Properties

Label 273.2.bw.a.11.1
Level $273$
Weight $2$
Character 273.11
Analytic conductor $2.180$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [273,2,Mod(11,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 8, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bw (of order \(12\), degree \(4\), 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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

Embedding invariants

Embedding label 11.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 273.11
Dual form 273.2.bw.a.149.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} +(1.73205 - 1.00000i) q^{4} +(-0.866025 + 2.50000i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} +(1.73205 - 1.00000i) q^{4} +(-0.866025 + 2.50000i) q^{7} +3.00000 q^{9} +(3.00000 - 1.73205i) q^{12} +(-3.46410 + 1.00000i) q^{13} +(2.00000 - 3.46410i) q^{16} +(-3.83013 - 3.83013i) q^{19} +(-1.50000 + 4.33013i) q^{21} +(-4.33013 - 2.50000i) q^{25} +5.19615 q^{27} +(1.00000 + 5.19615i) q^{28} +(-1.13397 + 0.303848i) q^{31} +(5.19615 - 3.00000i) q^{36} +(1.06218 + 3.96410i) q^{37} +(-6.00000 + 1.73205i) q^{39} +(9.00000 + 5.19615i) q^{43} +(3.46410 - 6.00000i) q^{48} +(-5.50000 - 4.33013i) q^{49} +(-5.00000 + 5.19615i) q^{52} +(-6.63397 - 6.63397i) q^{57} -15.5885 q^{61} +(-2.59808 + 7.50000i) q^{63} -8.00000i q^{64} +(-11.5622 - 11.5622i) q^{67} +(4.36603 + 16.2942i) q^{73} +(-7.50000 - 4.33013i) q^{75} +(-10.4641 - 2.80385i) q^{76} +(6.06218 + 10.5000i) q^{79} +9.00000 q^{81} +(1.73205 + 9.00000i) q^{84} +(0.500000 - 9.52628i) q^{91} +(-1.96410 + 0.526279i) q^{93} +(9.42820 - 2.52628i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} + 12 q^{12} + 8 q^{16} + 2 q^{19} - 6 q^{21} + 4 q^{28} - 8 q^{31} - 20 q^{37} - 24 q^{39} + 36 q^{43} - 22 q^{49} - 20 q^{52} - 30 q^{57} - 22 q^{67} + 14 q^{73} - 30 q^{75} - 28 q^{76} + 36 q^{81} + 2 q^{91} + 6 q^{93} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(3\) 1.73205 1.00000
\(4\) 1.73205 1.00000i 0.866025 0.500000i
\(5\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(6\) 0 0
\(7\) −0.866025 + 2.50000i −0.327327 + 0.944911i
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) 3.00000 1.73205i 0.866025 0.500000i
\(13\) −3.46410 + 1.00000i −0.960769 + 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 2.00000 3.46410i 0.500000 0.866025i
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −3.83013 3.83013i −0.878691 0.878691i 0.114708 0.993399i \(-0.463407\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) 0 0
\(21\) −1.50000 + 4.33013i −0.327327 + 0.944911i
\(22\) 0 0
\(23\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) −4.33013 2.50000i −0.866025 0.500000i
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 1.00000 + 5.19615i 0.188982 + 0.981981i
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) −1.13397 + 0.303848i −0.203668 + 0.0545726i −0.359211 0.933257i \(-0.616954\pi\)
0.155543 + 0.987829i \(0.450287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 5.19615 3.00000i 0.866025 0.500000i
\(37\) 1.06218 + 3.96410i 0.174621 + 0.651694i 0.996616 + 0.0821995i \(0.0261945\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) −6.00000 + 1.73205i −0.960769 + 0.277350i
\(40\) 0 0
\(41\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(42\) 0 0
\(43\) 9.00000 + 5.19615i 1.37249 + 0.792406i 0.991241 0.132068i \(-0.0421616\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(48\) 3.46410 6.00000i 0.500000 0.866025i
\(49\) −5.50000 4.33013i −0.785714 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) −5.00000 + 5.19615i −0.693375 + 0.720577i
