Properties

Label 576.2.i.a.193.1
Level $576$
Weight $2$
Character 576.193
Analytic conductor $4.599$
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] = [576,2,Mod(193,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(4.59938315643\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 576.193
Dual form 576.2.i.a.385.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-1.50000 + 0.866025i) q^{3} +(1.00000 + 1.73205i) q^{7} +(1.50000 - 2.59808i) q^{9} +(1.50000 + 2.59808i) q^{11} +(1.00000 - 1.73205i) q^{13} -3.00000 q^{17} -1.00000 q^{19} +(-3.00000 - 1.73205i) q^{21} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +5.19615i q^{27} +(3.00000 + 5.19615i) q^{29} +(-2.00000 + 3.46410i) q^{31} +(-4.50000 - 2.59808i) q^{33} +4.00000 q^{37} +3.46410i q^{39} +(-4.50000 + 7.79423i) q^{41} +(0.500000 + 0.866025i) q^{43} +(-3.00000 - 5.19615i) q^{47} +(1.50000 - 2.59808i) q^{49} +(4.50000 - 2.59808i) q^{51} -12.0000 q^{53} +(1.50000 - 0.866025i) q^{57} +(-1.50000 + 2.59808i) q^{59} +(4.00000 + 6.92820i) q^{61} +6.00000 q^{63} +(-2.50000 + 4.33013i) q^{67} -10.3923i q^{69} +12.0000 q^{71} +11.0000 q^{73} +(-7.50000 - 4.33013i) q^{75} +(-3.00000 + 5.19615i) q^{77} +(-2.00000 - 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-6.00000 - 10.3923i) q^{83} +(-9.00000 - 5.19615i) q^{87} +6.00000 q^{89} +4.00000 q^{91} -6.92820i q^{93} +(-2.50000 - 4.33013i) q^{97} +9.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 3 q^{11} + 2 q^{13} - 6 q^{17} - 2 q^{19} - 6 q^{21} - 6 q^{23} + 5 q^{25} + 6 q^{29} - 4 q^{31} - 9 q^{33} + 8 q^{37} - 9 q^{41} + q^{43} - 6 q^{47} + 3 q^{49} + 9 q^{51} - 24 q^{53} + 3 q^{57} - 3 q^{59} + 8 q^{61} + 12 q^{63} - 5 q^{67} + 24 q^{71} + 22 q^{73} - 15 q^{75} - 6 q^{77} - 4 q^{79} - 9 q^{81} - 12 q^{83} - 18 q^{87} + 12 q^{89} + 8 q^{91} - 5 q^{97} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.50000 + 0.866025i −0.866025 + 0.500000i
\(4\) 0 0
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 1.00000 + 1.73205i 0.377964 + 0.654654i 0.990766 0.135583i \(-0.0432908\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(8\) 0 0
\(9\) 1.50000 2.59808i 0.500000 0.866025i
\(10\) 0 0
\(11\) 1.50000 + 2.59808i 0.452267 + 0.783349i 0.998526 0.0542666i \(-0.0172821\pi\)
−0.546259 + 0.837616i \(0.683949\pi\)
\(12\) 0 0
\(13\) 1.00000 1.73205i 0.277350 0.480384i −0.693375 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −3.00000 1.73205i −0.654654 0.377964i
\(22\) 0 0
\(23\) −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i \(0.381789\pi\)
−0.988436 + 0.151642i \(0.951544\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 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
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) −4.50000 2.59808i −0.783349 0.452267i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 3.46410i 0.554700i
\(40\) 0 0
\(41\) −4.50000 + 7.79423i −0.702782 + 1.21725i 0.264704 + 0.964330i \(0.414726\pi\)
−0.967486 + 0.252924i \(0.918608\pi\)
\(42\) 0 0
\(43\) 0.500000 + 0.866025i 0.0762493 + 0.132068i 0.901629 0.432511i \(-0.142372\pi\)
−0.825380 + 0.564578i \(0.809039\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 5.19615i −0.437595 0.757937i 0.559908 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 4.50000 2.59808i 0.630126 0.363803i
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.50000 0.866025i 0.198680 0.114708i
\(58\) 0 0
\(59\) −1.50000 + 2.59808i −0.195283 + 0.338241i −0.946993 0.321253i \(-0.895896\pi\)
0.751710 + 0.659494i \(0.229229\pi\)
\(60\) 0 0
\(61\) 4.00000 + 6.92820i 0.512148 + 0.887066i 0.999901 + 0.0140840i \(0.00448323\pi\)
−0.487753 + 0.872982i \(0.662183\pi\)
\(62\) 0 0
\(63\) 6.00000 0.755929
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.50000 + 4.33013i −0.305424 + 0.529009i −0.977356 0.211604i \(-0.932131\pi\)
0.671932 + 0.740613i \(0.265465\pi\)
\(68\) 0 0
\(69\) 10.3923i 1.25109i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 11.0000 1.28745 0.643726 0.765256i \(-0.277388\pi\)
0.643726 + 0.765256i \(0.277388\pi\)
\(74\) 0 0
\(75\) −7.50000 4.33013i −0.866025 0.500000i
\(76\) 0 0
\(77\) −3.00000 + 5.19615i −0.341882 + 0.592157i
\(78\) 0 0
\(79\) −2.00000 3.46410i −0.225018 0.389742i 0.731307 0.682048i \(-0.238911\pi\)
−0.956325 + 0.292306i \(0.905577\pi\)
\(80\) 0 0
\(81\) −4.50000 7.79423i −0.500000 0.866025i
\(82\) 0 0
\(83\) −6.00000 10.3923i −0.658586 1.14070i −0.980982 0.194099i \(-0.937822\pi\)
0.322396 0.946605i \(-0.395512\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.00000 5.19615i −0.964901 0.557086i
