Properties

Label 336.2.bl.h.31.1
Level $336$
Weight $2$
Character 336.31
Analytic conductor $2.683$
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] = [336,2,Mod(31,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.bl (of order \(6\), degree \(2\), 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: \(2.68297350792\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 31.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 336.31
Dual form 336.2.bl.h.271.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{3} +(3.00000 - 1.73205i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{3} +(3.00000 - 1.73205i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-0.500000 - 0.866025i) q^{9} +(3.00000 + 1.73205i) q^{11} -1.73205i q^{13} -3.46410i q^{15} +(-2.50000 - 4.33013i) q^{19} +(2.00000 + 1.73205i) q^{21} +(-6.00000 + 3.46410i) q^{23} +(3.50000 - 6.06218i) q^{25} -1.00000 q^{27} +(2.50000 - 4.33013i) q^{31} +(3.00000 - 1.73205i) q^{33} +(3.00000 + 8.66025i) q^{35} +(5.50000 + 9.52628i) q^{37} +(-1.50000 - 0.866025i) q^{39} -3.46410i q^{41} -8.66025i q^{43} +(-3.00000 - 1.73205i) q^{45} +(3.00000 + 5.19615i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-6.00000 + 10.3923i) q^{53} +12.0000 q^{55} -5.00000 q^{57} +(-6.00000 + 10.3923i) q^{59} +(-12.0000 + 6.92820i) q^{61} +(2.50000 - 0.866025i) q^{63} +(-3.00000 - 5.19615i) q^{65} +(-7.50000 - 4.33013i) q^{67} +6.92820i q^{69} -3.46410i q^{71} +(-4.50000 - 2.59808i) q^{73} +(-3.50000 - 6.06218i) q^{75} +(-6.00000 + 6.92820i) q^{77} +(-10.5000 + 6.06218i) q^{79} +(-0.500000 + 0.866025i) q^{81} +18.0000 q^{83} +(6.00000 - 3.46410i) q^{89} +(4.50000 + 0.866025i) q^{91} +(-2.50000 - 4.33013i) q^{93} +(-15.0000 - 8.66025i) q^{95} +6.92820i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 6 q^{5} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 6 q^{5} - q^{7} - q^{9} + 6 q^{11} - 5 q^{19} + 4 q^{21} - 12 q^{23} + 7 q^{25} - 2 q^{27} + 5 q^{31} + 6 q^{33} + 6 q^{35} + 11 q^{37} - 3 q^{39} - 6 q^{45} + 6 q^{47} - 13 q^{49} - 12 q^{53} + 24 q^{55} - 10 q^{57} - 12 q^{59} - 24 q^{61} + 5 q^{63} - 6 q^{65} - 15 q^{67} - 9 q^{73} - 7 q^{75} - 12 q^{77} - 21 q^{79} - q^{81} + 36 q^{83} + 12 q^{89} + 9 q^{91} - 5 q^{93} - 30 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 0.500000 0.866025i 0.288675 0.500000i
\(4\) 0 0
\(5\) 3.00000 1.73205i 1.34164 0.774597i 0.354593 0.935021i \(-0.384620\pi\)
0.987048 + 0.160424i \(0.0512862\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) −0.500000 0.866025i −0.166667 0.288675i
\(10\) 0 0
\(11\) 3.00000 + 1.73205i 0.904534 + 0.522233i 0.878668 0.477432i \(-0.158432\pi\)
0.0258656 + 0.999665i \(0.491766\pi\)
\(12\) 0 0
\(13\) 1.73205i 0.480384i −0.970725 0.240192i \(-0.922790\pi\)
0.970725 0.240192i \(-0.0772105\pi\)
\(14\) 0 0
\(15\) 3.46410i 0.894427i
\(16\) 0 0
\(17\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(18\) 0 0
\(19\) −2.50000 4.33013i −0.573539 0.993399i −0.996199 0.0871106i \(-0.972237\pi\)
0.422659 0.906289i \(-0.361097\pi\)
\(20\) 0 0
\(21\) 2.00000 + 1.73205i 0.436436 + 0.377964i
\(22\) 0 0
\(23\) −6.00000 + 3.46410i −1.25109 + 0.722315i −0.971325 0.237754i \(-0.923589\pi\)
−0.279761 + 0.960070i \(0.590255\pi\)
\(24\) 0 0
\(25\) 3.50000 6.06218i 0.700000 1.21244i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.50000 4.33013i 0.449013 0.777714i −0.549309 0.835619i \(-0.685109\pi\)
0.998322 + 0.0579057i \(0.0184423\pi\)
\(32\) 0 0
\(33\) 3.00000 1.73205i 0.522233 0.301511i
\(34\) 0 0
\(35\) 3.00000 + 8.66025i 0.507093 + 1.46385i
\(36\) 0 0
\(37\) 5.50000 + 9.52628i 0.904194 + 1.56611i 0.821995 + 0.569495i \(0.192861\pi\)
0.0821995 + 0.996616i \(0.473806\pi\)
\(38\) 0 0
\(39\) −1.50000 0.866025i −0.240192 0.138675i
\(40\) 0 0
\(41\) 3.46410i 0.541002i −0.962720 0.270501i \(-0.912811\pi\)
0.962720 0.270501i \(-0.0871893\pi\)
\(42\) 0 0
\(43\) 8.66025i 1.32068i −0.750968 0.660338i \(-0.770413\pi\)
0.750968 0.660338i \(-0.229587\pi\)
\(44\) 0 0
\(45\) −3.00000 1.73205i −0.447214 0.258199i
\(46\) 0 0
\(47\) 3.00000 + 5.19615i 0.437595 + 0.757937i 0.997503 0.0706177i \(-0.0224970\pi\)
−0.559908 + 0.828554i \(0.689164\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 + 10.3923i −0.824163 + 1.42749i 0.0783936 + 0.996922i \(0.475021\pi\)
−0.902557 + 0.430570i \(0.858312\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) −5.00000 −0.662266
\(58\) 0 0
\(59\) −6.00000 + 10.3923i −0.781133 + 1.35296i 0.150148 + 0.988663i \(0.452025\pi\)
−0.931282 + 0.364299i \(0.881308\pi\)
\(60\) 0 0
\(61\) −12.0000 + 6.92820i −1.53644 + 0.887066i −0.537400 + 0.843328i \(0.680593\pi\)
−0.999043 + 0.0437377i \(0.986073\pi\)
\(62\) 0 0
\(63\) 2.50000 0.866025i 0.314970 0.109109i
\(64\) 0 0
\(65\) −3.00000 5.19615i −0.372104 0.644503i
\(66\) 0 0
\(67\) −7.50000 4.33013i −0.916271 0.529009i −0.0338274 0.999428i \(-0.510770\pi\)
−0.882443 + 0.470418i \(0.844103\pi\)
\(68\) 0 0
