Properties

Label 784.2.i.c.177.1
Level $784$
Weight $2$
Character 784.177
Analytic conductor $6.260$
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] = [784,2,Mod(177,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, 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("784.177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.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: \(6.26027151847\)
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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 177.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 784.177
Dual form 784.2.i.c.753.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{3} +(-0.500000 + 0.866025i) q^{9} -4.00000 q^{13} +(-3.00000 - 5.19615i) q^{17} +(1.00000 - 1.73205i) q^{19} +(2.50000 + 4.33013i) q^{25} -4.00000 q^{27} -6.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(-1.00000 + 1.73205i) q^{37} +(4.00000 + 6.92820i) q^{39} +6.00000 q^{41} -8.00000 q^{43} +(-6.00000 + 10.3923i) q^{47} +(-6.00000 + 10.3923i) q^{51} +(-3.00000 - 5.19615i) q^{53} -4.00000 q^{57} +(-3.00000 - 5.19615i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(-2.00000 - 3.46410i) q^{67} +(-1.00000 - 1.73205i) q^{73} +(5.00000 - 8.66025i) q^{75} +(4.00000 - 6.92820i) q^{79} +(5.50000 + 9.52628i) q^{81} +6.00000 q^{83} +(6.00000 + 10.3923i) q^{87} +(3.00000 - 5.19615i) q^{89} +(-4.00000 + 6.92820i) q^{93} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - q^{9} - 8 q^{13} - 6 q^{17} + 2 q^{19} + 5 q^{25} - 8 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{37} + 8 q^{39} + 12 q^{41} - 16 q^{43} - 12 q^{47} - 12 q^{51} - 6 q^{53} - 8 q^{57} - 6 q^{59} - 8 q^{61} - 4 q^{67} - 2 q^{73} + 10 q^{75} + 8 q^{79} + 11 q^{81} + 12 q^{83} + 12 q^{87} + 6 q^{89} - 8 q^{93} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 1.73205i −0.577350 1.00000i −0.995782 0.0917517i \(-0.970753\pi\)
0.418432 0.908248i \(-0.362580\pi\)
\(4\) 0 0
\(5\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 5.19615i −0.727607 1.26025i −0.957892 0.287129i \(-0.907299\pi\)
0.230285 0.973123i \(-0.426034\pi\)
\(18\) 0 0
\(19\) 1.00000 1.73205i 0.229416 0.397360i −0.728219 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 3.46410i −0.359211 0.622171i 0.628619 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 + 1.73205i −0.164399 + 0.284747i −0.936442 0.350823i \(-0.885902\pi\)
0.772043 + 0.635571i \(0.219235\pi\)
\(38\) 0 0
\(39\) 4.00000 + 6.92820i 0.640513 + 1.10940i
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.00000 + 10.3923i −0.875190 + 1.51587i −0.0186297 + 0.999826i \(0.505930\pi\)
−0.856560 + 0.516047i \(0.827403\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −6.00000 + 10.3923i −0.840168 + 1.45521i
\(52\) 0 0
\(53\) −3.00000 5.19615i −0.412082 0.713746i 0.583036 0.812447i \(-0.301865\pi\)
−0.995117 + 0.0987002i \(0.968532\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −3.00000 5.19615i −0.390567 0.676481i 0.601958 0.798528i \(-0.294388\pi\)
−0.992524 + 0.122047i \(0.961054\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −2.00000 3.46410i −0.244339 0.423207i 0.717607 0.696449i \(-0.245238\pi\)
−0.961946 + 0.273241i \(0.911904\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.00000 1.73205i −0.117041 0.202721i 0.801553 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) 5.00000 8.66025i 0.577350 1.00000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 6.92820i 0.450035 0.779484i −0.548352 0.836247i \(-0.684745\pi\)
0.998388 + 0.0567635i \(0.0180781\pi\)
\(80\) 0 0
\(81\) 5.50000 + 9.52628i 0.611111 + 1.05848i
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.00000 + 10.3923i 0.643268 + 1.11417i
\(88\) 0 0
\(89\) 3.00000 5.19615i 0.317999 0.550791i −0.662071 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.00000 + 6.92820i −0.414781 + 0.718421i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(102\) 0 0