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 6.92820i 0.718421i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.50000 4.33013i −0.253837 0.439658i 0.710742 0.703452i \(-0.248359\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) 9.00000 0.904534
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) 7.00000 12.1244i 0.689730 1.19465i −0.282194 0.959357i \(-0.591062\pi\)
0.971925 0.235291i \(-0.0756043\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.00000 0.290021 0.145010 0.989430i \(-0.453678\pi\)
0.145010 + 0.989430i \(0.453678\pi\)
\(108\) 0 0
\(109\) 16.0000 1.53252 0.766261 0.642529i \(-0.222115\pi\)
0.766261 + 0.642529i \(0.222115\pi\)
\(110\) 0 0
\(111\) −6.00000 + 3.46410i −0.569495 + 0.328798i
\(112\) 0 0
\(113\) −3.00000 + 5.19615i −0.282216 + 0.488813i −0.971930 0.235269i \(-0.924403\pi\)
0.689714 + 0.724082i \(0.257736\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.00000 5.19615i −0.277350 0.480384i
\(118\) 0 0
\(119\) −3.00000 5.19615i −0.275010 0.476331i
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 15.5885i 1.40556i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) −1.50000 0.866025i −0.132068 0.0762493i
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) −1.00000 1.73205i −0.0867110 0.150188i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.50000 + 2.59808i 0.128154 + 0.221969i 0.922961 0.384893i \(-0.125762\pi\)
−0.794808 + 0.606861i \(0.792428\pi\)
\(138\) 0 0
\(139\) 9.50000 16.4545i 0.805779 1.39565i −0.109984 0.993933i \(-0.535080\pi\)
0.915764 0.401718i \(-0.131587\pi\)
\(140\) 0 0
\(141\) 9.00000 + 5.19615i 0.757937 + 0.437595i
\(142\) 0 0
\(143\) 6.00000 0.501745
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.19615i 0.428571i
\(148\) 0 0
\(149\) −3.00000 + 5.19615i −0.245770 + 0.425685i −0.962348 0.271821i \(-0.912374\pi\)
0.716578 + 0.697507i \(0.245707\pi\)
\(150\) 0 0
\(151\) −5.00000 8.66025i −0.406894 0.704761i 0.587646 0.809118i \(-0.300055\pi\)
−0.994540 + 0.104357i \(0.966722\pi\)
\(152\) 0 0
\(153\) −4.50000 + 7.79423i −0.363803 + 0.630126i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −2.00000 + 3.46410i −0.159617 + 0.276465i −0.934731 0.355357i \(-0.884359\pi\)
0.775113 + 0.631822i \(0.217693\pi\)
\(158\) 0 0
\(159\) 18.0000 10.3923i 1.42749 0.824163i
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 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
\(171\) −1.50000 + 2.59808i −0.114708 + 0.198680i
\(172\) 0 0
\(173\) −3.00000 5.19615i −0.228086 0.395056i 0.729155 0.684349i \(-0.239913\pi\)
−0.957241 + 0.289292i \(0.906580\pi\)
\(174\) 0 0
\(175\) −5.00000 + 8.66025i −0.377964 + 0.654654i
\(176\) 0 0
\(177\) 5.19615i 0.390567i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 0 0
\(183\) −12.0000 6.92820i −0.887066 0.512148i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.50000 7.79423i −0.329073 0.569970i
\(188\) 0 0
\(189\) −9.00000 + 5.19615i −0.654654 + 0.377964i
\(190\) 0 0
\(191\) −9.00000 15.5885i −0.651217 1.12794i −0.982828 0.184525i \(-0.940925\pi\)
0.331611 0.943416i \(-0.392408\pi\)
\(192\) 0 0
\(193\) −2.50000 + 4.33013i −0.179954 + 0.311689i −0.941865 0.335993i \(-0.890928\pi\)
0.761911 + 0.647682i \(0.224262\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) 8.66025i 0.610847i
\(202\) 0 0
\(203\) −6.00000 + 10.3923i −0.421117 + 0.729397i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 9.00000 + 15.5885i 0.625543 + 1.08347i
\(208\) 0 0
\(209\) −1.50000 2.59808i −0.103757 0.179713i
\(210\) 0 0
\(211\) −10.0000 + 17.3205i −0.688428 + 1.19239i 0.283918 + 0.958849i \(0.408366\pi\)
−0.972346 + 0.233544i \(0.924968\pi\)
\(212\) 0 0
\(213\) −18.0000 + 10.3923i −1.23334 + 0.712069i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) −16.5000 + 9.52628i −1.11497 + 0.643726i
\(220\) 0 0
\(221\) −3.00000 + 5.19615i −0.201802 + 0.349531i
\(222\) 0 0
\(223\) 13.0000 + 22.5167i 0.870544 + 1.50783i 0.861435 + 0.507869i \(0.169566\pi\)
0.00910984 + 0.999959i \(0.497100\pi\)
\(224\) 0 0
\(225\) 15.0000 1.00000
\(226\) 0 0
\(227\) −10.5000 18.1865i −0.696909 1.20708i −0.969533 0.244962i \(-0.921225\pi\)
0.272623 0.962121i \(-0.412109\pi\)
\(228\) 0 0
\(229\) 7.00000 12.1244i 0.462573 0.801200i −0.536515 0.843891i \(-0.680260\pi\)
0.999088 + 0.0426906i \(0.0135930\pi\)
\(230\) 0 0
\(231\) 10.3923i 0.683763i
\(232\) 0 0
\(233\) 3.00000 0.196537 0.0982683 0.995160i \(-0.468670\pi\)
0.0982683 + 0.995160i \(0.468670\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.00000 + 3.46410i 0.389742 + 0.225018i
\(238\) 0 0
\(239\) 3.00000 5.19615i 0.194054 0.336111i −0.752536 0.658551i \(-0.771170\pi\)
0.946590 + 0.322440i \(0.104503\pi\)
\(240\) 0 0
\(241\) 3.50000 + 6.06218i 0.225455 + 0.390499i 0.956456 0.291877i \(-0.0942799\pi\)