\(53\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.63397 6.63397i −0.878691 0.878691i
\(58\) 0 0
\(59\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(60\) 0 0
\(61\) −15.5885 −1.99590 −0.997949 0.0640184i \(-0.979608\pi\)
−0.997949 + 0.0640184i \(0.979608\pi\)
\(62\) 0 0
\(63\) −2.59808 + 7.50000i −0.327327 + 0.944911i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −11.5622 11.5622i −1.41254 1.41254i −0.740613 0.671932i \(-0.765465\pi\)
−0.671932 0.740613i \(-0.734535\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(72\) 0 0
\(73\) 4.36603 + 16.2942i 0.511005 + 1.90710i 0.409644 + 0.912245i \(0.365653\pi\)
0.101361 + 0.994850i \(0.467680\pi\)
\(74\) 0 0
\(75\) −7.50000 4.33013i −0.866025 0.500000i
\(76\) −10.4641 2.80385i −1.20031 0.321623i
\(77\) 0 0
\(78\) 0 0
\(79\) 6.06218 + 10.5000i 0.682048 + 1.18134i 0.974355 + 0.225018i \(0.0722440\pi\)
−0.292306 + 0.956325i \(0.594423\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 1.73205 + 9.00000i 0.188982 + 0.981981i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(90\) 0 0
\(91\) 0.500000 9.52628i 0.0524142 0.998625i
\(92\) 0 0
\(93\) −1.96410 + 0.526279i −0.203668 + 0.0545726i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.42820 2.52628i 0.957289 0.256505i 0.253837 0.967247i \(-0.418307\pi\)
0.703452 + 0.710742i \(0.251641\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −10.0000 −1.00000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −3.00000 + 1.73205i −0.295599 + 0.170664i −0.640464 0.767988i \(-0.721258\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 9.00000 5.19615i 0.866025 0.500000i
\(109\) 15.5622 4.16987i 1.49059 0.399401i 0.580651 0.814152i \(-0.302798\pi\)
0.909935 + 0.414751i \(0.136131\pi\)
\(110\) 0 0
\(111\) 1.83975 + 6.86603i 0.174621 + 0.651694i
\(112\) 6.92820 + 8.00000i 0.654654 + 0.755929i
\(113\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −10.3923 + 3.00000i −0.960769 + 0.277350i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.66025 + 1.66025i −0.149095 + 0.149095i
\(125\) 0 0
\(126\) 0 0
\(127\) 17.3205 10.0000i 1.53695 0.887357i 0.537931 0.842989i \(-0.319206\pi\)
0.999015 0.0443678i \(-0.0141274\pi\)
\(128\) 0 0
\(129\) 15.5885 + 9.00000i 1.37249 + 0.792406i
\(130\) 0 0
\(131\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(132\) 0 0
\(133\) 12.8923 6.25833i 1.11790 0.542666i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(138\) 0 0
\(139\) 3.50000 6.06218i 0.296866 0.514187i −0.678551 0.734553i \(-0.737392\pi\)
0.975417 + 0.220366i \(0.0707252\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 6.00000 10.3923i 0.500000 0.866025i
\(145\) 0 0
\(146\) 0 0
\(147\) −9.52628 7.50000i −0.785714 0.618590i
\(148\) 5.80385 + 5.80385i 0.477073 + 0.477073i
\(149\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(150\) 0 0
\(151\) 3.70577 + 13.8301i 0.301571 + 1.12548i 0.935857 + 0.352381i \(0.114628\pi\)
−0.634285 + 0.773099i \(0.718706\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −8.66025 + 9.00000i −0.693375 + 0.720577i
\(157\) −12.5000 + 21.6506i −0.997609 + 1.72791i −0.438948 + 0.898513i \(0.644649\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 18.0263 18.0263i 1.41193 1.41193i 0.665771 0.746156i \(-0.268103\pi\)
0.746156 0.665771i \(-0.231897\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(168\) 0 0
\(169\) 11.0000 6.92820i 0.846154 0.532939i
\(170\) 0 0
\(171\) −11.4904 11.4904i −0.878691 0.878691i
\(172\) 20.7846 1.58481