\(103\) −2.00000 + 3.46410i −0.197066 + 0.341328i −0.947576 0.319531i \(-0.896475\pi\)
0.750510 + 0.660859i \(0.229808\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 10.3923i 0.580042 1.00466i −0.415432 0.909624i \(-0.636370\pi\)
0.995474 0.0950377i \(-0.0302972\pi\)
\(108\) 0 0
\(109\) −1.00000 1.73205i −0.0957826 0.165900i 0.814152 0.580651i \(-0.197202\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.00000 3.46410i 0.184900 0.320256i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.50000 9.52628i 0.500000 0.866025i
\(122\) 0 0
\(123\) −6.00000 10.3923i −0.541002 0.937043i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 8.00000 + 13.8564i 0.704361 + 1.21999i
\(130\) 0 0
\(131\) 9.00000 15.5885i 0.786334 1.36197i −0.141865 0.989886i \(-0.545310\pi\)
0.928199 0.372084i \(-0.121357\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 15.5885i −0.768922 1.33181i −0.938148 0.346235i \(-0.887460\pi\)
0.169226 0.985577i \(-0.445873\pi\)
\(138\) 0 0
\(139\) −14.0000 −1.18746 −0.593732 0.804663i \(-0.702346\pi\)
−0.593732 + 0.804663i \(0.702346\pi\)
\(140\) 0 0
\(141\) 24.0000 2.02116
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.00000 15.5885i 0.737309 1.27706i −0.216394 0.976306i \(-0.569430\pi\)
0.953703 0.300750i \(-0.0972370\pi\)
\(150\) 0 0
\(151\) 4.00000 + 6.92820i 0.325515 + 0.563809i 0.981617 0.190864i \(-0.0611289\pi\)
−0.656101 + 0.754673i \(0.727796\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 + 3.46410i 0.159617 + 0.276465i 0.934731 0.355357i \(-0.115641\pi\)
−0.775113 + 0.631822i \(0.782307\pi\)
\(158\) 0 0
\(159\) −6.00000 + 10.3923i −0.475831 + 0.824163i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −8.00000 + 13.8564i −0.626608 + 1.08532i 0.361619 + 0.932326i \(0.382224\pi\)
−0.988227 + 0.152992i \(0.951109\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) 1.00000 + 1.73205i 0.0764719 + 0.132453i
\(172\) 0 0
\(173\) 6.00000 10.3923i 0.456172 0.790112i −0.542583 0.840002i \(-0.682554\pi\)
0.998755 + 0.0498898i \(0.0158870\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.00000 + 10.3923i −0.450988 + 0.781133i
\(178\) 0 0
\(179\) −6.00000 10.3923i −0.448461 0.776757i 0.549825 0.835280i \(-0.314694\pi\)
−0.998286 + 0.0585225i \(0.981361\pi\)
\(180\) 0 0
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 0 0
\(183\) 16.0000 1.18275
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.0000 20.7846i 0.868290 1.50392i 0.00454614 0.999990i \(-0.498553\pi\)
0.863743 0.503932i \(-0.168114\pi\)
\(192\) 0 0
\(193\) −7.00000 12.1244i −0.503871 0.872730i −0.999990 0.00447566i \(-0.998575\pi\)
0.496119 0.868255i \(-0.334758\pi\)
\(194\) 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
\(199\) 10.0000 + 17.3205i 0.708881 + 1.22782i 0.965272 + 0.261245i \(0.0841331\pi\)
−0.256391 + 0.966573i \(0.582534\pi\)
\(200\) 0 0
\(201\) −4.00000 + 6.92820i −0.282138 + 0.488678i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2.00000 + 3.46410i −0.135147 + 0.234082i
\(220\) 0 0
\(221\) 12.0000 + 20.7846i 0.807207 + 1.39812i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) −5.00000 −0.333333
\(226\) 0 0
\(227\) 9.00000 + 15.5885i 0.597351 + 1.03464i 0.993210 + 0.116331i \(0.0371134\pi\)
−0.395860 + 0.918311i \(0.629553\pi\)
\(228\) 0 0
\(229\) 2.00000 3.46410i 0.132164 0.228914i −0.792347 0.610071i \(-0.791141\pi\)
0.924510 + 0.381157i \(0.124474\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.00000 5.19615i 0.196537 0.340411i −0.750867 0.660454i \(-0.770364\pi\)
0.947403 + 0.320043i \(0.103697\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −16.0000 −1.03931
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 5.00000 + 8.66025i 0.322078 + 0.557856i 0.980917 0.194429i \(-0.0622852\pi\)
−0.658838 + 0.752285i \(0.728952\pi\)
\(242\) 0 0