\(69\) 6.92820i 0.834058i
\(70\) 0 0
\(71\) 3.46410i 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) −4.50000 2.59808i −0.526685 0.304082i 0.212980 0.977056i \(-0.431683\pi\)
−0.739666 + 0.672975i \(0.765016\pi\)
\(74\) 0 0
\(75\) −3.50000 6.06218i −0.404145 0.700000i
\(76\) 0 0
\(77\) −6.00000 + 6.92820i −0.683763 + 0.789542i
\(78\) 0 0
\(79\) −10.5000 + 6.06218i −1.18134 + 0.682048i −0.956325 0.292306i \(-0.905577\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 18.0000 1.97576 0.987878 0.155230i \(-0.0496119\pi\)
0.987878 + 0.155230i \(0.0496119\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 3.46410i 0.635999 0.367194i −0.147073 0.989126i \(-0.546985\pi\)
0.783072 + 0.621932i \(0.213652\pi\)
\(90\) 0 0
\(91\) 4.50000 + 0.866025i 0.471728 + 0.0907841i
\(92\) 0 0
\(93\) −2.50000 4.33013i −0.259238 0.449013i
\(94\) 0 0
\(95\) −15.0000 8.66025i −1.53897 0.888523i
\(96\) 0 0
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) 0 0
\(99\) 3.46410i 0.348155i
\(100\) 0 0
\(101\) 9.00000 + 5.19615i 0.895533 + 0.517036i 0.875748 0.482768i \(-0.160368\pi\)
0.0197851 + 0.999804i \(0.493702\pi\)
\(102\) 0 0
\(103\) 2.50000 + 4.33013i 0.246332 + 0.426660i 0.962505 0.271263i \(-0.0874412\pi\)
−0.716173 + 0.697923i \(0.754108\pi\)
\(104\) 0 0
\(105\) 9.00000 + 1.73205i 0.878310 + 0.169031i
\(106\) 0 0
\(107\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(108\) 0 0
\(109\) 3.50000 6.06218i 0.335239 0.580651i −0.648292 0.761392i \(-0.724516\pi\)
0.983531 + 0.180741i \(0.0578495\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −12.0000 + 20.7846i −1.11901 + 1.93817i
\(116\) 0 0
\(117\) −1.50000 + 0.866025i −0.138675 + 0.0800641i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.500000 + 0.866025i 0.0454545 + 0.0787296i
\(122\) 0 0
\(123\) −3.00000 1.73205i −0.270501 0.156174i
\(124\) 0 0
\(125\) 6.92820i 0.619677i
\(126\) 0 0
\(127\) 12.1244i 1.07586i −0.842989 0.537931i \(-0.819206\pi\)
0.842989 0.537931i \(-0.180794\pi\)
\(128\) 0 0
\(129\) −7.50000 4.33013i −0.660338 0.381246i
\(130\) 0 0
\(131\) −3.00000 5.19615i −0.262111 0.453990i 0.704692 0.709514i \(-0.251085\pi\)
−0.966803 + 0.255524i \(0.917752\pi\)
\(132\) 0 0
\(133\) 12.5000 4.33013i 1.08389 0.375470i
\(134\) 0 0
\(135\) −3.00000 + 1.73205i −0.258199 + 0.149071i
\(136\) 0 0
\(137\) −6.00000 + 10.3923i −0.512615 + 0.887875i 0.487278 + 0.873247i \(0.337990\pi\)
−0.999893 + 0.0146279i \(0.995344\pi\)
\(138\) 0 0
\(139\) 7.00000 0.593732 0.296866 0.954919i \(-0.404058\pi\)
0.296866 + 0.954919i \(0.404058\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 3.00000 5.19615i 0.250873 0.434524i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.50000 + 4.33013i −0.453632 + 0.357143i
\(148\) 0 0
\(149\) −6.00000 10.3923i −0.491539 0.851371i 0.508413 0.861113i \(-0.330232\pi\)
−0.999953 + 0.00974235i \(0.996899\pi\)
\(150\) 0 0
\(151\) −3.00000 1.73205i −0.244137 0.140952i 0.372940 0.927855i \(-0.378350\pi\)
−0.617076 + 0.786903i \(0.711683\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 17.3205i 1.39122i
\(156\) 0 0
\(157\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(158\) 0 0
\(159\) 6.00000 + 10.3923i 0.475831 + 0.824163i
\(160\) 0 0
\(161\) −6.00000 17.3205i −0.472866 1.36505i
\(162\) 0 0
\(163\) 3.00000 1.73205i 0.234978 0.135665i −0.377888 0.925851i \(-0.623350\pi\)
0.612866 + 0.790186i \(0.290016\pi\)
\(164\) 0 0
\(165\) 6.00000 10.3923i 0.467099 0.809040i
\(166\) 0 0
\(167\) 6.00000 0.464294 0.232147 0.972681i \(-0.425425\pi\)
0.232147 + 0.972681i \(0.425425\pi\)
\(168\) 0 0
\(169\) 10.0000 0.769231
\(170\) 0 0
\(171\) −2.50000 + 4.33013i −0.191180 + 0.331133i
\(172\) 0 0
\(173\) 6.00000 3.46410i 0.456172 0.263371i −0.254262 0.967135i \(-0.581832\pi\)
0.710433 + 0.703765i \(0.248499\pi\)
\(174\) 0 0
\(175\) 14.0000 + 12.1244i 1.05830 + 0.916515i
\(176\) 0 0
\(177\) 6.00000 + 10.3923i 0.450988 + 0.781133i
\(178\) 0 0
\(179\) 9.00000 + 5.19615i 0.672692 + 0.388379i 0.797096 0.603853i \(-0.206369\pi\)
−0.124404 + 0.992232i \(0.539702\pi\)
\(180\) 0 0
\(181\) 5.19615i 0.386227i −0.981176 0.193113i \(-0.938141\pi\)
0.981176 0.193113i \(-0.0618586\pi\)
\(182\) 0 0
\(183\) 13.8564i 1.02430i
\(184\) 0 0
\(185\) 33.0000 + 19.0526i 2.42621 + 1.40077i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.500000 2.59808i 0.0363696 0.188982i
\(190\) 0 0
\(191\) −15.0000 + 8.66025i −1.08536 + 0.626634i −0.932338 0.361588i \(-0.882235\pi\)
−0.153024 + 0.988222i \(0.548901\pi\)
\(192\) 0 0
\(193\) 5.50000 9.52628i 0.395899 0.685717i −0.597317 0.802005i \(-0.703766\pi\)
0.993215 + 0.116289i \(0.0370998\pi\)
\(194\) 0 0
\(195\) −6.00000 −0.429669
\(196\) 0 0
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) 8.00000 13.8564i 0.567105 0.982255i −0.429745 0.902950i \(-0.641397\pi\)
0.996850 0.0793045i \(-0.0252700\pi\)
\(200\) 0 0
\(201\) −7.50000 + 4.33013i −0.529009 + 0.305424i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 10.3923i −0.419058 0.725830i
