Properties

Label 1216.2.n.a.255.1
Level $1216$
Weight $2$
Character 1216.255
Analytic conductor $9.710$
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] = [1216,2,Mod(255,1216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1216, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("1216.255");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1216.n (of order \(6\), 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: \(9.70980888579\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 304)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 255.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1216.255
Dual form 1216.2.n.a.639.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{5} +3.46410i q^{7} +(1.00000 + 1.73205i) q^{9} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{3} +(1.50000 - 2.59808i) q^{5} +3.46410i q^{7} +(1.00000 + 1.73205i) q^{9} +3.46410i q^{11} +(4.50000 - 2.59808i) q^{13} +(1.50000 + 2.59808i) q^{15} +(-1.50000 + 2.59808i) q^{17} +(-4.00000 - 1.73205i) q^{19} +(-3.00000 - 1.73205i) q^{21} +(4.50000 - 2.59808i) q^{23} +(-2.00000 - 3.46410i) q^{25} -5.00000 q^{27} +(-7.50000 + 4.33013i) q^{29} -4.00000 q^{31} +(-3.00000 - 1.73205i) q^{33} +(9.00000 + 5.19615i) q^{35} +5.19615i q^{39} +(7.50000 + 4.33013i) q^{41} +(10.5000 + 6.06218i) q^{43} +6.00000 q^{45} +(-1.50000 + 0.866025i) q^{47} -5.00000 q^{49} +(-1.50000 - 2.59808i) q^{51} +(-1.50000 + 0.866025i) q^{53} +(9.00000 + 5.19615i) q^{55} +(3.50000 - 2.59808i) q^{57} +(1.50000 - 2.59808i) q^{59} +(3.50000 + 6.06218i) q^{61} +(-6.00000 + 3.46410i) q^{63} -15.5885i q^{65} +(2.50000 + 4.33013i) q^{67} +5.19615i q^{69} +(4.50000 - 7.79423i) q^{71} +(-3.50000 + 6.06218i) q^{73} +4.00000 q^{75} -12.0000 q^{77} +(-3.50000 + 6.06218i) q^{79} +(-0.500000 + 0.866025i) q^{81} -3.46410i q^{83} +(4.50000 + 7.79423i) q^{85} -8.66025i q^{87} +(7.50000 - 4.33013i) q^{89} +(9.00000 + 15.5885i) q^{91} +(2.00000 - 3.46410i) q^{93} +(-10.5000 + 7.79423i) q^{95} +(7.50000 + 4.33013i) q^{97} +(-6.00000 + 3.46410i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 3 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 3 q^{5} + 2 q^{9} + 9 q^{13} + 3 q^{15} - 3 q^{17} - 8 q^{19} - 6 q^{21} + 9 q^{23} - 4 q^{25} - 10 q^{27} - 15 q^{29} - 8 q^{31} - 6 q^{33} + 18 q^{35} + 15 q^{41} + 21 q^{43} + 12 q^{45} - 3 q^{47} - 10 q^{49} - 3 q^{51} - 3 q^{53} + 18 q^{55} + 7 q^{57} + 3 q^{59} + 7 q^{61} - 12 q^{63} + 5 q^{67} + 9 q^{71} - 7 q^{73} + 8 q^{75} - 24 q^{77} - 7 q^{79} - q^{81} + 9 q^{85} + 15 q^{89} + 18 q^{91} + 4 q^{93} - 21 q^{95} + 15 q^{97} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(705\) \(837\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 + 0.866025i −0.288675 + 0.500000i −0.973494 0.228714i \(-0.926548\pi\)
0.684819 + 0.728714i \(0.259881\pi\)
\(4\) 0 0
\(5\) 1.50000 2.59808i 0.670820 1.16190i −0.306851 0.951757i \(-0.599275\pi\)
0.977672 0.210138i \(-0.0673912\pi\)
\(6\) 0 0
\(7\) 3.46410i 1.30931i 0.755929 + 0.654654i \(0.227186\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 0 0
\(9\) 1.00000 + 1.73205i 0.333333 + 0.577350i
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 4.50000 2.59808i 1.24808 0.720577i 0.277350 0.960769i \(-0.410544\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 1.50000 + 2.59808i 0.387298 + 0.670820i