\(243\) 5.00000 8.66025i 0.320750 0.555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.00000 + 6.92820i −0.254514 + 0.440831i
\(248\) 0 0
\(249\) −6.00000 10.3923i −0.380235 0.658586i
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.00000 + 15.5885i −0.561405 + 0.972381i 0.435970 + 0.899961i \(0.356405\pi\)
−0.997374 + 0.0724199i \(0.976928\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 5.19615i 0.185695 0.321634i
\(262\) 0 0
\(263\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −12.0000 −0.734388
\(268\) 0 0
\(269\) 6.00000 + 10.3923i 0.365826 + 0.633630i 0.988908 0.148527i \(-0.0474530\pi\)
−0.623082 + 0.782157i \(0.714120\pi\)
\(270\) 0 0
\(271\) −8.00000 + 13.8564i −0.485965 + 0.841717i −0.999870 0.0161307i \(-0.994865\pi\)
0.513905 + 0.857847i \(0.328199\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.00000 + 8.66025i 0.300421 + 0.520344i 0.976231 0.216731i \(-0.0695395\pi\)
−0.675810 + 0.737075i \(0.736206\pi\)
\(278\) 0 0
\(279\) 4.00000 0.239474
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) 0 0
\(283\) −11.0000 19.0526i −0.653882 1.13256i −0.982173 0.187980i \(-0.939806\pi\)
0.328291 0.944577i \(-0.393527\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.50000 + 16.4545i −0.558824 + 0.967911i
\(290\) 0 0
\(291\) 10.0000 + 17.3205i 0.586210 + 1.01535i
\(292\) 0 0
\(293\) 24.0000 1.40209 0.701047 0.713115i \(-0.252716\pi\)
0.701047 + 0.713115i \(0.252716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −12.0000 20.7846i −0.680458 1.17859i −0.974841 0.222900i \(-0.928448\pi\)
0.294384 0.955687i \(-0.404886\pi\)
\(312\) 0 0
\(313\) 5.00000 8.66025i 0.282617 0.489506i −0.689412 0.724370i \(-0.742131\pi\)
0.972028 + 0.234863i \(0.0754642\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.00000 + 5.19615i −0.168497 + 0.291845i −0.937892 0.346929i \(-0.887225\pi\)
0.769395 + 0.638774i \(0.220558\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −24.0000 −1.33955
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −10.0000 17.3205i −0.554700 0.960769i
\(326\) 0 0
\(327\) −2.00000 + 3.46410i −0.110600 + 0.191565i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 6.92820i 0.219860 0.380808i −0.734905 0.678170i \(-0.762773\pi\)
0.954765 + 0.297361i \(0.0961066\pi\)
\(332\) 0 0
\(333\) −1.00000 1.73205i −0.0547997 0.0949158i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) −6.00000 10.3923i −0.325875 0.564433i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −12.0000 20.7846i −0.644194 1.11578i −0.984487 0.175457i \(-0.943860\pi\)
0.340293 0.940319i \(-0.389474\pi\)
\(348\) 0 0
\(349\) −28.0000 −1.49881 −0.749403 0.662114i \(-0.769659\pi\)
−0.749403 + 0.662114i \(0.769659\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 0 0
\(353\) −9.00000 15.5885i −0.479022 0.829690i 0.520689 0.853746i \(-0.325675\pi\)
−0.999711 + 0.0240566i \(0.992342\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.0000 + 20.7846i −0.633336 + 1.09697i 0.353529 + 0.935423i \(0.384981\pi\)
−0.986865 + 0.161546i \(0.948352\pi\)
\(360\) 0 0
\(361\) 7.50000 + 12.9904i 0.394737 + 0.683704i
\(362\) 0 0
\(363\) −22.0000 −1.15470
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000 + 6.92820i 0.208798 + 0.361649i 0.951336 0.308155i \(-0.0997115\pi\)
−0.742538 + 0.669804i \(0.766378\pi\)
\(368\) 0 0
\(369\) −3.00000 + 5.19615i −0.156174 + 0.270501i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −7.00000 + 12.1244i −0.362446 + 0.627775i −0.988363 0.152115i \(-0.951392\pi\)
0.625917 + 0.779890i \(0.284725\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) −16.0000 27.7128i −0.819705 1.41977i
\(382\) 0 0
\(383\) 18.0000 31.1769i 0.919757 1.59307i 0.119974 0.992777i \(-0.461719\pi\)
0.799783 0.600289i \(-0.204948\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000 6.92820i 0.203331 0.352180i
\(388\) 0 0