\(206\) 0 0
\(207\) 6.00000 + 3.46410i 0.417029 + 0.240772i
\(208\) 0 0
\(209\) 17.3205i 1.19808i
\(210\) 0 0
\(211\) 3.46410i 0.238479i −0.992866 0.119239i \(-0.961954\pi\)
0.992866 0.119239i \(-0.0380456\pi\)
\(212\) 0 0
\(213\) −3.00000 1.73205i −0.205557 0.118678i
\(214\) 0 0
\(215\) −15.0000 25.9808i −1.02299 1.77187i
\(216\) 0 0
\(217\) 10.0000 + 8.66025i 0.678844 + 0.587896i
\(218\) 0 0
\(219\) −4.50000 + 2.59808i −0.304082 + 0.175562i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −7.00000 −0.466667
\(226\) 0 0
\(227\) 9.00000 15.5885i 0.597351 1.03464i −0.395860 0.918311i \(-0.629553\pi\)
0.993210 0.116331i \(-0.0371134\pi\)
\(228\) 0 0
\(229\) 7.50000 4.33013i 0.495614 0.286143i −0.231287 0.972886i \(-0.574293\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 3.00000 + 8.66025i 0.197386 + 0.569803i
\(232\) 0 0
\(233\) 3.00000 + 5.19615i 0.196537 + 0.340411i 0.947403 0.320043i \(-0.103697\pi\)
−0.750867 + 0.660454i \(0.770364\pi\)
\(234\) 0 0
\(235\) 18.0000 + 10.3923i 1.17419 + 0.677919i
\(236\) 0 0
\(237\) 12.1244i 0.787562i
\(238\) 0 0
\(239\) 10.3923i 0.672222i −0.941822 0.336111i \(-0.890888\pi\)
0.941822 0.336111i \(-0.109112\pi\)
\(240\) 0 0
\(241\) −12.0000 6.92820i −0.772988 0.446285i 0.0609515 0.998141i \(-0.480586\pi\)
−0.833939 + 0.551856i \(0.813920\pi\)
\(242\) 0 0
\(243\) 0.500000 + 0.866025i 0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) −24.0000 + 3.46410i −1.53330 + 0.221313i
\(246\) 0 0
\(247\) −7.50000 + 4.33013i −0.477214 + 0.275519i
\(248\) 0 0
\(249\) 9.00000 15.5885i 0.570352 0.987878i
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) −24.0000 −1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −15.0000 + 8.66025i −0.935674 + 0.540212i −0.888602 0.458680i \(-0.848323\pi\)
−0.0470726 + 0.998891i \(0.514989\pi\)
\(258\) 0 0
\(259\) −27.5000 + 9.52628i −1.70877 + 0.591934i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −12.0000 6.92820i −0.739952 0.427211i 0.0821001 0.996624i \(-0.473837\pi\)
−0.822052 + 0.569413i \(0.807171\pi\)
\(264\) 0 0
\(265\) 41.5692i 2.55358i
\(266\) 0 0
\(267\) 6.92820i 0.423999i
\(268\) 0 0
\(269\) −15.0000 8.66025i −0.914566 0.528025i −0.0326687 0.999466i \(-0.510401\pi\)
−0.881897 + 0.471441i \(0.843734\pi\)
\(270\) 0 0
\(271\) 4.00000 + 6.92820i 0.242983 + 0.420858i 0.961563 0.274586i \(-0.0885408\pi\)
−0.718580 + 0.695444i \(0.755208\pi\)
\(272\) 0 0
\(273\) 3.00000 3.46410i 0.181568 0.209657i
\(274\) 0 0
\(275\) 21.0000 12.1244i 1.26635 0.731126i
\(276\) 0 0
\(277\) −9.50000 + 16.4545i −0.570800 + 0.988654i 0.425684 + 0.904872i \(0.360033\pi\)
−0.996484 + 0.0837823i \(0.973300\pi\)
\(278\) 0 0
\(279\) −5.00000 −0.299342
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 3.50000 6.06218i 0.208053 0.360359i −0.743048 0.669238i \(-0.766621\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) 0 0
\(285\) −15.0000 + 8.66025i −0.888523 + 0.512989i
\(286\) 0 0
\(287\) 9.00000 + 1.73205i 0.531253 + 0.102240i
\(288\) 0 0
\(289\) −8.50000 14.7224i −0.500000 0.866025i
\(290\) 0 0
\(291\) 6.00000 + 3.46410i 0.351726 + 0.203069i
\(292\) 0 0
\(293\) 20.7846i 1.21425i −0.794606 0.607125i \(-0.792323\pi\)
0.794606 0.607125i \(-0.207677\pi\)
\(294\) 0 0
\(295\) 41.5692i 2.42025i
\(296\) 0 0
\(297\) −3.00000 1.73205i −0.174078 0.100504i
\(298\) 0 0
\(299\) 6.00000 + 10.3923i 0.346989 + 0.601003i
\(300\) 0 0
\(301\) 22.5000 + 4.33013i 1.29688 + 0.249584i
\(302\) 0 0
\(303\) 9.00000 5.19615i 0.517036 0.298511i
\(304\) 0 0
\(305\) −24.0000 + 41.5692i −1.37424 + 2.38025i
\(306\) 0 0
\(307\) 11.0000 0.627803 0.313902 0.949456i \(-0.398364\pi\)
0.313902 + 0.949456i \(0.398364\pi\)
\(308\) 0 0
\(309\) 5.00000 0.284440
\(310\) 0 0
\(311\) 9.00000 15.5885i 0.510343 0.883940i −0.489585 0.871956i \(-0.662852\pi\)
0.999928 0.0119847i \(-0.00381495\pi\)
\(312\) 0 0
\(313\) 13.5000 7.79423i 0.763065 0.440556i −0.0673300 0.997731i \(-0.521448\pi\)
0.830395 + 0.557175i \(0.188115\pi\)
\(314\) 0 0
\(315\) 6.00000 6.92820i 0.338062 0.390360i
\(316\) 0 0
\(317\) 6.00000 + 10.3923i 0.336994 + 0.583690i 0.983866 0.178908i \(-0.0572566\pi\)
−0.646872 + 0.762598i \(0.723923\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −10.5000 6.06218i −0.582435 0.336269i
\(326\) 0 0
\(327\) −3.50000 6.06218i −0.193550 0.335239i
\(328\) 0 0
\(329\) −15.0000 + 5.19615i −0.826977 + 0.286473i
\(330\) 0 0
\(331\) −13.5000 + 7.79423i −0.742027 + 0.428410i −0.822806 0.568323i \(-0.807593\pi\)
0.0807788 + 0.996732i \(0.474259\pi\)
\(332\) 0 0
\(333\) 5.50000 9.52628i 0.301398 0.522037i
\(334\) 0 0
\(335\) −30.0000 −1.63908
\(336\) 0 0
\(337\) −19.0000 −1.03500 −0.517498 0.855684i \(-0.673136\pi\)
−0.517498 + 0.855684i \(0.673136\pi\)
\(338\) 0 0
\(339\) −3.00000 + 5.19615i −0.162938 + 0.282216i
\(340\) 0 0
\(341\) 15.0000 8.66025i 0.812296 0.468979i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 12.0000 + 20.7846i 0.646058 + 1.11901i
