247272 + 0.968946i \(0.579534\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −60.0000 −2.27593
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) −0.500000 0.866025i −0.0188579 0.0326628i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 30.0000 + 25.9808i 1.12827 + 0.977107i
\(708\) 0 0
\(709\) −1.00000 −0.0375558 −0.0187779 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10.5000 18.1865i −0.393228 0.681091i
\(714\) 0 0
\(715\) 9.00000 15.5885i 0.336581 0.582975i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −10.5000 + 18.1865i −0.391584 + 0.678243i −0.992659 0.120950i \(-0.961406\pi\)
0.601075 + 0.799193i \(0.294739\pi\)
\(720\) 0 0
\(721\) 27.5000 9.52628i 1.02415 0.354777i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24.0000 0.891338
\(726\) 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
\(731\) −6.00000 + 10.3923i −0.221918 + 0.384373i
\(732\) 0 0
\(733\) 12.5000 + 21.6506i 0.461698 + 0.799684i 0.999046 0.0436764i \(-0.0139070\pi\)
−0.537348 + 0.843361i \(0.680574\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.5000 18.1865i 0.386772 0.669910i
\(738\) 0 0
\(739\) 9.50000 + 16.4545i 0.349463 + 0.605288i 0.986154 0.165831i \(-0.0530307\pi\)
−0.636691 + 0.771119i \(0.719697\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 + 41.5692i −0.880475 + 1.52503i −0.0296605 + 0.999560i \(0.509443\pi\)
−0.850814 + 0.525467i \(0.823891\pi\)
\(744\) 0 0
\(745\) −4.50000 + 7.79423i −0.164867 + 0.285558i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.0000 + 25.9808i 1.09618 + 0.949316i
\(750\) 0 0
\(751\) 12.5000 + 21.6506i 0.456131 + 0.790043i 0.998752 0.0499348i \(-0.0159013\pi\)
−0.542621 + 0.839978i \(0.682568\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −51.0000 −1.85608
\(756\) 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
\(761\) 1.50000 + 2.59808i 0.0543750 + 0.0941802i 0.891932 0.452170i \(-0.149350\pi\)
−0.837557 + 0.546350i \(0.816017\pi\)
\(762\) 0 0
\(763\) −2.50000 + 0.866025i −0.0905061 + 0.0313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.00000 15.5885i 0.324971 0.562867i
\(768\) 0 0
\(769\) 17.0000 29.4449i 0.613036 1.06181i −0.377690 0.925932i \(-0.623282\pi\)
0.990726 0.135877i \(-0.0433852\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −16.5000 28.5788i −0.593464 1.02791i −0.993762 0.111524i \(-0.964427\pi\)
0.400298 0.916385i \(-0.368907\pi\)
\(774\) 0 0
\(775\) 14.0000 24.2487i 0.502895 0.871039i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.00000 5.19615i −0.107486 0.186171i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −19.5000 33.7750i −0.695985 1.20548i
\(786\) 0 0
\(787\) −31.0000 −1.10503 −0.552515 0.833503i \(-0.686332\pi\)
−0.552515 + 0.833503i \(0.686332\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.0000 + 10.3923i 0.426671 + 0.369508i
\(792\) 0 0
\(793\) 1.00000 1.73205i 0.0355110 0.0615069i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.0000 36.3731i 0.743858 1.28840i −0.206868 0.978369i \(-0.566327\pi\)
0.950726 0.310031i \(-0.100340\pi\)
\(798\) 0 0