\(389\) −9.00000 15.5885i −0.456318 0.790366i 0.542445 0.840091i \(-0.317499\pi\)
−0.998763 + 0.0497253i \(0.984165\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −36.0000 −1.81596
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −10.0000 + 17.3205i −0.501886 + 0.869291i 0.498112 + 0.867113i \(0.334027\pi\)
−0.999998 + 0.00217869i \(0.999307\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 9.00000 15.5885i 0.449439 0.778450i −0.548911 0.835881i \(-0.684957\pi\)
0.998350 + 0.0574304i \(0.0182907\pi\)
\(402\) 0 0
\(403\) 8.00000 + 13.8564i 0.398508 + 0.690237i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −7.00000 12.1244i −0.346128 0.599511i 0.639430 0.768849i \(-0.279170\pi\)
−0.985558 + 0.169338i \(0.945837\pi\)
\(410\) 0 0
\(411\) −18.0000 + 31.1769i −0.887875 + 1.53784i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 14.0000 + 24.2487i 0.685583 + 1.18746i
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) −6.00000 10.3923i −0.291730 0.505291i
\(424\) 0 0
\(425\) 15.0000 25.9808i 0.727607 1.26025i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 + 20.7846i 0.578020 + 1.00116i 0.995706 + 0.0925683i \(0.0295076\pi\)
−0.417687 + 0.908591i \(0.637159\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 4.00000 6.92820i 0.190910 0.330665i −0.754642 0.656136i \(-0.772190\pi\)
0.945552 + 0.325471i \(0.105523\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.00000 + 10.3923i −0.285069 + 0.493753i −0.972626 0.232377i \(-0.925350\pi\)
0.687557 + 0.726130i \(0.258683\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −36.0000 −1.70274
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.00000 13.8564i 0.375873 0.651031i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.00000 8.66025i 0.233890 0.405110i −0.725059 0.688686i \(-0.758188\pi\)
0.958950 + 0.283577i \(0.0915211\pi\)
\(458\) 0 0
\(459\) 12.0000 + 20.7846i 0.560112 + 0.970143i
\(460\) 0 0
\(461\) 12.0000 0.558896 0.279448 0.960161i \(-0.409849\pi\)
0.279448 + 0.960161i \(0.409849\pi\)
\(462\) 0 0
\(463\) −32.0000 −1.48717 −0.743583 0.668644i \(-0.766875\pi\)
−0.743583 + 0.668644i \(0.766875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.00000 + 5.19615i −0.138823 + 0.240449i −0.927052 0.374934i \(-0.877665\pi\)
0.788228 + 0.615383i \(0.210999\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 4.00000 6.92820i 0.184310 0.319235i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 10.0000 0.458831
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −18.0000 31.1769i −0.822441 1.42451i −0.903859 0.427830i \(-0.859278\pi\)
0.0814184 0.996680i \(-0.474055\pi\)
\(480\) 0 0
\(481\) 4.00000 6.92820i 0.182384 0.315899i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −8.00000 13.8564i −0.362515 0.627894i 0.625859 0.779936i \(-0.284748\pi\)
−0.988374 + 0.152042i \(0.951415\pi\)
\(488\) 0 0
\(489\) 32.0000 1.44709
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 18.0000 + 31.1769i 0.810679 + 1.40414i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −2.00000 + 3.46410i −0.0895323 + 0.155074i −0.907314 0.420455i \(-0.861871\pi\)
0.817781 + 0.575529i \(0.195204\pi\)
\(500\) 0 0
\(501\) −12.0000 20.7846i −0.536120 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
\(507\) −3.00000 5.19615i −0.133235 0.230769i
\(508\) 0 0
\(509\) −18.0000 + 31.1769i −0.797836 + 1.38189i 0.123187 + 0.992384i \(0.460689\pi\)
−0.921023 + 0.389509i \(0.872645\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −4.00000 + 6.92820i −0.176604 + 0.305888i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) −3.00000 5.19615i −0.131432 0.227648i 0.792797 0.609486i \(-0.208624\pi\)
−0.924229 + 0.381839i \(0.875291\pi\)
\(522\) 0 0
\(523\) 1.00000 1.73205i 0.0437269 0.0757373i −0.843334 0.537390i \(-0.819410\pi\)
0.887061 + 0.461653i \(0.152744\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 + 20.7846i −0.522728 + 0.905392i