\(346\) 0 0
\(347\) 24.0000 + 13.8564i 1.28839 + 0.743851i 0.978367 0.206879i \(-0.0663306\pi\)
0.310021 + 0.950730i \(0.399664\pi\)
\(348\) 0 0
\(349\) 13.8564i 0.741716i −0.928689 0.370858i \(-0.879064\pi\)
0.928689 0.370858i \(-0.120936\pi\)
\(350\) 0 0
\(351\) 1.73205i 0.0924500i
\(352\) 0 0
\(353\) 21.0000 + 12.1244i 1.11772 + 0.645314i 0.940817 0.338914i \(-0.110060\pi\)
0.176900 + 0.984229i \(0.443393\pi\)
\(354\) 0 0
\(355\) −6.00000 10.3923i −0.318447 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 10.3923i 0.950004 0.548485i 0.0569216 0.998379i \(-0.481871\pi\)
0.893082 + 0.449894i \(0.148538\pi\)
\(360\) 0 0
\(361\) −3.00000 + 5.19615i −0.157895 + 0.273482i
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −18.0000 −0.942163
\(366\) 0 0
\(367\) −5.50000 + 9.52628i −0.287098 + 0.497268i −0.973116 0.230317i \(-0.926024\pi\)
0.686018 + 0.727585i \(0.259357\pi\)
\(368\) 0 0
\(369\) −3.00000 + 1.73205i −0.156174 + 0.0901670i
\(370\) 0 0
\(371\) −24.0000 20.7846i −1.24602 1.07908i
\(372\) 0 0
\(373\) 0.500000 + 0.866025i 0.0258890 + 0.0448411i 0.878680 0.477412i \(-0.158425\pi\)
−0.852791 + 0.522253i \(0.825092\pi\)
\(374\) 0 0
\(375\) −6.00000 3.46410i −0.309839 0.178885i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.66025i 0.444847i −0.974950 0.222424i \(-0.928603\pi\)
0.974950 0.222424i \(-0.0713968\pi\)
\(380\) 0 0
\(381\) −10.5000 6.06218i −0.537931 0.310575i
\(382\) 0 0
\(383\) 12.0000 + 20.7846i 0.613171 + 1.06204i 0.990702 + 0.136047i \(0.0434398\pi\)
−0.377531 + 0.925997i \(0.623227\pi\)
\(384\) 0 0
\(385\) −6.00000 + 31.1769i −0.305788 + 1.58892i
\(386\) 0 0
\(387\) −7.50000 + 4.33013i −0.381246 + 0.220113i
\(388\) 0 0
\(389\) 3.00000 5.19615i 0.152106 0.263455i −0.779895 0.625910i \(-0.784728\pi\)
0.932002 + 0.362454i \(0.118061\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −6.00000 −0.302660
\(394\) 0 0
\(395\) −21.0000 + 36.3731i −1.05662 + 1.83013i
\(396\) 0 0
\(397\) −1.50000 + 0.866025i −0.0752828 + 0.0434646i −0.537169 0.843475i \(-0.680506\pi\)
0.461886 + 0.886939i \(0.347173\pi\)
\(398\) 0 0
\(399\) 2.50000 12.9904i 0.125157 0.650332i
\(400\) 0 0
\(401\) −6.00000 10.3923i −0.299626 0.518967i 0.676425 0.736512i \(-0.263528\pi\)
−0.976050 + 0.217545i \(0.930195\pi\)
\(402\) 0 0
\(403\) −7.50000 4.33013i −0.373602 0.215699i
\(404\) 0 0
\(405\) 3.46410i 0.172133i
\(406\) 0 0
\(407\) 38.1051i 1.88880i
\(408\) 0 0
\(409\) −19.5000 11.2583i −0.964213 0.556689i −0.0667458 0.997770i \(-0.521262\pi\)
−0.897467 + 0.441081i \(0.854595\pi\)
\(410\) 0 0
\(411\) 6.00000 + 10.3923i 0.295958 + 0.512615i
\(412\) 0 0
\(413\) −24.0000 20.7846i −1.18096 1.02274i
\(414\) 0 0
\(415\) 54.0000 31.1769i 2.65076 1.53041i
\(416\) 0 0
\(417\) 3.50000 6.06218i 0.171396 0.296866i
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) 7.00000 0.341159 0.170580 0.985344i \(-0.445436\pi\)
0.170580 + 0.985344i \(0.445436\pi\)
\(422\) 0 0
\(423\) 3.00000 5.19615i 0.145865 0.252646i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −12.0000 34.6410i −0.580721 1.67640i
\(428\) 0 0
\(429\) −3.00000 5.19615i −0.144841 0.250873i
\(430\) 0 0
\(431\) −21.0000 12.1244i −1.01153 0.584010i −0.0998939 0.994998i \(-0.531850\pi\)
−0.911641 + 0.410988i \(0.865184\pi\)
\(432\) 0 0
\(433\) 1.73205i 0.0832370i −0.999134 0.0416185i \(-0.986749\pi\)
0.999134 0.0416185i \(-0.0132514\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.0000 + 17.3205i 1.43509 + 0.828552i
\(438\) 0 0
\(439\) −4.00000 6.92820i −0.190910 0.330665i 0.754642 0.656136i \(-0.227810\pi\)
−0.945552 + 0.325471i \(0.894477\pi\)
\(440\) 0 0
\(441\) 1.00000 + 6.92820i 0.0476190 + 0.329914i
\(442\) 0 0
\(443\) −12.0000 + 6.92820i −0.570137 + 0.329169i −0.757204 0.653178i \(-0.773435\pi\)
0.187067 + 0.982347i \(0.440102\pi\)
\(444\) 0 0
\(445\) 12.0000 20.7846i 0.568855 0.985285i
\(446\) 0 0
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) 6.00000 10.3923i 0.282529 0.489355i
\(452\) 0 0
\(453\) −3.00000 + 1.73205i −0.140952 + 0.0813788i
\(454\) 0 0
\(455\) 15.0000 5.19615i 0.703211 0.243599i
\(456\) 0 0
\(457\) 9.50000 + 16.4545i 0.444391 + 0.769708i 0.998010 0.0630623i \(-0.0200867\pi\)
−0.553618 + 0.832771i \(0.686753\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 13.8564i 0.645357i −0.946509 0.322679i \(-0.895417\pi\)
0.946509 0.322679i \(-0.104583\pi\)
\(462\) 0 0
\(463\) 15.5885i 0.724457i 0.932089 + 0.362229i \(0.117984\pi\)
−0.932089 + 0.362229i \(0.882016\pi\)
\(464\) 0 0
\(465\) −15.0000 8.66025i −0.695608 0.401610i
\(466\) 0 0
\(467\) 15.0000 + 25.9808i 0.694117 + 1.20225i 0.970477 + 0.241192i \(0.0775384\pi\)
−0.276360 + 0.961054i \(0.589128\pi\)
\(468\) 0 0
\(469\) 15.0000 17.3205i 0.692636 0.799787i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 15.0000 25.9808i 0.689701 1.19460i
\(474\) 0 0
\(475\) −35.0000 −1.60591