\(799\) 13.5000 + 23.3827i 0.477596 + 0.827220i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.00000 −0.105868
\(804\) 0 0
\(805\) −18.0000 15.5885i −0.634417 0.549421i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.5000 28.5788i −0.580109 1.00478i −0.995466 0.0951198i \(-0.969677\pi\)
0.415357 0.909659i \(-0.363657\pi\)
\(810\) 0 0
\(811\) −52.0000 −1.82597 −0.912983 0.407997i \(-0.866228\pi\)
−0.912983 + 0.407997i \(0.866228\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −33.0000 −1.15594
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −27.0000 −0.942306 −0.471153 0.882051i \(-0.656162\pi\)
−0.471153 + 0.882051i \(0.656162\pi\)
\(822\) 0 0
\(823\) 5.00000 0.174289 0.0871445 0.996196i \(-0.472226\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.0000 1.25184 0.625921 0.779886i \(-0.284723\pi\)
0.625921 + 0.779886i \(0.284723\pi\)
\(828\) 0 0
\(829\) 0.500000 + 0.866025i 0.0173657 + 0.0300783i 0.874578 0.484885i \(-0.161139\pi\)
−0.857212 + 0.514964i \(0.827805\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.00000 20.7846i −0.103944 0.720144i
\(834\) 0 0
\(835\) −36.0000 −1.24583
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12.0000 20.7846i −0.414286 0.717564i 0.581067 0.813856i \(-0.302635\pi\)
−0.995353 + 0.0962912i \(0.969302\pi\)
\(840\) 0 0
\(841\) −3.50000 + 6.06218i −0.120690 + 0.209041i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −13.5000 + 23.3827i −0.464414 + 0.804389i
\(846\) 0 0
\(847\) −5.00000 + 1.73205i −0.171802 + 0.0595140i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.00000 0.102839
\(852\) 0 0
\(853\) 11.0000 + 19.0526i 0.376633 + 0.652347i 0.990570 0.137008i \(-0.0437485\pi\)
−0.613937 + 0.789355i \(0.710415\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.50000 12.9904i 0.256195 0.443743i −0.709024 0.705184i \(-0.750864\pi\)
0.965219 + 0.261441i \(0.0841977\pi\)
\(858\) 0 0
\(859\) −11.5000 19.9186i −0.392375 0.679613i 0.600387 0.799709i \(-0.295013\pi\)
−0.992762 + 0.120096i \(0.961680\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.5000 44.1673i 0.868030 1.50347i 0.00402340 0.999992i \(-0.498719\pi\)
0.864007 0.503480i \(-0.167947\pi\)
\(864\) 0 0
\(865\) 13.5000 + 23.3827i 0.459014 + 0.795035i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 19.5000 33.7750i 0.661492 1.14574i
\(870\) 0 0
\(871\) 7.00000 12.1244i 0.237186 0.410818i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.50000 + 7.79423i −0.0507093 + 0.263493i
\(876\) 0 0
\(877\) 6.50000 + 11.2583i 0.219489 + 0.380167i 0.954652 0.297724i \(-0.0962275\pi\)
−0.735163 + 0.677891i \(0.762894\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19.5000 + 33.7750i 0.654746 + 1.13405i 0.981957 + 0.189102i \(0.0605577\pi\)
−0.327212 + 0.944951i \(0.606109\pi\)
\(888\) 0 0
\(889\) −4.00000 + 20.7846i −0.134156 + 0.697093i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.50000 7.79423i 0.150587 0.260824i
\(894\) 0 0
\(895\) −31.5000 + 54.5596i −1.05293 + 1.82373i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 21.0000 + 36.3731i 0.700389 + 1.21311i
\(900\) 0 0