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −12.0000 + 20.7846i −0.517838 + 0.896922i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −19.0000 + 32.9090i −0.816874 + 1.41487i 0.0911008 + 0.995842i \(0.470961\pi\)
−0.907975 + 0.419025i \(0.862372\pi\)
\(542\) 0 0
\(543\) −20.0000 34.6410i −0.858282 1.48659i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) −4.00000 6.92820i −0.170716 0.295689i
\(550\) 0 0
\(551\) −6.00000 + 10.3923i −0.255609 + 0.442727i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.00000 5.19615i −0.127114 0.220168i 0.795443 0.606028i \(-0.207238\pi\)
−0.922557 + 0.385860i \(0.873905\pi\)
\(558\) 0 0
\(559\) 32.0000 1.35346
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.0000 + 25.9808i 0.632175 + 1.09496i 0.987106 + 0.160066i \(0.0511708\pi\)
−0.354932 + 0.934892i \(0.615496\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −3.00000 + 5.19615i −0.125767 + 0.217834i −0.922032 0.387113i \(-0.873472\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) 16.0000 + 27.7128i 0.669579 + 1.15975i 0.978022 + 0.208502i \(0.0668588\pi\)
−0.308443 + 0.951243i \(0.599808\pi\)
\(572\) 0 0
\(573\) −48.0000 −2.00523
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.00000 1.73205i −0.0416305 0.0721062i 0.844459 0.535620i \(-0.179922\pi\)
−0.886090 + 0.463513i \(0.846589\pi\)
\(578\) 0 0
\(579\) −14.0000 + 24.2487i −0.581820 + 1.00774i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 42.0000 1.73353 0.866763 0.498721i \(-0.166197\pi\)
0.866763 + 0.498721i \(0.166197\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) 18.0000 + 31.1769i 0.740421 + 1.28245i
\(592\) 0 0
\(593\) 3.00000 5.19615i 0.123195 0.213380i −0.797831 0.602881i \(-0.794019\pi\)
0.921026 + 0.389501i \(0.127353\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 20.0000 34.6410i 0.818546 1.41776i
\(598\) 0 0
\(599\) −12.0000 20.7846i −0.490307 0.849236i 0.509631 0.860393i \(-0.329782\pi\)
−0.999938 + 0.0111569i \(0.996449\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.0000 27.7128i 0.649420 1.12483i −0.333842 0.942629i \(-0.608345\pi\)
0.983262 0.182199i \(-0.0583216\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 41.5692i 0.970936 1.68171i
\(612\) 0 0
\(613\) −1.00000 1.73205i −0.0403896 0.0699569i 0.845124 0.534570i \(-0.179527\pi\)
−0.885514 + 0.464614i \(0.846193\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.00000 0.241551 0.120775 0.992680i \(-0.461462\pi\)
0.120775 + 0.992680i \(0.461462\pi\)
\(618\) 0 0
\(619\) 13.0000 + 22.5167i 0.522514 + 0.905021i 0.999657 + 0.0261952i \(0.00833914\pi\)
−0.477143 + 0.878826i \(0.658328\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12.5000 + 21.6506i −0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.0000 0.478471
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) −4.00000 6.92820i −0.158986 0.275371i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.00000 + 15.5885i 0.355479 + 0.615707i 0.987200 0.159489i \(-0.0509845\pi\)
−0.631721 + 0.775196i \(0.717651\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.00000 10.3923i −0.235884 0.408564i 0.723645 0.690172i \(-0.242465\pi\)
−0.959529 + 0.281609i \(0.909132\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.00000 + 15.5885i −0.352197 + 0.610023i −0.986634 0.162951i \(-0.947899\pi\)
0.634437 + 0.772975i \(0.281232\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 20.0000 + 34.6410i 0.777910 + 1.34738i 0.933144 + 0.359502i \(0.117053\pi\)
−0.155235 + 0.987878i \(0.549613\pi\)
\(662\) 0 0
\(663\) 24.0000 41.5692i 0.932083 1.61441i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 8.00000 + 13.8564i 0.309298 + 0.535720i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 26.0000 1.00223 0.501113 0.865382i \(-0.332924\pi\)
0.501113 + 0.865382i \(0.332924\pi\)
\(674\) 0 0