−0.731001 + 0.682376i \(0.760947\pi\)
\(242\) 0 0
\(243\) 13.5000 + 7.79423i 0.866025 + 0.500000i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.00000 + 1.73205i −0.0636285 + 0.110208i
\(248\) 0 0
\(249\) 18.0000 + 10.3923i 1.14070 + 0.658586i
\(250\) 0 0
\(251\) 21.0000 1.32551 0.662754 0.748837i \(-0.269387\pi\)
0.662754 + 0.748837i \(0.269387\pi\)
\(252\) 0 0
\(253\) −18.0000 −1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.5000 18.1865i 0.654972 1.13444i −0.326929 0.945049i \(-0.606014\pi\)
0.981901 0.189396i \(-0.0606529\pi\)
\(258\) 0 0
\(259\) 4.00000 + 6.92820i 0.248548 + 0.430498i
\(260\) 0 0
\(261\) 18.0000 1.11417
\(262\) 0 0
\(263\) 9.00000 + 15.5885i 0.554964 + 0.961225i 0.997906 + 0.0646755i \(0.0206012\pi\)
−0.442943 + 0.896550i \(0.646065\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −9.00000 + 5.19615i −0.550791 + 0.317999i
\(268\) 0 0
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 0 0
\(273\) −6.00000 + 3.46410i −0.363137 + 0.209657i
\(274\) 0 0
\(275\) −7.50000 + 12.9904i −0.452267 + 0.783349i
\(276\) 0 0
\(277\) −5.00000 8.66025i −0.300421 0.520344i 0.675810 0.737075i \(-0.263794\pi\)
−0.976231 + 0.216731i \(0.930460\pi\)
\(278\) 0 0
\(279\) 6.00000 + 10.3923i 0.359211 + 0.622171i
\(280\) 0 0
\(281\) −3.00000 5.19615i −0.178965 0.309976i 0.762561 0.646916i \(-0.223942\pi\)
−0.941526 + 0.336939i \(0.890608\pi\)
\(282\) 0 0
\(283\) 2.00000 3.46410i 0.118888 0.205919i −0.800439 0.599414i \(-0.795400\pi\)
0.919327 + 0.393494i \(0.128734\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −18.0000 −1.06251
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 7.50000 + 4.33013i 0.439658 + 0.253837i
\(292\) 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
\(297\) −13.5000 + 7.79423i −0.783349 + 0.452267i
\(298\) 0 0
\(299\) 6.00000 + 10.3923i 0.346989 + 0.601003i
\(300\) 0 0
\(301\) −1.00000 + 1.73205i −0.0576390 + 0.0998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 24.2487i 1.37946i
\(310\) 0 0
\(311\) −9.00000 + 15.5885i −0.510343 + 0.883940i 0.489585 + 0.871956i \(0.337148\pi\)
−0.999928 + 0.0119847i \(0.996185\pi\)
\(312\) 0 0
\(313\) −14.5000 25.1147i −0.819588 1.41957i −0.905986 0.423308i \(-0.860869\pi\)
0.0863973 0.996261i \(-0.472465\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.00000 15.5885i −0.505490 0.875535i −0.999980 0.00635137i \(-0.997978\pi\)
0.494489 0.869184i \(-0.335355\pi\)
\(318\) 0 0
\(319\) −9.00000 + 15.5885i −0.503903 + 0.872786i
\(320\) 0 0
\(321\) −4.50000 + 2.59808i −0.251166 + 0.145010i
\(322\) 0 0
\(323\) 3.00000 0.166924
\(324\) 0 0
\(325\) 10.0000 0.554700
\(326\) 0 0
\(327\) −24.0000 + 13.8564i −1.32720 + 0.766261i
\(328\) 0 0
\(329\) 6.00000 10.3923i 0.330791 0.572946i
\(330\) 0 0
\(331\) 2.00000 + 3.46410i 0.109930 + 0.190404i 0.915742 0.401768i \(-0.131604\pi\)
−0.805812 + 0.592172i \(0.798271\pi\)
\(332\) 0 0
\(333\) 6.00000 10.3923i 0.328798 0.569495i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.500000 0.866025i 0.0272367 0.0471754i −0.852086 0.523402i \(-0.824663\pi\)
0.879322 + 0.476227i \(0.157996\pi\)
\(338\) 0 0
\(339\) 10.3923i 0.564433i
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.5000 + 28.5788i −0.885766 + 1.53419i −0.0409337 + 0.999162i \(0.513033\pi\)
−0.844833 + 0.535031i \(0.820300\pi\)
\(348\) 0 0
\(349\) −8.00000 13.8564i −0.428230 0.741716i 0.568486 0.822693i \(-0.307529\pi\)
−0.996716 + 0.0809766i \(0.974196\pi\)
\(350\) 0 0
\(351\) 9.00000 + 5.19615i 0.480384 + 0.277350i
\(352\) 0 0
\(353\) 10.5000 + 18.1865i 0.558859 + 0.967972i 0.997592 + 0.0693543i \(0.0220939\pi\)
−0.438733 + 0.898617i \(0.644573\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 9.00000 + 5.19615i 0.476331 + 0.275010i
\(358\) 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
\(363\) 3.46410i 0.181818i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −14.0000 24.2487i −0.730794 1.26577i −0.956544 0.291587i \(-0.905817\pi\)
0.225750 0.974185i \(-0.427517\pi\)
\(368\) 0 0
\(369\) 13.5000 + 23.3827i 0.702782 + 1.21725i
\(370\) 0 0
\(371\) −12.0000 20.7846i −0.623009 1.07908i
\(372\) 0 0
\(373\) −17.0000 + 29.4449i −0.880227 + 1.52460i −0.0291379 + 0.999575i \(0.509276\pi\)
−0.851089 + 0.525022i \(0.824057\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) 23.0000 1.18143 0.590715 0.806880i \(-0.298846\pi\)
0.590715 + 0.806880i \(0.298846\pi\)
\(380\) 0 0
\(381\) 3.00000 1.73205i 0.153695 0.0887357i
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3.00000 0.152499
\(388\) 0 0
\(389\) 9.00000 + 15.5885i 0.456318 + 0.790366i 0.998763 0.0497253i \(-0.0158346\pi\)
−0.542445 + 0.840091i \(0.682501\pi\)
\(390\) 0 0