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 10.0000 8.66025i 0.755929 0.654654i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 19.0526i 1.41617i 0.706129 + 0.708083i \(0.250440\pi\)
−0.706129 + 0.708083i \(0.749560\pi\)
\(182\) 0 0
\(183\) −27.0000 −1.99590
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −4.50000 + 12.9904i −0.327327 + 0.944911i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 13.8564i 1.00000i
\(193\) −14.8564 + 14.8564i −1.06939 + 1.06939i −0.0719816 + 0.997406i \(0.522932\pi\)
−0.997406 + 0.0719816i \(0.977068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −13.8564 2.00000i −0.989743 0.142857i
\(197\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(198\) 0 0
\(199\) 9.52628 5.50000i 0.675300 0.389885i −0.122782 0.992434i \(-0.539182\pi\)
0.798082 + 0.602549i \(0.205848\pi\)
\(200\) 0 0
\(201\) −20.0263 20.0263i −1.41254 1.41254i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −3.46410 + 14.0000i −0.240192 + 0.970725i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.866025 + 1.50000i 0.0596196 + 0.103264i 0.894295 0.447478i \(-0.147678\pi\)
−0.834675 + 0.550743i \(0.814345\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.222432 3.09808i 0.0150997 0.210311i
\(218\) 0 0
\(219\) 7.56218 + 28.2224i 0.511005 + 1.90710i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 7.02628 26.2224i 0.470514 1.75598i −0.167412 0.985887i \(-0.553541\pi\)
0.637927 0.770097i \(-0.279792\pi\)
\(224\) 0 0
\(225\) −12.9904 7.50000i −0.866025 0.500000i
\(226\) 0 0
\(227\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(228\) −18.1244 4.85641i −1.20031 0.321623i
\(229\) 25.7224 + 6.89230i 1.69979 + 0.455456i 0.972886 0.231287i \(-0.0742935\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 10.5000 + 18.1865i 0.682048 + 1.18134i
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) −28.4904 7.63397i −1.83523 0.491748i −0.836784 0.547533i \(-0.815567\pi\)
−0.998443 + 0.0557856i \(0.982234\pi\)
\(242\) 0 0
\(243\) 15.5885 1.00000
\(244\) −27.0000 + 15.5885i −1.72850 + 0.997949i
\(245\) 0 0
\(246\) 0 0
\(247\) 17.0981 + 9.43782i 1.08792 + 0.600514i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(252\) 3.00000 + 15.5885i 0.188982 + 0.981981i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(258\) 0 0
\(259\) −10.8301 0.777568i −0.672951 0.0483157i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −31.5885 8.46410i −1.92957 0.517027i
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) −6.20577 23.1603i −0.376974 1.40689i −0.850439 0.526073i \(-0.823664\pi\)
0.473466 0.880812i \(-0.343003\pi\)
\(272\) 0 0
\(273\) 0.866025 16.5000i 0.0524142 0.998625i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −28.5000 + 16.4545i −1.71240 + 0.988654i −0.781094 + 0.624413i \(0.785338\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) 0 0
\(279\) −3.40192 + 0.911543i −0.203668 + 0.0545726i
\(280\) 0 0
\(281\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(282\) 0 0
\(283\) 32.0000i 1.90220i −0.308879 0.951101i \(-0.599954\pi\)
0.308879 0.951101i \(-0.400046\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) 16.3301 4.37564i 0.957289 0.256505i
\(292\) 23.8564 + 23.8564i 1.39609 + 1.39609i
\(293\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −17.3205 −1.00000
\(301\) −20.7846 + 18.0000i −1.19800 + 1.03750i
\(302\) 0 0
\(303\) 0 0
\(304\) −20.9282 + 5.60770i −1.20031 + 0.321623i
\(305\) 0 0
\(306\) 0 0
\(307\) 18.3660 18.3660i 1.04820 1.04820i 0.0494267 0.998778i \(-0.484261\pi\)
0.998778 0.0494267i \(-0.0157394\pi\)
\(308\) 0 0
\(309\) −5.19615 + 3.00000i −0.295599 + 0.170664i