\(901\) −4.50000 + 7.79423i −0.149917 + 0.259663i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −15.0000 25.9808i −0.498617 0.863630i
\(906\) 0 0
\(907\) −20.5000 + 35.5070i −0.680691 + 1.17899i 0.294079 + 0.955781i \(0.404987\pi\)
−0.974770 + 0.223211i \(0.928346\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.0000 + 41.5692i 0.795155 + 1.37725i 0.922740 + 0.385422i \(0.125944\pi\)
−0.127585 + 0.991828i \(0.540723\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7.50000 + 2.59808i −0.247672 + 0.0857960i
\(918\) 0 0
\(919\) 0.500000 0.866025i 0.0164935 0.0285675i −0.857661 0.514216i \(-0.828083\pi\)
0.874154 + 0.485648i \(0.161416\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000 + 3.46410i 0.0657596 + 0.113899i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.00000 0.295280 0.147640 0.989041i \(-0.452832\pi\)
0.147640 + 0.989041i \(0.452832\pi\)
\(930\) 0 0
\(931\) −5.50000 + 4.33013i −0.180255 + 0.141914i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −13.5000 23.3827i −0.441497 0.764696i
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −27.0000 −0.880175 −0.440087 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 0 0
\(943\) 18.0000 0.586161
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 45.0000 1.46230 0.731152 0.682215i \(-0.238983\pi\)
0.731152 + 0.682215i \(0.238983\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.0000 0.583077 0.291539 0.956559i \(-0.405833\pi\)
0.291539 + 0.956559i \(0.405833\pi\)
\(954\) 0 0
\(955\) 13.5000 + 23.3827i 0.436850 + 0.756646i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −42.0000 36.3731i −1.35625 1.17455i
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 16.5000 + 28.5788i 0.531154 + 0.919985i
\(966\) 0 0
\(967\) 8.00000 13.8564i 0.257263 0.445592i −0.708245 0.705967i \(-0.750513\pi\)
0.965508 + 0.260375i \(0.0838461\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 25.5000 44.1673i 0.818334 1.41740i −0.0885751 0.996070i \(-0.528231\pi\)
0.906909 0.421326i \(-0.138435\pi\)
\(972\) 0 0
\(973\) −40.0000 34.6410i −1.28234 1.11054i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.0000 −0.863807 −0.431903 0.901920i \(-0.642158\pi\)
−0.431903 + 0.901920i \(0.642158\pi\)
\(978\) 0 0
\(979\) 22.5000 + 38.9711i 0.719103 + 1.24552i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 16.5000 28.5788i 0.526268 0.911523i −0.473263 0.880921i \(-0.656924\pi\)
0.999532 0.0306024i \(-0.00974257\pi\)
\(984\) 0 0
\(985\) −27.0000 46.7654i −0.860292 1.49007i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −6.00000 + 10.3923i −0.190789 + 0.330456i
\(990\) 0 0
\(991\) 9.50000 + 16.4545i 0.301777 + 0.522694i 0.976539 0.215342i \(-0.0690867\pi\)
−0.674761 + 0.738036i \(0.735753\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −10.5000 + 18.1865i −0.332872 + 0.576552i
\(996\) 0 0
\(997\) −29.5000 + 51.0955i −0.934274 + 1.61821i −0.158352 + 0.987383i \(0.550618\pi\)
−0.775923 + 0.630828i \(0.782715\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2268.2.i.h.2053.1 2
3.2 odd 2 2268.2.i.a.2053.1 2
7.4 even 3 2268.2.l.a.109.1 2
9.2 odd 6 2268.2.l.h.541.1 2