\(391\) 9.00000 15.5885i 0.455150 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −20.0000 −1.00377 −0.501886 0.864934i \(-0.667360\pi\)
−0.501886 + 0.864934i \(0.667360\pi\)
\(398\) 0 0
\(399\) 3.00000 + 1.73205i 0.150188 + 0.0867110i
\(400\) 0 0
\(401\) 13.5000 23.3827i 0.674158 1.16768i −0.302556 0.953131i \(-0.597840\pi\)
0.976714 0.214544i \(-0.0688266\pi\)
\(402\) 0 0
\(403\) 4.00000 + 6.92820i 0.199254 + 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000 + 10.3923i 0.297409 + 0.515127i
\(408\) 0 0
\(409\) −8.50000 + 14.7224i −0.420298 + 0.727977i −0.995968 0.0897044i \(-0.971408\pi\)
0.575670 + 0.817682i \(0.304741\pi\)
\(410\) 0 0
\(411\) −4.50000 2.59808i −0.221969 0.128154i
\(412\) 0 0
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 32.9090i 1.61156i
\(418\) 0 0
\(419\) 6.00000 10.3923i 0.293119 0.507697i −0.681426 0.731887i \(-0.738640\pi\)
0.974546 + 0.224189i \(0.0719734\pi\)
\(420\) 0 0
\(421\) 10.0000 + 17.3205i 0.487370 + 0.844150i 0.999895 0.0145228i \(-0.00462290\pi\)
−0.512524 + 0.858673i \(0.671290\pi\)
\(422\) 0 0
\(423\) −18.0000 −0.875190
\(424\) 0 0
\(425\) −7.50000 12.9904i −0.363803 0.630126i
\(426\) 0 0
\(427\) −8.00000 + 13.8564i −0.387147 + 0.670559i
\(428\) 0 0
\(429\) −9.00000 + 5.19615i −0.434524 + 0.250873i
\(430\) 0 0
\(431\) 30.0000 1.44505 0.722525 0.691345i \(-0.242982\pi\)
0.722525 + 0.691345i \(0.242982\pi\)
\(432\) 0 0
\(433\) −7.00000 −0.336399 −0.168199 0.985753i \(-0.553795\pi\)
−0.168199 + 0.985753i \(0.553795\pi\)
\(434\) 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 0.945552 0.325471i \(-0.105523\pi\)
−0.754642 + 0.656136i \(0.772190\pi\)
\(440\) 0 0
\(441\) −4.50000 7.79423i −0.214286 0.371154i
\(442\) 0 0
\(443\) −1.50000 2.59808i −0.0712672 0.123438i 0.828190 0.560448i \(-0.189371\pi\)
−0.899457 + 0.437009i \(0.856038\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 10.3923i 0.491539i
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) 0 0
\(453\) 15.0000 + 8.66025i 0.704761 + 0.406894i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.50000 14.7224i −0.397613 0.688686i 0.595818 0.803120i \(-0.296828\pi\)
−0.993431 + 0.114433i \(0.963495\pi\)
\(458\) 0 0
\(459\) 15.5885i 0.727607i
\(460\) 0 0
\(461\) −15.0000 25.9808i −0.698620 1.21004i −0.968945 0.247276i \(-0.920465\pi\)
0.270326 0.962769i \(-0.412869\pi\)
\(462\) 0 0
\(463\) 10.0000 17.3205i 0.464739 0.804952i −0.534450 0.845200i \(-0.679481\pi\)
0.999190 + 0.0402476i \(0.0128147\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −15.0000 −0.694117 −0.347059 0.937843i \(-0.612820\pi\)
−0.347059 + 0.937843i \(0.612820\pi\)
\(468\) 0 0
\(469\) −10.0000 −0.461757
\(470\) 0 0
\(471\) 6.92820i 0.319235i
\(472\) 0 0
\(473\) −1.50000 + 2.59808i −0.0689701 + 0.119460i
\(474\) 0 0
\(475\) −2.50000 4.33013i −0.114708 0.198680i
\(476\) 0 0
\(477\) −18.0000 + 31.1769i −0.824163 + 1.42749i
\(478\) 0 0
\(479\) −21.0000 36.3731i −0.959514 1.66193i −0.723681 0.690134i \(-0.757551\pi\)
−0.235833 0.971794i \(-0.575782\pi\)
\(480\) 0 0
\(481\) 4.00000 6.92820i 0.182384 0.315899i
\(482\) 0 0
\(483\) 18.0000 10.3923i 0.819028 0.472866i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) 6.00000 3.46410i 0.271329 0.156652i
\(490\) 0 0
\(491\) 7.50000 12.9904i 0.338470 0.586248i −0.645675 0.763612i \(-0.723424\pi\)
0.984145 + 0.177365i \(0.0567572\pi\)
\(492\) 0 0
\(493\) −9.00000 15.5885i −0.405340 0.702069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 12.0000 + 20.7846i 0.538274 + 0.932317i
\(498\) 0 0
\(499\) 6.50000 11.2583i 0.290980 0.503992i −0.683062 0.730361i \(-0.739352\pi\)
0.974042 + 0.226369i \(0.0726854\pi\)
\(500\) 0 0
\(501\) 20.7846i 0.928588i
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −13.5000 7.79423i −0.599556 0.346154i
\(508\) 0 0
\(509\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(510\) 0 0
\(511\) 11.0000 + 19.0526i 0.486611 + 0.842836i
\(512\) 0 0
\(513\) 5.19615i 0.229416i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 9.00000 15.5885i 0.395820 0.685580i
\(518\) 0 0
\(519\) 9.00000 + 5.19615i 0.395056 + 0.228086i
\(520\) 0 0
\(521\) −3.00000 −0.131432 −0.0657162 0.997838i \(-0.520933\pi\)
−0.0657162 + 0.997838i \(0.520933\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 0 0
\(525\) 17.3205i 0.755929i
\(526\) 0 0
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 4.50000 + 7.79423i 0.195283 + 0.338241i
\(532\) 0 0
\(533\) 9.00000 + 15.5885i 0.389833 + 0.675211i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −18.0000 + 10.3923i −0.776757 + 0.448461i
\(538\) 0 0
\(539\) 9.00000 0.387657
\(540\) 0 0
\(541\) 4.00000 0.171973 0.0859867 0.996296i \(-0.472596\pi\)
0.0859867 + 0.996296i \(0.472596\pi\)