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −6.00000 + 10.3923i −0.274147 + 0.474837i −0.969920 0.243426i \(-0.921729\pi\)
0.695773 + 0.718262i \(0.255062\pi\)
\(480\) 0 0
\(481\) 16.5000 9.52628i 0.752335 0.434361i
\(482\) 0 0
\(483\) −18.0000 3.46410i −0.819028 0.157622i
\(484\) 0 0
\(485\) 12.0000 + 20.7846i 0.544892 + 0.943781i
\(486\) 0 0
\(487\) −22.5000 12.9904i −1.01957 0.588650i −0.105592 0.994410i \(-0.533674\pi\)
−0.913980 + 0.405759i \(0.867007\pi\)
\(488\) 0 0
\(489\) 3.46410i 0.156652i
\(490\) 0 0
\(491\) 34.6410i 1.56333i 0.623700 + 0.781664i \(0.285629\pi\)
−0.623700 + 0.781664i \(0.714371\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −6.00000 10.3923i −0.269680 0.467099i
\(496\) 0 0
\(497\) 9.00000 + 1.73205i 0.403705 + 0.0776931i
\(498\) 0 0
\(499\) 28.5000 16.4545i 1.27584 0.736604i 0.299755 0.954016i \(-0.403095\pi\)
0.976080 + 0.217412i \(0.0697616\pi\)
\(500\) 0 0
\(501\) 3.00000 5.19615i 0.134030 0.232147i
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) 36.0000 1.60198
\(506\) 0 0
\(507\) 5.00000 8.66025i 0.222058 0.384615i
\(508\) 0 0
\(509\) 27.0000 15.5885i 1.19675 0.690946i 0.236924 0.971528i \(-0.423861\pi\)
0.959830 + 0.280582i \(0.0905275\pi\)
\(510\) 0 0
\(511\) 9.00000 10.3923i 0.398137 0.459728i
\(512\) 0 0
\(513\) 2.50000 + 4.33013i 0.110378 + 0.191180i
\(514\) 0 0
\(515\) 15.0000 + 8.66025i 0.660979 + 0.381616i
\(516\) 0 0
\(517\) 20.7846i 0.914106i
\(518\) 0 0
\(519\) 6.92820i 0.304114i
\(520\) 0 0
\(521\) −12.0000 6.92820i −0.525730 0.303530i 0.213546 0.976933i \(-0.431499\pi\)
−0.739276 + 0.673403i \(0.764832\pi\)
\(522\) 0 0
\(523\) 5.50000 + 9.52628i 0.240498 + 0.416555i 0.960856 0.277047i \(-0.0893559\pi\)
−0.720358 + 0.693602i \(0.756023\pi\)
\(524\) 0 0
\(525\) 17.5000 6.06218i 0.763763 0.264575i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 12.5000 21.6506i 0.543478 0.941332i
\(530\) 0 0
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9.00000 5.19615i 0.388379 0.224231i
\(538\) 0 0
\(539\) −15.0000 19.0526i −0.646096 0.820652i
\(540\) 0 0
\(541\) 2.50000 + 4.33013i 0.107483 + 0.186167i 0.914750 0.404020i \(-0.132387\pi\)
−0.807267 + 0.590187i \(0.799054\pi\)
\(542\) 0 0
\(543\) −4.50000 2.59808i −0.193113 0.111494i
\(544\) 0 0
\(545\) 24.2487i 1.03870i
\(546\) 0 0
\(547\) 3.46410i 0.148114i 0.997254 + 0.0740571i \(0.0235947\pi\)
−0.997254 + 0.0740571i \(0.976405\pi\)
\(548\) 0 0
\(549\) 12.0000 + 6.92820i 0.512148 + 0.295689i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −10.5000 30.3109i −0.446505 1.28895i
\(554\) 0 0
\(555\) 33.0000 19.0526i 1.40077 0.808736i
\(556\) 0 0
\(557\) −21.0000 + 36.3731i −0.889799 + 1.54118i −0.0496855 + 0.998765i \(0.515822\pi\)
−0.840113 + 0.542411i \(0.817511\pi\)
\(558\) 0 0
\(559\) −15.0000 −0.634432
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.00000 5.19615i 0.126435 0.218992i −0.795858 0.605483i \(-0.792980\pi\)
0.922293 + 0.386492i \(0.126313\pi\)
\(564\) 0 0
\(565\) −18.0000 + 10.3923i −0.757266 + 0.437208i
\(566\) 0 0
\(567\) −2.00000 1.73205i −0.0839921 0.0727393i
\(568\) 0 0
\(569\) −21.0000 36.3731i −0.880366 1.52484i −0.850935 0.525271i \(-0.823964\pi\)
−0.0294311 0.999567i \(-0.509370\pi\)
\(570\) 0 0
\(571\) 40.5000 + 23.3827i 1.69487 + 0.978535i 0.950477 + 0.310796i \(0.100596\pi\)
0.744396 + 0.667739i \(0.232738\pi\)
\(572\) 0 0
\(573\) 17.3205i 0.723575i
\(574\) 0 0
\(575\) 48.4974i 2.02248i
\(576\) 0 0
\(577\) 10.5000 + 6.06218i 0.437121 + 0.252372i 0.702376 0.711807i \(-0.252123\pi\)
−0.265255 + 0.964178i \(0.585456\pi\)
\(578\) 0 0
\(579\) −5.50000 9.52628i −0.228572 0.395899i
\(580\) 0 0
\(581\) −9.00000 + 46.7654i −0.373383 + 1.94015i
\(582\) 0 0
\(583\) −36.0000 + 20.7846i −1.49097 + 0.860811i
\(584\) 0 0
\(585\) −3.00000 + 5.19615i −0.124035 + 0.214834i
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −25.0000 −1.03011
\(590\) 0 0
\(591\) 12.0000 20.7846i 0.493614 0.854965i
\(592\) 0 0
\(593\) 3.00000 1.73205i 0.123195 0.0711268i −0.437136 0.899395i \(-0.644007\pi\)
0.560331 + 0.828269i \(0.310674\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.00000 13.8564i −0.327418 0.567105i
\(598\) 0 0
\(599\) 42.0000 + 24.2487i 1.71607 + 0.990775i 0.925794 + 0.378030i \(0.123398\pi\)
0.790280 + 0.612746i \(0.209935\pi\)
\(600\) 0 0
\(601\) 39.8372i 1.62499i −0.582967 0.812496i \(-0.698108\pi\)
0.582967 0.812496i \(-0.301892\pi\)
\(602\) 0 0
\(603\) 8.66025i 0.352673i
\(604\) 0 0
\(605\) 3.00000 + 1.73205i 0.121967 + 0.0704179i
\(606\) 0 0
\(607\) 6.50000 + 11.2583i 0.263827 + 0.456962i 0.967256 0.253804i \(-0.0816819\pi\)
−0.703429 + 0.710766i \(0.748349\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.00000 5.19615i 0.364101 0.210214i
\(612\) 0 0
\(613\) −1.00000 + 1.73205i −0.0403896 + 0.0699569i −0.885514 0.464614i \(-0.846193\pi\)
0.845124 + 0.534570i \(0.179527\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −0.500000 + 0.866025i −0.0200967 + 0.0348085i −0.875899 0.482495i \(-0.839731\pi\)