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 13.8564 + 24.0000i 0.783210 + 1.35656i 0.930062 + 0.367402i \(0.119753\pi\)
−0.146852 + 0.989158i \(0.546914\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 21.0000 + 12.1244i 1.18134 + 0.682048i
\(317\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 15.5885 9.00000i 0.866025 0.500000i
\(325\) 17.5000 + 4.33013i 0.970725 + 0.240192i
\(326\) 0 0
\(327\) 26.9545 7.22243i 1.49059 0.399401i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −24.6603 24.6603i −1.35545 1.35545i −0.879440 0.476011i \(-0.842082\pi\)
−0.476011 0.879440i \(-0.657918\pi\)
\(332\) 0 0
\(333\) 3.18653 + 11.8923i 0.174621 + 0.651694i
\(334\) 0 0
\(335\) 0 0
\(336\) 12.0000 + 13.8564i 0.654654 + 0.755929i
\(337\) 34.0000i 1.85210i 0.377403 + 0.926049i \(0.376817\pi\)
−0.377403 + 0.926049i \(0.623183\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 15.5885 10.0000i 0.841698 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) −21.7224 5.82051i −1.16278 0.311565i −0.374701 0.927146i \(-0.622255\pi\)
−0.788074 + 0.615581i \(0.788921\pi\)
\(350\) 0 0
\(351\) −18.0000 + 5.19615i −0.960769 + 0.277350i
\(352\) 0 0
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(360\) 0 0
\(361\) 10.3397i 0.544197i
\(362\) 0 0
\(363\) 19.0526i 1.00000i
\(364\) −8.66025 17.0000i −0.453921 0.891042i
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.87564 + 2.87564i −0.149095 + 0.149095i
\(373\) −36.3731 −1.88333 −0.941663 0.336557i \(-0.890737\pi\)
−0.941663 + 0.336557i \(0.890737\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −8.99038 + 33.5526i −0.461805 + 1.72348i 0.205466 + 0.978664i \(0.434129\pi\)
−0.667271 + 0.744815i \(0.732538\pi\)
\(380\) 0 0
\(381\) 30.0000 17.3205i 1.53695 0.887357i
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 27.0000 + 15.5885i 1.37249 + 0.792406i
\(388\) 13.8038 13.8038i 0.700784 0.700784i
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 27.3923 27.3923i 1.37478 1.37478i 0.521575 0.853206i \(-0.325345\pi\)
0.853206 0.521575i \(-0.174655\pi\)
\(398\) 0 0
\(399\) 22.3301 10.8397i 1.11790 0.542666i
\(400\) −17.3205 + 10.0000i −0.866025 + 0.500000i
\(401\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(402\) 0 0
\(403\) 3.62436 2.18653i 0.180542 0.108919i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.41858 + 16.4904i −0.218485 + 0.815397i 0.766426 + 0.642333i \(0.222033\pi\)
−0.984911 + 0.173064i \(0.944633\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.46410 + 6.00000i −0.170664 + 0.295599i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.06218 10.5000i 0.296866 0.514187i
\(418\) 0 0
\(419\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(420\) 0 0
\(421\) −8.68653 8.68653i −0.423356 0.423356i 0.463002 0.886357i \(-0.346772\pi\)
−0.886357 + 0.463002i \(0.846772\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 13.5000 38.9711i 0.653311 1.88595i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(432\) 10.3923 18.0000i 0.500000 0.866025i
\(433\) 30.3109 + 17.5000i 1.45665 + 0.840996i 0.998845 0.0480569i \(-0.0153029\pi\)
0.457804 + 0.889053i \(0.348636\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 22.7846 22.7846i 1.09118 1.09118i
\(437\) 0 0
\(438\) 0 0
\(439\) 7.50000 + 4.33013i 0.357955 + 0.206666i 0.668184 0.743996i \(-0.267072\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) −16.5000 12.9904i −0.785714 0.618590i
\(442\) 0 0
\(443\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(444\) 10.0526 + 10.0526i 0.477073 + 0.477073i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 20.0000 + 6.92820i 0.944911 + 0.327327i