\(542\) 0 0
\(543\) 21.0000 12.1244i 0.901196 0.520306i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.500000 + 0.866025i 0.0213785 + 0.0370286i 0.876517 0.481371i \(-0.159861\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) 24.0000 1.02430
\(550\) 0 0
\(551\) −3.00000 5.19615i −0.127804 0.221364i
\(552\) 0 0
\(553\) 4.00000 6.92820i 0.170097 0.294617i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 13.5000 + 7.79423i 0.569970 + 0.329073i
\(562\) 0 0
\(563\) 19.5000 33.7750i 0.821827 1.42345i −0.0824933 0.996592i \(-0.526288\pi\)
0.904320 0.426855i \(-0.140378\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 9.00000 15.5885i 0.377964 0.654654i
\(568\) 0 0
\(569\) −22.5000 38.9711i −0.943249 1.63376i −0.759220 0.650835i \(-0.774419\pi\)
−0.184030 0.982921i \(-0.558914\pi\)
\(570\) 0 0
\(571\) 18.5000 32.0429i 0.774201 1.34096i −0.161042 0.986948i \(-0.551485\pi\)
0.935243 0.354008i \(-0.115181\pi\)
\(572\) 0 0
\(573\) 27.0000 + 15.5885i 1.12794 + 0.651217i
\(574\) 0 0
\(575\) −30.0000 −1.25109
\(576\) 0 0
\(577\) 11.0000 0.457936 0.228968 0.973434i \(-0.426465\pi\)
0.228968 + 0.973434i \(0.426465\pi\)
\(578\) 0 0
\(579\) 8.66025i 0.359908i
\(580\) 0 0
\(581\) 12.0000 20.7846i 0.497844 0.862291i
\(582\) 0 0
\(583\) −18.0000 31.1769i −0.745484 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.50000 + 7.79423i 0.185735 + 0.321702i 0.943824 0.330449i \(-0.107200\pi\)
−0.758089 + 0.652151i \(0.773867\pi\)
\(588\) 0 0
\(589\) 2.00000 3.46410i 0.0824086 0.142736i
\(590\) 0 0
\(591\) −18.0000 + 10.3923i −0.740421 + 0.427482i
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −15.0000 + 8.66025i −0.613909 + 0.354441i
\(598\) 0 0
\(599\) 6.00000 10.3923i 0.245153 0.424618i −0.717021 0.697051i \(-0.754495\pi\)
0.962175 + 0.272433i \(0.0878284\pi\)
\(600\) 0 0
\(601\) 18.5000 + 32.0429i 0.754631 + 1.30706i 0.945558 + 0.325455i \(0.105517\pi\)
−0.190927 + 0.981604i \(0.561149\pi\)
\(602\) 0 0
\(603\) 7.50000 + 12.9904i 0.305424 + 0.529009i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −14.0000 + 24.2487i −0.568242 + 0.984225i 0.428497 + 0.903543i \(0.359043\pi\)
−0.996740 + 0.0806818i \(0.974290\pi\)
\(608\) 0 0
\(609\) 20.7846i 0.842235i
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 16.0000 0.646234 0.323117 0.946359i \(-0.395269\pi\)
0.323117 + 0.946359i \(0.395269\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.5000 + 23.3827i −0.543490 + 0.941351i 0.455211 + 0.890384i \(0.349564\pi\)
−0.998700 + 0.0509678i \(0.983769\pi\)
\(618\) 0 0
\(619\) −17.5000 30.3109i −0.703384 1.21830i −0.967271 0.253744i \(-0.918338\pi\)
0.263887 0.964554i \(-0.414995\pi\)
\(620\) 0 0
\(621\) −27.0000 15.5885i −1.08347 0.625543i
\(622\) 0 0
\(623\) 6.00000 + 10.3923i 0.240385 + 0.416359i
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 4.50000 + 2.59808i 0.179713 + 0.103757i
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 34.6410i 1.37686i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.00000 5.19615i −0.118864 0.205879i
\(638\) 0 0
\(639\) 18.0000 31.1769i 0.712069 1.23334i
\(640\) 0 0
\(641\) 1.50000 + 2.59808i 0.0592464 + 0.102618i 0.894127 0.447813i \(-0.147797\pi\)
−0.834881 + 0.550431i \(0.814464\pi\)
\(642\) 0 0
\(643\) −11.5000 + 19.9186i −0.453516 + 0.785512i −0.998602 0.0528680i \(-0.983164\pi\)
0.545086 + 0.838380i \(0.316497\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) −9.00000 −0.353281
\(650\) 0 0
\(651\) 12.0000 6.92820i 0.470317 0.271538i
\(652\) 0 0
\(653\) −3.00000 + 5.19615i −0.117399 + 0.203341i −0.918736 0.394872i \(-0.870789\pi\)
0.801337 + 0.598213i \(0.204122\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 16.5000 28.5788i 0.643726 1.11497i
\(658\) 0 0
\(659\) 18.0000 + 31.1769i 0.701180 + 1.21448i 0.968052 + 0.250748i \(0.0806766\pi\)
−0.266872 + 0.963732i \(0.585990\pi\)
\(660\) 0 0
\(661\) −2.00000 + 3.46410i −0.0777910 + 0.134738i −0.902297 0.431116i \(-0.858120\pi\)
0.824506 + 0.565854i \(0.191453\pi\)
\(662\) 0 0
\(663\) 10.3923i 0.403604i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −36.0000 −1.39393
\(668\) 0 0
\(669\) −39.0000 22.5167i −1.50783 0.870544i
\(670\) 0 0
\(671\) −12.0000 + 20.7846i −0.463255 + 0.802381i
\(672\) 0 0
\(673\) 11.0000 + 19.0526i 0.424019 + 0.734422i 0.996328 0.0856156i \(-0.0272857\pi\)
−0.572309 + 0.820038i \(0.693952\pi\)
\(674\) 0 0
\(675\) −22.5000 + 12.9904i −0.866025 + 0.500000i
\(676\) 0 0
\(677\) 18.0000 + 31.1769i 0.691796 + 1.19823i 0.971249 + 0.238067i \(0.0765137\pi\)
−0.279453 + 0.960159i \(0.590153\pi\)
\(678\) 0 0
\(679\) 5.00000 8.66025i 0.191882 0.332350i
\(680\) 0 0
\(681\) 31.5000 + 18.1865i 1.20708 + 0.696909i