\(675\) −10.0000 17.3205i −0.384900 0.666667i
\(676\) 0 0
\(677\) 6.00000 10.3923i 0.230599 0.399409i −0.727386 0.686229i \(-0.759265\pi\)
0.957984 + 0.286820i \(0.0925982\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 18.0000 31.1769i 0.689761 1.19470i
\(682\) 0 0
\(683\) −6.00000 10.3923i −0.229584 0.397650i 0.728101 0.685470i \(-0.240403\pi\)
−0.957685 + 0.287819i \(0.907070\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −8.00000 −0.305219
\(688\) 0 0
\(689\) 12.0000 + 20.7846i 0.457164 + 0.791831i
\(690\) 0 0
\(691\) −23.0000 + 39.8372i −0.874961 + 1.51548i −0.0181572 + 0.999835i \(0.505780\pi\)
−0.856804 + 0.515642i \(0.827553\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −18.0000 31.1769i −0.681799 1.18091i
\(698\) 0 0
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 0 0
\(703\) 2.00000 + 3.46410i 0.0754314 + 0.130651i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 23.0000 39.8372i 0.863783 1.49612i −0.00446726 0.999990i \(-0.501422\pi\)
0.868250 0.496126i \(-0.165245\pi\)
\(710\) 0 0
\(711\) 4.00000 + 6.92820i 0.150012 + 0.259828i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 24.0000 + 41.5692i 0.896296 + 1.55243i
\(718\) 0 0
\(719\) 6.00000 10.3923i 0.223762 0.387568i −0.732185 0.681106i \(-0.761499\pi\)
0.955947 + 0.293538i \(0.0948328\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 10.0000 17.3205i 0.371904 0.644157i
\(724\) 0 0
\(725\) −15.0000 25.9808i −0.557086 0.964901i
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 24.0000 + 41.5692i 0.887672 + 1.53749i
\(732\) 0 0
\(733\) 20.0000 34.6410i 0.738717 1.27950i −0.214356 0.976756i \(-0.568765\pi\)
0.953073 0.302740i \(-0.0979013\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −8.00000 13.8564i −0.294285 0.509716i 0.680534 0.732717i \(-0.261748\pi\)
−0.974818 + 0.223001i \(0.928415\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −3.00000 + 5.19615i −0.109764 + 0.190117i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −20.0000 + 34.6410i −0.729810 + 1.26407i 0.227153 + 0.973859i \(0.427058\pi\)
−0.956963 + 0.290209i \(0.906275\pi\)
\(752\) 0 0
\(753\) −18.0000 31.1769i −0.655956 1.13615i
\(754\) 0 0
\(755\) 0 0
\(756\) 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
\(761\) 9.00000 15.5885i 0.326250 0.565081i −0.655515 0.755182i \(-0.727548\pi\)
0.981764 + 0.190101i \(0.0608816\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 12.0000 + 20.7846i 0.433295 + 0.750489i
\(768\) 0 0
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 36.0000 1.29651
\(772\) 0 0
\(773\) −12.0000 20.7846i −0.431610 0.747570i 0.565402 0.824815i \(-0.308721\pi\)
−0.997012 + 0.0772449i \(0.975388\pi\)
\(774\) 0 0
\(775\) 10.0000 17.3205i 0.359211 0.622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 10.3923i 0.214972 0.372343i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −11.0000 19.0526i −0.392108 0.679150i 0.600620 0.799535i \(-0.294921\pi\)
−0.992727 + 0.120384i \(0.961587\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 16.0000 27.7128i 0.568177 0.984111i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.0000 −0.425062 −0.212531 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) 3.00000 + 5.19615i 0.106000 + 0.183597i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12.0000 20.7846i 0.422420 0.731653i
\(808\) 0 0
\(809\) −3.00000 5.19615i −0.105474 0.182687i 0.808458 0.588555i \(-0.200303\pi\)
−0.913932 + 0.405868i \(0.866969\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) 32.0000 1.12229
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −8.00000 + 13.8564i −0.279885 + 0.484774i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.00000 + 5.19615i −0.104701 + 0.181347i −0.913616 0.406578i \(-0.866722\pi\)
0.808915 + 0.587925i \(0.200055\pi\)
\(822\) 0 0
\(823\) −20.0000 34.6410i −0.697156 1.20751i −0.969448 0.245295i \(-0.921115\pi\)