\(449\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 6.41858 + 23.9545i 0.301571 + 1.12548i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.7224 + 5.28461i −0.922576 + 0.247204i −0.688686 0.725059i \(-0.741812\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(462\) 0 0
\(463\) −9.05256 + 9.05256i −0.420708 + 0.420708i −0.885448 0.464739i \(-0.846148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(468\) −15.0000 + 15.5885i −0.693375 + 0.720577i
\(469\) 38.9186 18.8923i 1.79709 0.872366i
\(470\) 0 0
\(471\) −21.6506 + 37.5000i −0.997609 + 1.72791i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.00962 + 26.1603i 0.321623 + 1.20031i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(480\) 0 0
\(481\) −7.64359 12.6699i −0.348518 0.577696i
\(482\) 0 0
\(483\) 0 0
\(484\) −11.0000 19.0526i −0.500000 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −10.7679 + 40.1865i −0.487942 + 1.82103i 0.0784867 + 0.996915i \(0.474991\pi\)
−0.566429 + 0.824110i \(0.691675\pi\)
\(488\) 0 0
\(489\) 31.2224 31.2224i 1.41193 1.41193i
\(490\) 0 0
\(491\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.21539 + 4.53590i −0.0545726 + 0.203668i
\(497\) 0 0
\(498\) 0 0
\(499\) −5.91154 + 22.0622i −0.264637 + 0.987639i 0.697835 + 0.716258i \(0.254147\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 19.0526 12.0000i 0.846154 0.532939i
\(508\) 20.0000 34.6410i 0.887357 1.53695i
\(509\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(510\) 0 0
\(511\) −44.5167 3.19615i −1.96930 0.141389i
\(512\) 0 0
\(513\) −19.9019 19.9019i −0.878691 0.878691i
\(514\) 0 0
\(515\) 0 0
\(516\) 36.0000 1.58481
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) −21.5000 + 37.2391i −0.940129 + 1.62835i −0.174908 + 0.984585i \(0.555963\pi\)
−0.765222 + 0.643767i \(0.777371\pi\)
\(524\) 0 0
\(525\) 17.3205 15.0000i 0.755929 0.654654i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 11.5000 + 19.9186i 0.500000 + 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) 16.0718 23.7321i 0.696801 1.02891i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −36.1506 9.68653i −1.55424 0.416457i −0.623404 0.781900i \(-0.714251\pi\)
−0.930834 + 0.365444i \(0.880917\pi\)
\(542\) 0 0
\(543\) 33.0000i 1.41617i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −0.0427569 −0.0213785 0.999771i \(-0.506805\pi\)
−0.0213785 + 0.999771i \(0.506805\pi\)
\(548\) 0 0
\(549\) −46.7654 −1.99590
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −31.5000 + 6.06218i −1.33952 + 0.257790i
\(554\) 0 0
\(555\) 0 0
\(556\) 14.0000i 0.593732i
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) −36.3731 9.00000i −1.53842 0.380659i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.79423 + 22.5000i −0.327327 + 0.944911i
\(568\) 0 0
\(569\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) −40.7032 23.5000i −1.70338 0.983444i −0.942293 0.334790i \(-0.891335\pi\)
−0.761083 0.648655i \(-0.775332\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 24.0000i 1.00000i
\(577\) −39.4545 + 10.5718i −1.64251 + 0.440110i −0.957503 0.288425i \(-0.906868\pi\)
−0.685009 + 0.728535i \(0.740202\pi\)
\(578\) 0 0
\(579\) −25.7321 + 25.7321i −1.06939 + 1.06939i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(588\) −24.0000 3.46410i −0.989743 0.142857i
\(589\) 5.50704 + 3.17949i 0.226914 + 0.131009i
\(590\) 0 0
\(591\) 0 0
\(592\) 15.8564 + 4.24871i 0.651694 + 0.174621i
\(593\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 16.5000 9.52628i 0.675300 0.389885i