9.4 even 3 252.2.k.c.37.1 2
9.5 odd 6 28.2.e.a.9.1 2
9.7 even 3 2268.2.l.a.541.1 2
21.11 odd 6 2268.2.l.h.109.1 2
36.23 even 6 112.2.i.b.65.1 2
36.31 odd 6 1008.2.s.p.289.1 2
45.14 odd 6 700.2.i.c.401.1 2
45.23 even 12 700.2.r.b.149.2 4
45.32 even 12 700.2.r.b.149.1 4
63.4 even 3 252.2.k.c.109.1 2
63.5 even 6 196.2.a.a.1.1 1
63.11 odd 6 2268.2.i.a.865.1 2
63.13 odd 6 1764.2.k.b.1549.1 2
63.23 odd 6 196.2.a.b.1.1 1
63.25 even 3 inner 2268.2.i.h.865.1 2
63.31 odd 6 1764.2.k.b.361.1 2
63.32 odd 6 28.2.e.a.25.1 yes 2
63.40 odd 6 1764.2.a.j.1.1 1
63.41 even 6 196.2.e.a.177.1 2
63.58 even 3 1764.2.a.a.1.1 1
63.59 even 6 196.2.e.a.165.1 2
72.5 odd 6 448.2.i.e.65.1 2
72.59 even 6 448.2.i.c.65.1 2
252.23 even 6 784.2.a.d.1.1 1
252.59 odd 6 784.2.i.d.753.1 2
252.67 odd 6 1008.2.s.p.865.1 2
252.95 even 6 112.2.i.b.81.1 2
252.103 even 6 7056.2.a.bw.1.1 1
252.131 odd 6 784.2.a.g.1.1 1
252.167 odd 6 784.2.i.d.177.1 2
252.247 odd 6 7056.2.a.f.1.1 1
315.23 even 12 4900.2.e.i.2549.2 2
315.32 even 12 700.2.r.b.249.2 4
315.68 odd 12 4900.2.e.h.2549.1 2
315.149 odd 6 4900.2.a.g.1.1 1
315.158 even 12 700.2.r.b.249.1 4
315.194 even 6 4900.2.a.n.1.1 1
315.212 even 12 4900.2.e.i.2549.1 2
315.257 odd 12 4900.2.e.h.2549.2 2
315.284 odd 6 700.2.i.c.501.1 2
504.5 even 6 3136.2.a.v.1.1 1
504.131 odd 6 3136.2.a.k.1.1 1
504.149 odd 6 3136.2.a.h.1.1 1
504.221 odd 6 448.2.i.e.193.1 2
504.275 even 6 3136.2.a.s.1.1 1
504.347 even 6 448.2.i.c.193.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
28.2.e.a.9.1 2 9.5 odd 6
28.2.e.a.25.1 yes 2 63.32 odd 6
112.2.i.b.65.1 2 36.23 even 6
112.2.i.b.81.1 2 252.95 even 6
196.2.a.a.1.1 1 63.5 even 6
196.2.a.b.1.1 1 63.23 odd 6
196.2.e.a.165.1 2 63.59 even 6
196.2.e.a.177.1 2 63.41 even 6
252.2.k.c.37.1 2 9.4 even 3
252.2.k.c.109.1 2 63.4 even 3
448.2.i.c.65.1 2 72.59 even 6
448.2.i.c.193.1 2 504.347 even 6
448.2.i.e.65.1 2 72.5 odd 6
448.2.i.e.193.1 2 504.221 odd 6
700.2.i.c.401.1 2 45.14 odd 6
700.2.i.c.501.1 2 315.284 odd 6
700.2.r.b.149.1 4 45.32 even 12
700.2.r.b.149.2 4 45.23 even 12
700.2.r.b.249.1 4 315.158 even 12
700.2.r.b.249.2 4 315.32 even 12
784.2.a.d.1.1 1 252.23 even 6
784.2.a.g.1.1 1 252.131 odd 6
784.2.i.d.177.1 2 252.167 odd 6
784.2.i.d.753.1 2 252.59 odd 6
1008.2.s.p.289.1 2 36.31 odd 6
1008.2.s.p.865.1 2 252.67 odd 6
1764.2.a.a.1.1 1 63.58 even 3
1764.2.a.j.1.1 1 63.40 odd 6
1764.2.k.b.361.1 2 63.31 odd 6
1764.2.k.b.1549.1 2 63.13 odd 6
2268.2.i.a.865.1 2 63.11 odd 6
2268.2.i.a.2053.1 2 3.2 odd 2
2268.2.i.h.865.1 2 63.25 even 3 inner
2268.2.i.h.2053.1 2 1.1 even 1 trivial
2268.2.l.a.109.1 2 7.4 even 3
2268.2.l.a.541.1 2 9.7 even 3
2268.2.l.h.109.1 2 21.11 odd 6
2268.2.l.h.541.1 2 9.2 odd 6
3136.2.a.h.1.1 1 504.149 odd 6
3136.2.a.k.1.1 1 504.131 odd 6
3136.2.a.s.1.1 1 504.275 even 6
3136.2.a.v.1.1 1 504.5 even 6
4900.2.a.g.1.1 1 315.149 odd 6
4900.2.a.n.1.1 1 315.194 even 6
4900.2.e.h.2549.1 2 315.68 odd 12
4900.2.e.h.2549.2 2 315.257 odd 12
4900.2.e.i.2549.1 2 315.212 even 12
4900.2.e.i.2549.2 2 315.23 even 12
7056.2.a.f.1.1 1 252.247 odd 6
7056.2.a.bw.1.1 1 252.103 even 6