\(682\) 0 0
\(683\) −9.00000 −0.344375 −0.172188 0.985064i \(-0.555084\pi\)
−0.172188 + 0.985064i \(0.555084\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 24.2487i 0.925146i
\(688\) 0 0
\(689\) −12.0000 + 20.7846i −0.457164 + 0.791831i
\(690\) 0 0
\(691\) −4.00000 6.92820i −0.152167 0.263561i 0.779857 0.625958i \(-0.215292\pi\)
−0.932024 + 0.362397i \(0.881959\pi\)
\(692\) 0 0
\(693\) 9.00000 + 15.5885i 0.341882 + 0.592157i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13.5000 23.3827i 0.511349 0.885682i
\(698\) 0 0
\(699\) −4.50000 + 2.59808i −0.170206 + 0.0982683i
\(700\) 0 0
\(701\) −30.0000 −1.13308 −0.566542 0.824033i \(-0.691719\pi\)
−0.566542 + 0.824033i \(0.691719\pi\)
\(702\) 0 0
\(703\) −4.00000 −0.150863
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.00000 3.46410i −0.0751116 0.130097i 0.826023 0.563636i \(-0.190598\pi\)
−0.901135 + 0.433539i \(0.857265\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) −12.0000 20.7846i −0.449404 0.778390i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 10.3923i 0.388108i
\(718\) 0 0
\(719\) 36.0000 1.34257 0.671287 0.741198i \(-0.265742\pi\)
0.671287 + 0.741198i \(0.265742\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) −10.5000 6.06218i −0.390499 0.225455i
\(724\) 0 0
\(725\) −15.0000 + 25.9808i −0.557086 + 0.964901i
\(726\) 0 0
\(727\) 13.0000 + 22.5167i 0.482143 + 0.835097i 0.999790 0.0204978i \(-0.00652512\pi\)
−0.517647 + 0.855595i \(0.673192\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −1.50000 2.59808i −0.0554795 0.0960933i
\(732\) 0 0
\(733\) 7.00000 12.1244i 0.258551 0.447823i −0.707303 0.706910i \(-0.750088\pi\)
0.965854 + 0.259087i \(0.0834217\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −15.0000 −0.552532
\(738\) 0 0
\(739\) 47.0000 1.72892 0.864461 0.502699i \(-0.167660\pi\)
0.864461 + 0.502699i \(0.167660\pi\)
\(740\) 0 0
\(741\) 3.46410i 0.127257i
\(742\) 0 0
\(743\) 3.00000 5.19615i 0.110059 0.190628i −0.805735 0.592277i \(-0.798229\pi\)
0.915794 + 0.401648i \(0.131563\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −36.0000 −1.31717
\(748\) 0 0
\(749\) 3.00000 + 5.19615i 0.109618 + 0.189863i
\(750\) 0 0
\(751\) 4.00000 6.92820i 0.145962 0.252814i −0.783769 0.621052i \(-0.786706\pi\)
0.929731 + 0.368238i \(0.120039\pi\)
\(752\) 0 0
\(753\) −31.5000 + 18.1865i −1.14792 + 0.662754i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.00000 −0.0726912 −0.0363456 0.999339i \(-0.511572\pi\)
−0.0363456 + 0.999339i \(0.511572\pi\)
\(758\) 0 0
\(759\) 27.0000 15.5885i 0.980038 0.565825i
\(760\) 0 0
\(761\) −21.0000 + 36.3731i −0.761249 + 1.31852i 0.180957 + 0.983491i \(0.442080\pi\)
−0.942207 + 0.335032i \(0.891253\pi\)
\(762\) 0 0
\(763\) 16.0000 + 27.7128i 0.579239 + 1.00327i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.00000 + 5.19615i 0.108324 + 0.187622i
\(768\) 0 0
\(769\) −1.00000 + 1.73205i −0.0360609 + 0.0624593i −0.883493 0.468445i \(-0.844814\pi\)
0.847432 + 0.530904i \(0.178148\pi\)
\(770\) 0 0
\(771\) 36.3731i 1.30994i
\(772\) 0 0
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) −20.0000 −0.718421
\(776\) 0 0
\(777\) −12.0000 6.92820i −0.430498 0.248548i
\(778\) 0 0
\(779\) 4.50000 7.79423i 0.161229 0.279257i
\(780\) 0 0
\(781\) 18.0000 + 31.1769i 0.644091 + 1.11560i
\(782\) 0 0
\(783\) −27.0000 + 15.5885i −0.964901 + 0.557086i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.00000 3.46410i 0.0712923 0.123482i −0.828176 0.560469i \(-0.810621\pi\)
0.899468 + 0.436987i \(0.143954\pi\)
\(788\) 0 0
\(789\) −27.0000 15.5885i −0.961225 0.554964i
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 16.0000 0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −6.00000 + 10.3923i −0.212531 + 0.368114i −0.952506 0.304520i \(-0.901504\pi\)
0.739975 + 0.672634i \(0.234837\pi\)
\(798\) 0 0
\(799\) 9.00000 + 15.5885i 0.318397 + 0.551480i
\(800\) 0 0
\(801\) 9.00000 15.5885i 0.317999 0.550791i
\(802\) 0 0
\(803\) 16.5000 + 28.5788i 0.582272 + 1.00853i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −36.0000 + 20.7846i −1.26726 + 0.731653i
\(808\) 0 0
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) 0 0
\(811\) −7.00000 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(812\) 0 0
\(813\) 30.0000 17.3205i 1.05215 0.607457i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −0.500000 0.866025i −0.0174928 0.0302984i
\(818\) 0 0
\(819\) 6.00000 10.3923i 0.209657 0.363137i
\(820\) 0 0
\(821\) −9.00000 15.5885i −0.314102 0.544041i 0.665144 0.746715i \(-0.268370\pi\)
−0.979246 + 0.202674i \(0.935037\pi\)
\(822\) 0 0
\(823\) −8.00000 + 13.8564i −0.278862 + 0.483004i −0.971102 0.238664i \(-0.923291\pi\)