\(598\) 0 0
\(599\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(600\) 0 0
\(601\) −21.6506 37.5000i −0.883148 1.52966i −0.847822 0.530281i \(-0.822086\pi\)
−0.0353259 0.999376i \(-0.511247\pi\)
\(602\) 0 0
\(603\) −34.6865 34.6865i −1.41254 1.41254i
\(604\) 20.2487 + 20.2487i 0.823908 + 0.823908i
\(605\) 0 0
\(606\) 0 0
\(607\) −29.0000 −1.17707 −0.588537 0.808470i \(-0.700296\pi\)
−0.588537 + 0.808470i \(0.700296\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 31.2942 + 31.2942i 1.26396 + 1.26396i 0.949156 + 0.314806i \(0.101939\pi\)
0.314806 + 0.949156i \(0.398061\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(618\) 0 0
\(619\) −12.8301 47.8827i −0.515686 1.92457i −0.341644 0.939829i \(-0.610984\pi\)
−0.174042 0.984738i \(-0.555683\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −6.00000 + 24.2487i −0.240192 + 0.970725i
\(625\) 12.5000 + 21.6506i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 50.0000i 1.99522i
\(629\) 0 0
\(630\) 0 0
\(631\) 3.11474 + 11.6244i 0.123996 + 0.462758i 0.999802 0.0199047i \(-0.00633628\pi\)
−0.875806 + 0.482663i \(0.839670\pi\)
\(632\) 0 0
\(633\) 1.50000 + 2.59808i 0.0596196 + 0.103264i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 23.3827 + 9.50000i 0.926456 + 0.376404i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(642\) 0 0
\(643\) −6.02628 + 1.61474i −0.237653 + 0.0636790i −0.375680 0.926750i \(-0.622591\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.385263 5.36603i 0.0150997 0.210311i
\(652\) 13.1962 49.2487i 0.516801 1.92873i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 13.0981 + 48.8827i 0.511005 + 1.90710i
\(658\) 0 0
\(659\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(660\) 0 0
\(661\) −16.7058 + 16.7058i −0.649779 + 0.649779i −0.952940 0.303160i \(-0.901958\pi\)
0.303160 + 0.952940i \(0.401958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 12.1699 45.4186i 0.470514 1.75598i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 12.0000 6.92820i 0.462566 0.267063i −0.250557 0.968102i \(-0.580614\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) 0 0
\(675\) −22.5000 12.9904i −0.866025 0.500000i
\(676\) 12.1244 23.0000i 0.466321 0.884615i
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) −1.84936 + 25.7583i −0.0709721 + 0.988514i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(684\) −31.3923 8.41154i −1.20031 0.321623i
\(685\) 0 0
\(686\) 0 0
\(687\) 44.5526 + 11.9378i 1.69979 + 0.455456i
\(688\) 36.0000 20.7846i 1.37249 0.792406i
\(689\) 0 0
\(690\) 0 0
\(691\) 30.0263 8.04552i 1.14225 0.306066i 0.362397 0.932024i \(-0.381959\pi\)
0.779857 + 0.625958i \(0.215292\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 8.66025 25.0000i 0.327327 0.944911i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 11.1147 19.2513i 0.419200 0.726076i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.15064 6.15064i 0.230992 0.230992i −0.582115 0.813107i \(-0.697775\pi\)
0.813107 + 0.582115i \(0.197775\pi\)
\(710\) 0 0
\(711\) 18.1865 + 31.5000i 0.682048 + 1.18134i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −1.73205 9.00000i −0.0645049 0.335178i
\(722\) 0 0
\(723\) −49.3468 13.2224i −1.83523 0.491748i
\(724\) 19.0526 + 33.0000i 0.708083 + 1.22644i
\(725\) 0 0
\(726\) 0 0
\(727\) 49.0000i 1.81731i 0.417548 + 0.908655i \(0.362889\pi\)
−0.417548 + 0.908655i \(0.637111\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) −46.7654 + 27.0000i −1.72850 + 0.997949i
\(733\) 8.54552 31.8923i 0.315636 1.17797i −0.607760 0.794121i \(-0.707932\pi\)