0.855802 + 0.517303i \(0.173064\pi\)
\(620\) 0 0
\(621\) 6.00000 3.46410i 0.240772 0.139010i
\(622\) 0 0
\(623\) 6.00000 + 17.3205i 0.240385 + 0.693932i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) −15.0000 8.66025i −0.599042 0.345857i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 3.46410i 0.137904i −0.997620 0.0689519i \(-0.978035\pi\)
0.997620 0.0689519i \(-0.0219655\pi\)
\(632\) 0 0
\(633\) −3.00000 1.73205i −0.119239 0.0688428i
\(634\) 0 0
\(635\) −21.0000 36.3731i −0.833360 1.44342i
\(636\) 0 0
\(637\) −4.50000 + 11.2583i −0.178296 + 0.446071i
\(638\) 0 0
\(639\) −3.00000 + 1.73205i −0.118678 + 0.0685189i
\(640\) 0 0
\(641\) −12.0000 + 20.7846i −0.473972 + 0.820943i −0.999556 0.0297987i \(-0.990513\pi\)
0.525584 + 0.850741i \(0.323847\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) 0 0
\(645\) −30.0000 −1.18125
\(646\) 0 0
\(647\) 15.0000 25.9808i 0.589711 1.02141i −0.404559 0.914512i \(-0.632575\pi\)
0.994270 0.106897i \(-0.0340916\pi\)
\(648\) 0 0
\(649\) −36.0000 + 20.7846i −1.41312 + 0.815867i
\(650\) 0 0
\(651\) 12.5000 4.33013i 0.489914 0.169711i
\(652\) 0 0
\(653\) −3.00000 5.19615i −0.117399 0.203341i 0.801337 0.598213i \(-0.204122\pi\)
−0.918736 + 0.394872i \(0.870789\pi\)
\(654\) 0 0
\(655\) −18.0000 10.3923i −0.703318 0.406061i
\(656\) 0 0
\(657\) 5.19615i 0.202721i
\(658\) 0 0
\(659\) 13.8564i 0.539769i −0.962893 0.269884i \(-0.913014\pi\)
0.962893 0.269884i \(-0.0869855\pi\)
\(660\) 0 0
\(661\) 22.5000 + 12.9904i 0.875149 + 0.505267i 0.869056 0.494714i \(-0.164727\pi\)
0.00609283 + 0.999981i \(0.498061\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 30.0000 34.6410i 1.16335 1.34332i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −8.00000 + 13.8564i −0.309298 + 0.535720i
\(670\) 0 0
\(671\) −48.0000 −1.85302
\(672\) 0 0
\(673\) 1.00000 0.0385472 0.0192736 0.999814i \(-0.493865\pi\)
0.0192736 + 0.999814i \(0.493865\pi\)
\(674\) 0 0
\(675\) −3.50000 + 6.06218i −0.134715 + 0.233333i
\(676\) 0 0
\(677\) −6.00000 + 3.46410i −0.230599 + 0.133136i −0.610848 0.791748i \(-0.709171\pi\)
0.380250 + 0.924884i \(0.375838\pi\)
\(678\) 0 0
\(679\) −18.0000 3.46410i −0.690777 0.132940i
\(680\) 0 0
\(681\) −9.00000 15.5885i −0.344881 0.597351i
\(682\) 0 0
\(683\) −6.00000 3.46410i −0.229584 0.132550i 0.380796 0.924659i \(-0.375650\pi\)
−0.610380 + 0.792109i \(0.708983\pi\)
\(684\) 0 0
\(685\) 41.5692i 1.58828i
\(686\) 0 0
\(687\) 8.66025i 0.330409i
\(688\) 0 0
\(689\) 18.0000 + 10.3923i 0.685745 + 0.395915i
\(690\) 0 0
\(691\) 11.5000 + 19.9186i 0.437481 + 0.757739i 0.997494 0.0707446i \(-0.0225375\pi\)
−0.560014 + 0.828483i \(0.689204\pi\)
\(692\) 0 0
\(693\) 9.00000 + 1.73205i 0.341882 + 0.0657952i
\(694\) 0 0
\(695\) 21.0000 12.1244i 0.796575 0.459903i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 6.00000 0.226941
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) 0 0
\(703\) 27.5000 47.6314i 1.03718 1.79645i
\(704\) 0 0
\(705\) 18.0000 10.3923i 0.677919 0.391397i
\(706\) 0 0
\(707\) −18.0000 + 20.7846i −0.676960 + 0.781686i
\(708\) 0 0
\(709\) 5.00000 + 8.66025i 0.187779 + 0.325243i 0.944509 0.328484i \(-0.106538\pi\)
−0.756730 + 0.653727i \(0.773204\pi\)
\(710\) 0 0
\(711\) 10.5000 + 6.06218i 0.393781 + 0.227349i
\(712\) 0 0
\(713\) 34.6410i 1.29732i
\(714\) 0 0
\(715\) 20.7846i 0.777300i
\(716\) 0 0
\(717\) −9.00000 5.19615i −0.336111 0.194054i
\(718\) 0 0
\(719\) 15.0000 + 25.9808i 0.559406 + 0.968919i 0.997546 + 0.0700124i \(0.0223039\pi\)
−0.438141 + 0.898906i \(0.644363\pi\)
\(720\) 0 0
\(721\) −12.5000 + 4.33013i −0.465524 + 0.161262i
\(722\) 0 0
\(723\) −12.0000 + 6.92820i −0.446285 + 0.257663i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −23.0000 −0.853023 −0.426511 0.904482i \(-0.640258\pi\)
−0.426511 + 0.904482i \(0.640258\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −25.5000 + 14.7224i −0.941864 + 0.543785i −0.890544 0.454897i \(-0.849676\pi\)
−0.0513199 + 0.998682i \(0.516343\pi\)
\(734\) 0 0
\(735\) −9.00000 + 22.5167i −0.331970 + 0.830540i
\(736\) 0 0
\(737\) −15.0000 25.9808i −0.552532 0.957014i
\(738\) 0 0
\(739\) 46.5000 + 26.8468i 1.71053 + 0.987575i 0.933839 + 0.357693i \(0.116437\pi\)
0.776691 + 0.629882i \(0.216897\pi\)
\(740\) 0 0
\(741\) 8.66025i 0.318142i
\(742\) 0 0
\(743\) 3.46410i 0.127086i −0.997979 0.0635428i \(-0.979760\pi\)
0.997979 0.0635428i \(-0.0202399\pi\)
\(744\) 0 0
\(745\) −36.0000 20.7846i −1.31894 0.761489i
\(746\) 0 0
\(747\) −9.00000 15.5885i −0.329293 0.570352i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.50000 2.59808i 0.164207 0.0948051i −0.415644 0.909527i \(-0.636444\pi\)
0.579852 + 0.814722i \(0.303111\pi\)
\(752\) 0 0
\(753\) −6.00000 + 10.3923i −0.218652 + 0.378717i
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0 0
\(759\) −12.0000 + 20.7846i −0.435572 + 0.754434i
\(760\) 0 0
\(761\) −36.0000 + 20.7846i −1.30500 + 0.753442i −0.981257 0.192704i \(-0.938274\pi\)