0.272292 0.962215i \(-0.412218\pi\)
\(824\) 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
\(829\) −28.0000 48.4974i −0.972480 1.68439i −0.688012 0.725700i \(-0.741516\pi\)
−0.284469 0.958685i \(-0.591817\pi\)
\(830\) 0 0
\(831\) 10.0000 17.3205i 0.346896 0.600842i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000 + 13.8564i 0.276520 + 0.478947i
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 6.00000 + 10.3923i 0.206651 + 0.357930i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −22.0000 + 38.1051i −0.755038 + 1.30776i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 44.0000 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.00000 + 15.5885i 0.307434 + 0.532492i 0.977800 0.209539i \(-0.0671963\pi\)
−0.670366 + 0.742030i \(0.733863\pi\)
\(858\) 0 0
\(859\) 7.00000 12.1244i 0.238837 0.413678i −0.721544 0.692369i \(-0.756567\pi\)
0.960381 + 0.278691i \(0.0899005\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −12.0000 + 20.7846i −0.408485 + 0.707516i −0.994720 0.102624i \(-0.967276\pi\)
0.586235 + 0.810141i \(0.300609\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 38.0000 1.29055
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.00000 + 13.8564i 0.271070 + 0.469506i
\(872\) 0 0
\(873\) 5.00000 8.66025i 0.169224 0.293105i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11.0000 19.0526i 0.371444 0.643359i −0.618344 0.785907i \(-0.712196\pi\)
0.989788 + 0.142548i \(0.0455296\pi\)
\(878\) 0 0
\(879\) −24.0000 41.5692i −0.809500 1.40209i
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.0000 + 31.1769i −0.604381 + 1.04682i 0.387768 + 0.921757i \(0.373246\pi\)
−0.992149 + 0.125061i \(0.960087\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.0000 + 20.7846i 0.401565 + 0.695530i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000 + 20.7846i 0.400222 + 0.693206i
\(900\) 0 0
\(901\) −18.0000 + 31.1769i −0.599667 + 1.03865i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.0000 + 38.1051i 0.730498 + 1.26526i 0.956671 + 0.291172i \(0.0940453\pi\)
−0.226173 + 0.974087i \(0.572621\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 28.0000 48.4974i 0.923635 1.59978i 0.129893 0.991528i \(-0.458537\pi\)
0.793742 0.608254i \(-0.208130\pi\)
\(920\) 0 0
\(921\) 2.00000 + 3.46410i 0.0659022 + 0.114146i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) −2.00000 3.46410i −0.0656886 0.113776i
\(928\) 0 0
\(929\) −3.00000 + 5.19615i −0.0984268 + 0.170480i −0.911034 0.412332i \(-0.864714\pi\)
0.812607 + 0.582812i \(0.198048\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24.0000 + 41.5692i −0.785725 + 1.36092i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) −20.0000 −0.652675
\(940\) 0 0
\(941\) 12.0000 + 20.7846i 0.391189 + 0.677559i 0.992607 0.121376i \(-0.0387306\pi\)
−0.601418 + 0.798935i \(0.705397\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 20.7846i 0.389948 0.675409i −0.602494 0.798123i \(-0.705826\pi\)
0.992442 + 0.122714i \(0.0391598\pi\)
\(948\) 0 0
\(949\) 4.00000 + 6.92820i 0.129845 + 0.224899i
\(950\) 0 0
\(951\) 12.0000 0.389127
\(952\) 0 0
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 6.00000 + 10.3923i 0.193347 + 0.334887i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −32.0000 −1.02905 −0.514525 0.857475i \(-0.672032\pi\)
−0.514525 + 0.857475i \(0.672032\pi\)
\(968\) 0 0
\(969\) 12.0000 + 20.7846i 0.385496 + 0.667698i
\(970\) 0 0
\(971\) −3.00000 + 5.19615i −0.0962746 + 0.166752i −0.910140 0.414301i \(-0.864026\pi\)
0.813865 + 0.581054i \(0.197359\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −20.0000 + 34.6410i −0.640513 + 1.10940i
\(976\) 0 0
\(977\) 3.00000 + 5.19615i 0.0959785 + 0.166240i 0.910017 0.414572i \(-0.136069\pi\)
−0.814038 + 0.580812i \(0.802735\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −18.0000 31.1769i −0.574111 0.994389i −0.996138 0.0878058i \(-0.972015\pi\)