0.923396 0.383849i \(-0.125402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 38.4186 38.4186i 1.41325 1.41325i 0.680534 0.732717i \(-0.261748\pi\)
0.732717 0.680534i \(-0.238252\pi\)
\(740\) 0 0
\(741\) 29.6147 + 16.3468i 1.08792 + 0.600514i
\(742\) 0 0
\(743\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 46.5000 + 26.8468i 1.69681 + 0.979653i 0.948753 + 0.316017i \(0.102346\pi\)
0.748056 + 0.663636i \(0.230988\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 5.19615 + 27.0000i 0.188982 + 0.981981i
\(757\) −24.2487 42.0000i −0.881334 1.52652i −0.849858 0.527011i \(-0.823312\pi\)
−0.0314762 0.999505i \(-0.510021\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(762\) 0 0
\(763\) −3.05256 + 42.5167i −0.110510 + 1.53921i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −13.8564 24.0000i −0.500000 0.866025i
\(769\) 3.21281 11.9904i 0.115857 0.432384i −0.883493 0.468445i \(-0.844814\pi\)
0.999350 + 0.0360609i \(0.0114810\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.8756 + 40.5885i −0.391423 + 1.46081i
\(773\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(774\) 0 0
\(775\) 5.66987 + 1.51924i 0.203668 + 0.0545726i
\(776\) 0 0
\(777\) −18.7583 1.34679i −0.672951 0.0483157i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −26.0000 + 10.3923i −0.928571 + 0.371154i
\(785\) 0 0
\(786\) 0 0
\(787\) 53.1147 + 14.2321i 1.89334 + 0.507318i 0.998092 + 0.0617409i \(0.0196653\pi\)
0.895244 + 0.445577i \(0.147001\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 54.0000 15.5885i 1.91760 0.553562i
\(794\) 0 0
\(795\) 0 0
\(796\) 11.0000 19.0526i 0.389885 0.675300i
\(797\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −54.7128 14.6603i −1.92957 0.517027i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) −3.15064 3.15064i −0.110634 0.110634i 0.649623 0.760257i \(-0.274927\pi\)
−0.760257 + 0.649623i \(0.774927\pi\)
\(812\) 0 0
\(813\) −10.7487 40.1147i −0.376974 1.40689i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −14.5692 54.3731i −0.509712 1.90227i
\(818\) 0 0
\(819\) 1.50000 28.5788i 0.0524142 0.998625i
\(820\) 0 0
\(821\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(822\) 0 0
\(823\) −21.0000 + 12.1244i −0.732014 + 0.422628i −0.819159 0.573567i \(-0.805559\pi\)
0.0871445 + 0.996196i \(0.472226\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(828\) 0 0
\(829\) 46.0000i 1.59765i 0.601566 + 0.798823i \(0.294544\pi\)
−0.601566 + 0.798823i \(0.705456\pi\)
\(830\) 0 0
\(831\) −49.3634 + 28.5000i −1.71240 + 0.988654i
\(832\) 8.00000 + 27.7128i 0.277350 + 0.960769i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −5.89230 + 1.57884i −0.203668 + 0.0545726i
\(838\) 0 0
\(839\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(840\) 0 0
\(841\) −14.5000 + 25.1147i −0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0 0
\(844\) 3.00000 + 1.73205i 0.103264 + 0.0596196i
\(845\) 0 0
\(846\) 0 0
\(847\) 27.5000 + 9.52628i 0.944911 + 0.327327i
\(848\) 0 0
\(849\) 55.4256i 1.90220i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 40.8827 40.8827i 1.39980 1.39980i 0.599189 0.800608i \(-0.295490\pi\)
0.800608 0.599189i \(-0.204510\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(858\) 0 0
\(859\) −19.9186 34.5000i −0.679613 1.17712i −0.975097 0.221777i \(-0.928814\pi\)
0.295484 0.955348i \(-0.404519\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.7224 25.5000i 0.500000 0.866025i
\(868\) −2.71281 5.58846i −0.0920789 0.189685i
\(869\) 0 0
\(870\) 0 0
\(871\) 51.6147 + 28.4904i 1.74890 + 0.965360i
\(872\) 0 0
\(873\) 28.2846 7.57884i 0.957289 0.256505i