0.692240 + 0.721668i \(0.256624\pi\)
\(824\) 0 0
\(825\) 25.9808i 0.904534i
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 10.0000 0.347314 0.173657 0.984806i \(-0.444442\pi\)
0.173657 + 0.984806i \(0.444442\pi\)
\(830\) 0 0
\(831\) 15.0000 + 8.66025i 0.520344 + 0.300421i
\(832\) 0 0
\(833\) −4.50000 + 7.79423i −0.155916 + 0.270054i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −18.0000 10.3923i −0.622171 0.359211i
\(838\) 0 0
\(839\) 6.00000 + 10.3923i 0.207143 + 0.358782i 0.950813 0.309764i \(-0.100250\pi\)
−0.743670 + 0.668546i \(0.766917\pi\)
\(840\) 0 0
\(841\) −3.50000 + 6.06218i −0.120690 + 0.209041i
\(842\) 0 0
\(843\) 9.00000 + 5.19615i 0.309976 + 0.178965i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 4.00000 0.137442
\(848\) 0 0
\(849\) 6.92820i 0.237775i
\(850\) 0 0
\(851\) −12.0000 + 20.7846i −0.411355 + 0.712487i
\(852\) 0 0
\(853\) 13.0000 + 22.5167i 0.445112 + 0.770956i 0.998060 0.0622597i \(-0.0198307\pi\)
−0.552948 + 0.833215i \(0.686497\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.0000 + 36.3731i 0.717346 + 1.24248i 0.962048 + 0.272882i \(0.0879768\pi\)
−0.244701 + 0.969599i \(0.578690\pi\)
\(858\) 0 0
\(859\) −17.5000 + 30.3109i −0.597092 + 1.03419i 0.396156 + 0.918183i \(0.370344\pi\)
−0.993248 + 0.116011i \(0.962989\pi\)
\(860\) 0 0
\(861\) 27.0000 15.5885i 0.920158 0.531253i
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 12.0000 6.92820i 0.407541 0.235294i
\(868\) 0 0
\(869\) 6.00000 10.3923i 0.203536 0.352535i
\(870\) 0 0
\(871\) 5.00000 + 8.66025i 0.169419 + 0.293442i
\(872\) 0 0
\(873\) −15.0000 −0.507673
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 4.00000 6.92820i 0.135070 0.233949i −0.790554 0.612392i \(-0.790207\pi\)
0.925624 + 0.378444i \(0.123541\pi\)
\(878\) 0 0
\(879\) 51.9615i 1.75262i
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) 0 0
\(883\) −19.0000 −0.639401 −0.319700 0.947519i \(-0.603582\pi\)
−0.319700 + 0.947519i \(0.603582\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) −2.00000 3.46410i −0.0670778 0.116182i
\(890\) 0 0
\(891\) 13.5000 23.3827i 0.452267 0.783349i
\(892\) 0 0
\(893\) 3.00000 + 5.19615i 0.100391 + 0.173883i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −18.0000 10.3923i −0.601003 0.346989i
\(898\) 0 0
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 3.46410i 0.115278i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 15.5000 + 26.8468i 0.514669 + 0.891433i 0.999855 + 0.0170220i \(0.00541854\pi\)
−0.485186 + 0.874411i \(0.661248\pi\)
\(908\) 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
\(913\) 18.0000 31.1769i 0.595713 1.03181i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −38.0000 −1.25350 −0.626752 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 0 0
\(921\) 10.5000 6.06218i 0.345987 0.199756i
\(922\) 0 0
\(923\) 12.0000 20.7846i 0.394985 0.684134i
\(924\) 0 0
\(925\) 10.0000 + 17.3205i 0.328798 + 0.569495i
\(926\) 0 0
\(927\) −21.0000 36.3731i −0.689730 1.19465i
\(928\) 0 0
\(929\) 3.00000 + 5.19615i 0.0984268 + 0.170480i 0.911034 0.412332i \(-0.135286\pi\)
−0.812607 + 0.582812i \(0.801952\pi\)
\(930\) 0 0
\(931\) −1.50000 + 2.59808i −0.0491605 + 0.0851485i
\(932\) 0 0
\(933\) 31.1769i 1.02069i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 43.5000 + 25.1147i 1.41957 + 0.819588i
\(940\) 0 0
\(941\) −30.0000 + 51.9615i −0.977972 + 1.69390i −0.308215 + 0.951317i \(0.599732\pi\)
−0.669757 + 0.742581i \(0.733602\pi\)
\(942\) 0 0
\(943\) −27.0000 46.7654i −0.879241 1.52289i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.50000 2.59808i −0.0487435 0.0844261i 0.840624 0.541619i \(-0.182188\pi\)
−0.889368 + 0.457193i \(0.848855\pi\)
\(948\) 0 0
\(949\) 11.0000 19.0526i 0.357075 0.618472i
\(950\) 0 0
\(951\) 27.0000 + 15.5885i 0.875535 + 0.505490i
\(952\) 0 0
\(953\) 9.00000 0.291539 0.145769 0.989319i \(-0.453434\pi\)
0.145769 + 0.989319i \(0.453434\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 31.1769i 1.00781i
\(958\) 0 0
\(959\) −3.00000 + 5.19615i −0.0968751 + 0.167793i
\(960\) 0 0
\(961\) 7.50000 + 12.9904i 0.241935 + 0.419045i
\(962\) 0 0
\(963\) 4.50000 7.79423i 0.145010 0.251166i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −11.0000 + 19.0526i −0.353736 + 0.612689i −0.986901 0.161328i \(-0.948422\pi\)
0.633165 + 0.774017i \(0.281756\pi\)
\(968\) 0 0
\(969\) −4.50000 + 2.59808i −0.144561 + 0.0834622i
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 38.0000 1.21822
\(974\) 0 0
\(975\) −15.0000 + 8.66025i −0.480384 + 0.277350i
\(976\) 0 0
\(977\) 25.5000 44.1673i 0.815817 1.41304i −0.0929223 0.995673i \(-0.529621\pi\)
0.908740 0.417364i \(-0.137046\pi\)
\(978\) 0 0