0.422027 0.906583i \(-0.361319\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −8.00000 13.8564i −0.254128 0.440163i 0.710530 0.703667i \(-0.248455\pi\)
−0.964658 + 0.263504i \(0.915122\pi\)
\(992\) 0 0
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −4.00000 6.92820i −0.126681 0.219418i 0.795708 0.605681i \(-0.207099\pi\)
−0.922389 + 0.386263i \(0.873766\pi\)
\(998\) 0 0
\(999\) 4.00000 6.92820i 0.126554 0.219199i
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 784.2.i.c.177.1 2
4.3 odd 2 98.2.c.b.79.1 2
7.2 even 3 112.2.a.c.1.1 1
7.3 odd 6 784.2.i.i.753.1 2
7.4 even 3 inner 784.2.i.c.753.1 2
7.5 odd 6 784.2.a.b.1.1 1
7.6 odd 2 784.2.i.i.177.1 2
12.11 even 2 882.2.g.c.667.1 2
21.2 odd 6 1008.2.a.h.1.1 1
21.5 even 6 7056.2.a.bd.1.1 1
28.3 even 6 98.2.c.a.67.1 2
28.11 odd 6 98.2.c.b.67.1 2
28.19 even 6 98.2.a.a.1.1 1
28.23 odd 6 14.2.a.a.1.1 1
28.27 even 2 98.2.c.a.79.1 2
35.2 odd 12 2800.2.g.h.449.1 2
35.9 even 6 2800.2.a.g.1.1 1
35.23 odd 12 2800.2.g.h.449.2 2
56.5 odd 6 3136.2.a.z.1.1 1
56.19 even 6 3136.2.a.e.1.1 1
56.37 even 6 448.2.a.a.1.1 1
56.51 odd 6 448.2.a.g.1.1 1
84.11 even 6 882.2.g.c.361.1 2
84.23 even 6 126.2.a.b.1.1 1
84.47 odd 6 882.2.a.i.1.1 1
84.59 odd 6 882.2.g.d.361.1 2
84.83 odd 2 882.2.g.d.667.1 2
112.37 even 12 1792.2.b.g.897.1 2
112.51 odd 12 1792.2.b.c.897.1 2
112.93 even 12 1792.2.b.g.897.2 2
112.107 odd 12 1792.2.b.c.897.2 2
140.19 even 6 2450.2.a.t.1.1 1
140.23 even 12 350.2.c.d.99.2 2
140.47 odd 12 2450.2.c.c.99.1 2
140.79 odd 6 350.2.a.f.1.1 1
140.103 odd 12 2450.2.c.c.99.2 2
140.107 even 12 350.2.c.d.99.1 2
168.107 even 6 4032.2.a.w.1.1 1
168.149 odd 6 4032.2.a.r.1.1 1
252.23 even 6 1134.2.f.f.379.1 2
252.79 odd 6 1134.2.f.l.757.1 2
252.191 even 6 1134.2.f.f.757.1 2
252.247 odd 6 1134.2.f.l.379.1 2
308.219 even 6 1694.2.a.e.1.1 1
364.51 odd 6 2366.2.a.j.1.1 1
364.135 even 12 2366.2.d.b.337.2 2
364.359 even 12 2366.2.d.b.337.1 2
420.23 odd 12 3150.2.g.j.2899.1 2
420.107 odd 12 3150.2.g.j.2899.2 2
420.359 even 6 3150.2.a.i.1.1 1
476.135 odd 6 4046.2.a.f.1.1 1
532.303 even 6 5054.2.a.c.1.1 1
644.275 even 6 7406.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.2.a.a.1.1 1 28.23 odd 6
98.2.a.a.1.1 1 28.19 even 6
98.2.c.a.67.1 2 28.3 even 6
98.2.c.a.79.1 2 28.27 even 2
98.2.c.b.67.1 2 28.11 odd 6
98.2.c.b.79.1 2 4.3 odd 2
112.2.a.c.1.1 1 7.2 even 3
126.2.a.b.1.1 1 84.23 even 6
350.2.a.f.1.1 1 140.79 odd 6
350.2.c.d.99.1 2 140.107 even 12
350.2.c.d.99.2 2 140.23 even 12
448.2.a.a.1.1 1 56.37 even 6
448.2.a.g.1.1 1 56.51 odd 6
784.2.a.b.1.1 1 7.5 odd 6
784.2.i.c.177.1 2 1.1 even 1 trivial
784.2.i.c.753.1 2 7.4 even 3 inner
784.2.i.i.177.1 2 7.6 odd 2
784.2.i.i.753.1 2 7.3 odd 6
882.2.a.i.1.1 1 84.47 odd 6
882.2.g.c.361.1 2 84.11 even 6
882.2.g.c.667.1 2 12.11 even 2
882.2.g.d.361.1 2 84.59 odd 6
882.2.g.d.667.1 2 84.83 odd 2
1008.2.a.h.1.1 1 21.2 odd 6
1134.2.f.f.379.1 2 252.23 even 6
1134.2.f.f.757.1 2 252.191 even 6
1134.2.f.l.379.1 2 252.247 odd 6
1134.2.f.l.757.1 2 252.79 odd 6
1694.2.a.e.1.1 1 308.219 even 6
1792.2.b.c.897.1 2 112.51 odd 12
1792.2.b.c.897.2 2 112.107 odd 12
1792.2.b.g.897.1 2 112.37 even 12
1792.2.b.g.897.2 2 112.93 even 12
2366.2.a.j.1.1 1 364.51 odd 6
2366.2.d.b.337.1 2 364.359 even 12
2366.2.d.b.337.2 2 364.135 even 12
2450.2.a.t.1.1 1 140.19 even 6
2450.2.c.c.99.1 2 140.47 odd 12
2450.2.c.c.99.2 2 140.103 odd 12
2800.2.a.g.1.1 1 35.9 even 6
2800.2.g.h.449.1 2 35.2 odd 12
2800.2.g.h.449.2 2 35.23 odd 12
3136.2.a.e.1.1 1 56.19 even 6
3136.2.a.z.1.1 1 56.5 odd 6
3150.2.a.i.1.1 1 420.359 even 6
3150.2.g.j.2899.1 2 420.23 odd 12
3150.2.g.j.2899.2 2 420.107 odd 12
4032.2.a.r.1.1 1 168.149 odd 6
4032.2.a.w.1.1 1 168.107 even 6
4046.2.a.f.1.1 1 476.135 odd 6
5054.2.a.c.1.1 1 532.303 even 6
7056.2.a.bd.1.1 1 21.5 even 6
7406.2.a.a.1.1 1 644.275 even 6