−0.323742 + 0.946145i \(0.604941\pi\)
\(762\) 0 0
\(763\) 14.0000 + 12.1244i 0.506834 + 0.438931i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.0000 + 10.3923i 0.649942 + 0.375244i
\(768\) 0 0
\(769\) 15.5885i 0.562134i −0.959688 0.281067i \(-0.909312\pi\)
0.959688 0.281067i \(-0.0906883\pi\)
\(770\) 0 0
\(771\) 17.3205i 0.623783i
\(772\) 0 0
\(773\) −39.0000 22.5167i −1.40273 0.809868i −0.408060 0.912955i \(-0.633795\pi\)
−0.994672 + 0.103087i \(0.967128\pi\)
\(774\) 0 0
\(775\) −17.5000 30.3109i −0.628619 1.08880i
\(776\) 0 0
\(777\) −5.50000 + 28.5788i −0.197311 + 1.02526i
\(778\) 0 0
\(779\) −15.0000 + 8.66025i −0.537431 + 0.310286i
\(780\) 0 0
\(781\) 6.00000 10.3923i 0.214697 0.371866i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −8.00000 + 13.8564i −0.285169 + 0.493928i −0.972650 0.232275i \(-0.925383\pi\)
0.687481 + 0.726202i \(0.258716\pi\)
\(788\) 0 0
\(789\) −12.0000 + 6.92820i −0.427211 + 0.246651i
\(790\) 0 0
\(791\) 3.00000 15.5885i 0.106668 0.554262i
\(792\) 0 0
\(793\) 12.0000 + 20.7846i 0.426132 + 0.738083i
\(794\) 0 0
\(795\) 36.0000 + 20.7846i 1.27679 + 0.737154i
\(796\) 0 0
\(797\) 34.6410i 1.22705i 0.789676 + 0.613524i \(0.210249\pi\)
−0.789676 + 0.613524i \(0.789751\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 3.46410i −0.212000 0.122398i
\(802\) 0 0
\(803\) −9.00000 15.5885i −0.317603 0.550105i
\(804\) 0 0
\(805\) −48.0000 41.5692i −1.69178 1.46512i
\(806\) 0 0
\(807\) −15.0000 + 8.66025i −0.528025 + 0.304855i
\(808\) 0 0
\(809\) −9.00000 + 15.5885i −0.316423 + 0.548061i −0.979739 0.200279i \(-0.935815\pi\)
0.663316 + 0.748340i \(0.269149\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 0 0
\(813\) 8.00000 0.280572
\(814\) 0 0
\(815\) 6.00000 10.3923i 0.210171 0.364027i
\(816\) 0 0
\(817\) −37.5000 + 21.6506i −1.31196 + 0.757460i
\(818\) 0 0
\(819\) −1.50000 4.33013i −0.0524142 0.151307i
\(820\) 0 0
\(821\) 9.00000 + 15.5885i 0.314102 + 0.544041i 0.979246 0.202674i \(-0.0649632\pi\)
−0.665144 + 0.746715i \(0.731630\pi\)
\(822\) 0 0
\(823\) 3.00000 + 1.73205i 0.104573 + 0.0603755i 0.551375 0.834258i \(-0.314104\pi\)
−0.446801 + 0.894633i \(0.647437\pi\)
\(824\) 0 0
\(825\) 24.2487i 0.844232i
\(826\) 0 0
\(827\) 38.1051i 1.32504i −0.749042 0.662522i \(-0.769486\pi\)
0.749042 0.662522i \(-0.230514\pi\)
\(828\) 0 0
\(829\) 22.5000 + 12.9904i 0.781457 + 0.451175i 0.836947 0.547285i \(-0.184339\pi\)
−0.0554892 + 0.998459i \(0.517672\pi\)
\(830\) 0 0
\(831\) 9.50000 + 16.4545i 0.329551 + 0.570800i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 18.0000 10.3923i 0.622916 0.359641i
\(836\) 0 0
\(837\) −2.50000 + 4.33013i −0.0864126 + 0.149671i
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 30.0000 17.3205i 1.03203 0.595844i
\(846\) 0 0
\(847\) −2.50000 + 0.866025i −0.0859010 + 0.0297570i
\(848\) 0 0
\(849\) −3.50000 6.06218i −0.120120 0.208053i
\(850\) 0 0
\(851\) −66.0000 38.1051i −2.26245 1.30623i
\(852\) 0 0
\(853\) 5.19615i 0.177913i −0.996036 0.0889564i \(-0.971647\pi\)
0.996036 0.0889564i \(-0.0283532\pi\)
\(854\) 0 0
\(855\) 17.3205i 0.592349i
\(856\) 0 0
\(857\) 12.0000 + 6.92820i 0.409912 + 0.236663i 0.690752 0.723092i \(-0.257280\pi\)
−0.280840 + 0.959755i \(0.590613\pi\)
\(858\) 0 0
\(859\) −16.0000 27.7128i −0.545913 0.945549i −0.998549 0.0538535i \(-0.982850\pi\)
0.452636 0.891695i \(-0.350484\pi\)
\(860\) 0 0
\(861\) 6.00000 6.92820i 0.204479 0.236113i
\(862\) 0 0
\(863\) −9.00000 + 5.19615i −0.306364 + 0.176879i −0.645298 0.763931i \(-0.723267\pi\)
0.338935 + 0.940810i \(0.389933\pi\)
\(864\) 0 0
\(865\) 12.0000 20.7846i 0.408012 0.706698i
\(866\) 0 0
\(867\) −17.0000 −0.577350
\(868\) 0 0
\(869\) −42.0000 −1.42475
\(870\) 0 0
\(871\) −7.50000 + 12.9904i −0.254128 + 0.440162i
\(872\) 0 0
\(873\) 6.00000 3.46410i 0.203069 0.117242i
\(874\) 0 0
\(875\) 18.0000 + 3.46410i 0.608511 + 0.117108i
\(876\) 0 0
\(877\) −1.00000 1.73205i −0.0337676 0.0584872i 0.848648 0.528958i \(-0.177417\pi\)
−0.882415 + 0.470471i \(0.844084\pi\)
\(878\) 0 0
\(879\) −18.0000 10.3923i −0.607125 0.350524i
\(880\) 0 0
\(881\) 6.92820i 0.233417i 0.993166 + 0.116709i \(0.0372343\pi\)
−0.993166 + 0.116709i \(0.962766\pi\)
\(882\) 0 0
\(883\) 46.7654i 1.57378i 0.617093 + 0.786890i \(0.288310\pi\)
−0.617093 + 0.786890i \(0.711690\pi\)
\(884\) 0 0
\(885\) 36.0000 + 20.7846i 1.21013 + 0.698667i
\(886\) 0 0
\(887\) 9.00000 + 15.5885i 0.302190 + 0.523409i 0.976632 0.214919i \(-0.0689488\pi\)
−0.674441 + 0.738328i \(0.735615\pi\)
\(888\) 0 0
\(889\) 31.5000 + 6.06218i 1.05648 + 0.203319i
\(890\) 0 0
\(891\) −3.00000 + 1.73205i −0.100504 + 0.0580259i
\(892\) 0 0
\(893\) 15.0000 25.9808i 0.501956 0.869413i
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 15.0000 17.3205i 0.499169 0.576390i
\(904\) 0 0
\(905\) −9.00000 15.5885i −0.299170 0.518178i
\(906\) 0 0
\(907\) −28.5000 16.4545i −0.946327 0.546362i −0.0543890 0.998520i \(-0.517321\pi\)
−0.891938 + 0.452158i \(0.850654\pi\)