\(979\) 9.00000 + 15.5885i 0.287641 + 0.498209i
\(980\) 0 0
\(981\) 24.0000 41.5692i 0.766261 1.32720i
\(982\) 0 0
\(983\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 20.7846i 0.661581i
\(988\) 0 0
\(989\) −6.00000 −0.190789
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) −6.00000 3.46410i −0.190404 0.109930i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −14.0000 24.2487i −0.443384 0.767964i 0.554554 0.832148i \(-0.312889\pi\)
−0.997938 + 0.0641836i \(0.979556\pi\)
\(998\) 0 0
\(999\) 20.7846i 0.657596i
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 576.2.i.a.193.1 2
3.2 odd 2 1728.2.i.f.577.1 2
4.3 odd 2 576.2.i.g.193.1 2
8.3 odd 2 18.2.c.a.13.1 yes 2
8.5 even 2 144.2.i.c.49.1 2
9.2 odd 6 1728.2.i.f.1153.1 2
9.4 even 3 5184.2.a.o.1.1 1
9.5 odd 6 5184.2.a.p.1.1 1
9.7 even 3 inner 576.2.i.a.385.1 2
12.11 even 2 1728.2.i.e.577.1 2
24.5 odd 2 432.2.i.b.145.1 2
24.11 even 2 54.2.c.a.37.1 2
36.7 odd 6 576.2.i.g.385.1 2
36.11 even 6 1728.2.i.e.1153.1 2
36.23 even 6 5184.2.a.q.1.1 1
36.31 odd 6 5184.2.a.r.1.1 1
40.3 even 4 450.2.j.e.49.1 4
40.19 odd 2 450.2.e.i.301.1 2
40.27 even 4 450.2.j.e.49.2 4
56.3 even 6 882.2.e.g.373.1 2
56.11 odd 6 882.2.e.i.373.1 2
56.19 even 6 882.2.h.b.67.1 2
56.27 even 2 882.2.f.d.589.1 2
56.51 odd 6 882.2.h.c.67.1 2
72.5 odd 6 1296.2.a.f.1.1 1
72.11 even 6 54.2.c.a.19.1 2
72.13 even 6 1296.2.a.g.1.1 1
72.29 odd 6 432.2.i.b.289.1 2
72.43 odd 6 18.2.c.a.7.1 2
72.59 even 6 162.2.a.b.1.1 1
72.61 even 6 144.2.i.c.97.1 2
72.67 odd 6 162.2.a.c.1.1 1
120.59 even 2 1350.2.e.c.901.1 2
120.83 odd 4 1350.2.j.a.199.2 4
120.107 odd 4 1350.2.j.a.199.1 4
168.11 even 6 2646.2.e.b.1549.1 2
168.59 odd 6 2646.2.e.c.1549.1 2
168.83 odd 2 2646.2.f.g.1765.1 2
168.107 even 6 2646.2.h.h.361.1 2
168.131 odd 6 2646.2.h.i.361.1 2
360.43 even 12 450.2.j.e.349.2 4
360.59 even 6 4050.2.a.v.1.1 1
360.67 even 12 4050.2.c.c.649.2 2
360.83 odd 12 1350.2.j.a.1099.1 4
360.139 odd 6 4050.2.a.c.1.1 1
360.187 even 12 450.2.j.e.349.1 4
360.203 odd 12 4050.2.c.r.649.2 2
360.227 odd 12 1350.2.j.a.1099.2 4
360.259 odd 6 450.2.e.i.151.1 2
360.283 even 12 4050.2.c.c.649.1 2
360.299 even 6 1350.2.e.c.451.1 2
360.347 odd 12 4050.2.c.r.649.1 2
504.11 even 6 2646.2.h.h.667.1 2
504.83 odd 6 2646.2.f.g.883.1 2
504.115 even 6 882.2.h.b.79.1 2
504.139 even 6 7938.2.a.x.1.1 1
504.187 even 6 882.2.e.g.655.1 2
504.227 odd 6 2646.2.h.i.667.1 2
504.299 odd 6 2646.2.e.c.2125.1 2
504.331 odd 6 882.2.e.i.655.1 2
504.403 odd 6 882.2.h.c.79.1 2
504.419 odd 6 7938.2.a.i.1.1 1
504.443 even 6 2646.2.e.b.2125.1 2
504.475 even 6 882.2.f.d.295.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.2.c.a.7.1 2 72.43 odd 6
18.2.c.a.13.1 yes 2 8.3 odd 2
54.2.c.a.19.1 2 72.11 even 6
54.2.c.a.37.1 2 24.11 even 2
144.2.i.c.49.1 2 8.5 even 2
144.2.i.c.97.1 2 72.61 even 6
162.2.a.b.1.1 1 72.59 even 6
162.2.a.c.1.1 1 72.67 odd 6
432.2.i.b.145.1 2 24.5 odd 2
432.2.i.b.289.1 2 72.29 odd 6
450.2.e.i.151.1 2 360.259 odd 6
450.2.e.i.301.1 2 40.19 odd 2
450.2.j.e.49.1 4 40.3 even 4
450.2.j.e.49.2 4 40.27 even 4
450.2.j.e.349.1 4 360.187 even 12
450.2.j.e.349.2 4 360.43 even 12
576.2.i.a.193.1 2 1.1 even 1 trivial
576.2.i.a.385.1 2 9.7 even 3 inner
576.2.i.g.193.1 2 4.3 odd 2
576.2.i.g.385.1 2 36.7 odd 6
882.2.e.g.373.1 2 56.3 even 6
882.2.e.g.655.1 2 504.187 even 6
882.2.e.i.373.1 2 56.11 odd 6
882.2.e.i.655.1 2 504.331 odd 6
882.2.f.d.295.1 2 504.475 even 6
882.2.f.d.589.1 2 56.27 even 2
882.2.h.b.67.1 2 56.19 even 6
882.2.h.b.79.1 2 504.115 even 6
882.2.h.c.67.1 2 56.51 odd 6
882.2.h.c.79.1 2 504.403 odd 6
1296.2.a.f.1.1 1 72.5 odd 6
1296.2.a.g.1.1 1 72.13 even 6
1350.2.e.c.451.1 2 360.299 even 6
1350.2.e.c.901.1 2 120.59 even 2
1350.2.j.a.199.1 4 120.107 odd 4
1350.2.j.a.199.2 4 120.83 odd 4
1350.2.j.a.1099.1 4 360.83 odd 12
1350.2.j.a.1099.2 4 360.227 odd 12
1728.2.i.e.577.1 2 12.11 even 2
1728.2.i.e.1153.1 2 36.11 even 6
1728.2.i.f.577.1 2 3.2 odd 2
1728.2.i.f.1153.1 2 9.2 odd 6
2646.2.e.b.1549.1 2 168.11 even 6
2646.2.e.b.2125.1 2 504.443 even 6
2646.2.e.c.1549.1 2 168.59 odd 6
2646.2.e.c.2125.1 2 504.299 odd 6
2646.2.f.g.883.1 2 504.83 odd 6
2646.2.f.g.1765.1 2 168.83 odd 2
2646.2.h.h.361.1 2 168.107 even 6
2646.2.h.h.667.1 2 504.11 even 6
2646.2.h.i.361.1 2 168.131 odd 6
2646.2.h.i.667.1 2 504.227 odd 6
4050.2.a.c.1.1 1 360.139 odd 6
4050.2.a.v.1.1 1 360.59 even 6
4050.2.c.c.649.1 2 360.283 even 12
4050.2.c.c.649.2 2 360.67 even 12
4050.2.c.r.649.1 2 360.347 odd 12
4050.2.c.r.649.2 2 360.203 odd 12
5184.2.a.o.1.1 1 9.4 even 3
5184.2.a.p.1.1 1 9.5 odd 6
5184.2.a.q.1.1 1 36.23 even 6
5184.2.a.r.1.1 1 36.31 odd 6
7938.2.a.i.1.1 1 504.419 odd 6
7938.2.a.x.1.1 1 504.139 even 6