\(874\) 0 0
\(875\) 0 0
\(876\) 41.3205 + 41.3205i 1.39609 + 1.39609i
\(877\) 7.24871 + 7.24871i 0.244772 + 0.244772i 0.818821 0.574049i \(-0.194628\pi\)
−0.574049 + 0.818821i \(0.694628\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(882\) 0 0
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 10.0000 + 51.9615i 0.335389 + 1.74273i
\(890\) 0 0
\(891\) 0 0
\(892\) −14.0526 52.4449i −0.470514 1.75598i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −30.0000 −1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) −36.0000 + 31.1769i −1.19800 + 1.03750i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 19.0000i 0.630885i −0.948945 0.315442i \(-0.897847\pi\)
0.948945 0.315442i \(-0.102153\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −36.2487 + 9.71281i −1.20031 + 0.321623i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 51.4449 13.7846i 1.69979 0.455456i
\(917\) 0 0
\(918\) 0 0
\(919\) 29.4449 0.971296 0.485648 0.874154i \(-0.338584\pi\)
0.485648 + 0.874154i \(0.338584\pi\)
\(920\) 0 0
\(921\) 31.8109 31.8109i 1.04820 1.04820i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 5.31089 19.8205i 0.174621 0.651694i
\(926\) 0 0
\(927\) −9.00000 + 5.19615i −0.295599 + 0.170664i
\(928\) 0 0
\(929\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(930\) 0 0
\(931\) 4.48076 + 37.6506i 0.146851 + 1.23395i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −5.19615 −0.169751 −0.0848755 0.996392i \(-0.527049\pi\)
−0.0848755 + 0.996392i \(0.527049\pi\)
\(938\) 0 0
\(939\) 24.0000 + 41.5692i 0.783210 + 1.35656i
\(940\) 0 0
\(941\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(948\) 36.3731 + 21.0000i 1.18134 + 0.682048i
\(949\) −31.4186 52.0788i −1.01989 1.69055i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −25.6532 + 14.8109i −0.827523 + 0.477771i
\(962\) 0 0
\(963\) 0 0
\(964\) −56.9808 + 15.2679i −1.83523 + 0.491748i
\(965\) 0 0
\(966\) 0 0
\(967\) 2.88269 + 2.88269i 0.0927009 + 0.0927009i 0.751936 0.659236i \(-0.229120\pi\)
−0.659236 + 0.751936i \(0.729120\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 27.0000 15.5885i 0.866025 0.500000i
\(973\) 12.1244 + 14.0000i 0.388689 + 0.448819i
\(974\) 0 0
\(975\) 30.3109 + 7.50000i 0.970725 + 0.240192i
\(976\) −31.1769 + 54.0000i −0.997949 + 1.72850i
\(977\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 46.6865 12.5096i 1.49059 0.399401i
\(982\) 0 0
\(983\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 39.0526 0.751289i 1.24243 0.0239017i
\(989\) 0 0
\(990\) 0 0
\(991\) −61.0000 −1.93773 −0.968864 0.247592i \(-0.920361\pi\)
−0.968864 + 0.247592i \(0.920361\pi\)
\(992\) 0 0
\(993\) −42.7128 42.7128i −1.35545 1.35545i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −5.00000 8.66025i −0.158352 0.274273i 0.775923 0.630828i \(-0.217285\pi\)
−0.934274 + 0.356555i \(0.883951\pi\)
\(998\) 0 0
\(999\) 5.51924 + 20.5981i 0.174621 + 0.651694i
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 273.2.bw.a.11.1 yes 4
3.2 odd 2 CM 273.2.bw.a.11.1 yes 4
7.2 even 3 273.2.bv.a.128.1 yes 4
13.6 odd 12 273.2.bv.a.32.1 4
21.2 odd 6 273.2.bv.a.128.1 yes 4
39.32 even 12 273.2.bv.a.32.1 4
91.58 odd 12 inner 273.2.bw.a.149.1 yes 4
273.149 even 12 inner 273.2.bw.a.149.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.bv.a.32.1 4 13.6 odd 12
273.2.bv.a.32.1 4 39.32 even 12
273.2.bv.a.128.1 yes 4 7.2 even 3
273.2.bv.a.128.1 yes 4 21.2 odd 6
273.2.bw.a.11.1 yes 4 1.1 even 1 trivial
273.2.bw.a.11.1 yes 4 3.2 odd 2 CM
273.2.bw.a.149.1 yes 4 91.58 odd 12 inner
273.2.bw.a.149.1 yes 4 273.149 even 12 inner