\(908\) 0 0
\(909\) 10.3923i 0.344691i
\(910\) 0 0
\(911\) 41.5692i 1.37725i −0.725118 0.688625i \(-0.758215\pi\)
0.725118 0.688625i \(-0.241785\pi\)
\(912\) 0 0
\(913\) 54.0000 + 31.1769i 1.78714 + 1.03181i
\(914\) 0 0
\(915\) 24.0000 + 41.5692i 0.793416 + 1.37424i
\(916\) 0 0
\(917\) 15.0000 5.19615i 0.495344 0.171592i
\(918\) 0 0
\(919\) −37.5000 + 21.6506i −1.23701 + 0.714189i −0.968482 0.249083i \(-0.919871\pi\)
−0.268529 + 0.963272i \(0.586537\pi\)
\(920\) 0 0
\(921\) 5.50000 9.52628i 0.181231 0.313902i
\(922\) 0 0
\(923\) −6.00000 −0.197492
\(924\) 0 0
\(925\) 77.0000 2.53174
\(926\) 0 0
\(927\) 2.50000 4.33013i 0.0821108 0.142220i
\(928\) 0 0
\(929\) 9.00000 5.19615i 0.295280 0.170480i −0.345040 0.938588i \(-0.612135\pi\)
0.640321 + 0.768108i \(0.278801\pi\)
\(930\) 0 0
\(931\) 5.00000 + 34.6410i 0.163868 + 1.13531i
\(932\) 0 0
\(933\) −9.00000 15.5885i −0.294647 0.510343i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 53.6936i 1.75409i −0.480406 0.877046i \(-0.659511\pi\)
0.480406 0.877046i \(-0.340489\pi\)
\(938\) 0 0
\(939\) 15.5885i 0.508710i
\(940\) 0 0
\(941\) −12.0000 6.92820i −0.391189 0.225853i 0.291486 0.956575i \(-0.405850\pi\)
−0.682675 + 0.730722i \(0.739184\pi\)
\(942\) 0 0
\(943\) 12.0000 + 20.7846i 0.390774 + 0.676840i
\(944\) 0 0
\(945\) −3.00000 8.66025i −0.0975900 0.281718i
\(946\) 0 0
\(947\) 39.0000 22.5167i 1.26733 0.731693i 0.292848 0.956159i \(-0.405397\pi\)
0.974482 + 0.224466i \(0.0720637\pi\)
\(948\) 0 0
\(949\) −4.50000 + 7.79423i −0.146076 + 0.253011i
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) 24.0000 0.777436 0.388718 0.921357i \(-0.372918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(954\) 0 0
\(955\) −30.0000 + 51.9615i −0.970777 + 1.68144i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −24.0000 20.7846i −0.775000 0.671170i
\(960\) 0 0
\(961\) 3.00000 + 5.19615i 0.0967742 + 0.167618i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 38.1051i 1.22665i
\(966\) 0 0
\(967\) 8.66025i 0.278495i 0.990258 + 0.139247i \(0.0444684\pi\)
−0.990258 + 0.139247i \(0.955532\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −24.0000 41.5692i −0.770197 1.33402i −0.937455 0.348107i \(-0.886825\pi\)
0.167258 0.985913i \(-0.446509\pi\)
\(972\) 0 0
\(973\) −3.50000 + 18.1865i −0.112205 + 0.583033i
\(974\) 0 0
\(975\) −10.5000 + 6.06218i −0.336269 + 0.194145i
\(976\) 0 0
\(977\) 21.0000 36.3731i 0.671850 1.16368i −0.305530 0.952183i \(-0.598833\pi\)
0.977379 0.211495i \(-0.0678332\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −7.00000 −0.223493
\(982\) 0 0
\(983\) 6.00000 10.3923i 0.191370 0.331463i −0.754334 0.656490i \(-0.772040\pi\)
0.945705 + 0.325027i \(0.105374\pi\)
\(984\) 0 0
\(985\) 72.0000 41.5692i 2.29411 1.32451i
\(986\) 0 0
\(987\) −3.00000 + 15.5885i −0.0954911 + 0.496186i
\(988\) 0 0
\(989\) 30.0000 + 51.9615i 0.953945 + 1.65228i
\(990\) 0 0
\(991\) −34.5000 19.9186i −1.09593 0.632735i −0.160780 0.986990i \(-0.551401\pi\)
−0.935149 + 0.354256i \(0.884734\pi\)
\(992\) 0 0
\(993\) 15.5885i 0.494685i
\(994\) 0 0
\(995\) 55.4256i 1.75711i
\(996\) 0 0
\(997\) −16.5000 9.52628i −0.522560 0.301700i 0.215421 0.976521i \(-0.430888\pi\)
−0.737982 + 0.674821i \(0.764221\pi\)
\(998\) 0 0
\(999\) −5.50000 9.52628i −0.174012 0.301398i
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 336.2.bl.h.31.1 yes 2
3.2 odd 2 1008.2.cs.a.703.1 2
4.3 odd 2 336.2.bl.d.31.1 2
7.2 even 3 2352.2.bl.g.607.1 2
7.3 odd 6 2352.2.b.e.1567.1 2
7.4 even 3 2352.2.b.d.1567.2 2
7.5 odd 6 336.2.bl.d.271.1 yes 2
7.6 odd 2 2352.2.bl.a.31.1 2
8.3 odd 2 1344.2.bl.e.703.1 2
8.5 even 2 1344.2.bl.a.703.1 2
12.11 even 2 1008.2.cs.b.703.1 2
21.5 even 6 1008.2.cs.b.271.1 2
21.11 odd 6 7056.2.b.k.1567.1 2
21.17 even 6 7056.2.b.d.1567.2 2
28.3 even 6 2352.2.b.d.1567.1 2
28.11 odd 6 2352.2.b.e.1567.2 2
28.19 even 6 inner 336.2.bl.h.271.1 yes 2
28.23 odd 6 2352.2.bl.a.607.1 2
28.27 even 2 2352.2.bl.g.31.1 2
56.5 odd 6 1344.2.bl.e.1279.1 2
56.19 even 6 1344.2.bl.a.1279.1 2
84.11 even 6 7056.2.b.d.1567.1 2
84.47 odd 6 1008.2.cs.a.271.1 2
84.59 odd 6 7056.2.b.k.1567.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.2.bl.d.31.1 2 4.3 odd 2
336.2.bl.d.271.1 yes 2 7.5 odd 6
336.2.bl.h.31.1 yes 2 1.1 even 1 trivial
336.2.bl.h.271.1 yes 2 28.19 even 6 inner
1008.2.cs.a.271.1 2 84.47 odd 6
1008.2.cs.a.703.1 2 3.2 odd 2
1008.2.cs.b.271.1 2 21.5 even 6
1008.2.cs.b.703.1 2 12.11 even 2
1344.2.bl.a.703.1 2 8.5 even 2
1344.2.bl.a.1279.1 2 56.19 even 6
1344.2.bl.e.703.1 2 8.3 odd 2
1344.2.bl.e.1279.1 2 56.5 odd 6
2352.2.b.d.1567.1 2 28.3 even 6
2352.2.b.d.1567.2 2 7.4 even 3
2352.2.b.e.1567.1 2 7.3 odd 6
2352.2.b.e.1567.2 2 28.11 odd 6
2352.2.bl.a.31.1 2 7.6 odd 2
2352.2.bl.a.607.1 2 28.23 odd 6
2352.2.bl.g.31.1 2 28.27 even 2
2352.2.bl.g.607.1 2 7.2 even 3
7056.2.b.d.1567.1 2 84.11 even 6
7056.2.b.d.1567.2 2 21.17 even 6
7056.2.b.k.1567.1 2 21.11 odd 6
7056.2.b.k.1567.2 2 84.59 odd 6