\(16\) 0 0
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 0 0
\(19\) −4.00000 1.73205i −0.917663 0.397360i
\(20\) 0 0
\(21\) −3.00000 1.73205i −0.654654 0.377964i
\(22\) 0 0
\(23\) 4.50000 2.59808i 0.938315 0.541736i 0.0488832 0.998805i \(-0.484434\pi\)
0.889432 + 0.457068i \(0.151100\pi\)
\(24\) 0 0
\(25\) −2.00000 3.46410i −0.400000 0.692820i
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) −7.50000 + 4.33013i −1.39272 + 0.804084i −0.993615 0.112823i \(-0.964011\pi\)
−0.399100 + 0.916907i \(0.630677\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −3.00000 1.73205i −0.522233 0.301511i
\(34\) 0 0
\(35\) 9.00000 + 5.19615i 1.52128 + 0.878310i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 5.19615i 0.832050i
\(40\) 0 0
\(41\) 7.50000 + 4.33013i 1.17130 + 0.676252i 0.953987 0.299849i \(-0.0969363\pi\)
0.217317 + 0.976101i \(0.430270\pi\)
\(42\) 0 0
\(43\) 10.5000 + 6.06218i 1.60123 + 0.924473i 0.991241 + 0.132068i \(0.0421616\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 6.00000 0.894427
\(46\) 0 0
\(47\) −1.50000 + 0.866025i −0.218797 + 0.126323i −0.605393 0.795926i \(-0.706984\pi\)
0.386596 + 0.922249i \(0.373651\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) −1.50000 2.59808i −0.210042 0.363803i
\(52\) 0 0
\(53\) −1.50000 + 0.866025i −0.206041 + 0.118958i −0.599470 0.800397i \(-0.704622\pi\)
0.393429 + 0.919355i \(0.371289\pi\)
\(54\) 0 0
\(55\) 9.00000 + 5.19615i 1.21356 + 0.700649i
\(56\) 0 0
\(57\) 3.50000 2.59808i 0.463586 0.344124i
\(58\) 0 0
\(59\) 1.50000 2.59808i 0.195283 0.338241i −0.751710 0.659494i \(-0.770771\pi\)
0.946993 + 0.321253i \(0.104104\pi\)
\(60\) 0 0
\(61\) 3.50000 + 6.06218i 0.448129 + 0.776182i 0.998264 0.0588933i \(-0.0187572\pi\)
−0.550135 + 0.835076i \(0.685424\pi\)
\(62\) 0 0
\(63\) −6.00000 + 3.46410i −0.755929 + 0.436436i
\(64\) 0 0
\(65\) 15.5885i 1.93351i
\(66\) 0 0
\(67\) 2.50000 + 4.33013i 0.305424 + 0.529009i 0.977356 0.211604i \(-0.0678686\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0 0
\(69\) 5.19615i 0.625543i
\(70\) 0 0
\(71\) 4.50000 7.79423i 0.534052 0.925005i −0.465157 0.885228i \(-0.654002\pi\)
0.999209 0.0397765i \(-0.0126646\pi\)
\(72\) 0 0
\(73\) −3.50000 + 6.06218i −0.409644 + 0.709524i −0.994850 0.101361i \(-0.967680\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 4.00000 0.461880
\(76\) 0 0
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −3.50000 + 6.06218i −0.393781 + 0.682048i −0.992945 0.118578i \(-0.962166\pi\)
0.599164 + 0.800626i \(0.295500\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 3.46410i 0.380235i −0.981761 0.190117i \(-0.939113\pi\)
0.981761 0.190117i \(-0.0608868\pi\)
\(84\) 0 0
\(85\) 4.50000 + 7.79423i 0.488094 + 0.845403i
\(86\) 0 0
\(87\) 8.66025i 0.928477i
\(88\) 0 0
\(89\) 7.50000 4.33013i 0.794998 0.458993i −0.0467209 0.998908i \(-0.514877\pi\)
0.841719 + 0.539915i \(0.181544\pi\)
\(90\) 0 0
\(91\) 9.00000 + 15.5885i 0.943456 + 1.63411i
\(92\) 0 0
\(93\) 2.00000 3.46410i 0.207390 0.359211i
\(94\) 0 0
\(95\) −10.5000 + 7.79423i −1.07728 + 0.799671i
\(96\) 0 0
\(97\) 7.50000 + 4.33013i 0.761510 + 0.439658i 0.829837 0.558005i \(-0.188433\pi\)
−0.0683279 + 0.997663i \(0.521766\pi\)
\(98\) 0 0
\(99\) −6.00000 + 3.46410i −0.603023 + 0.348155i
\(100\) 0 0
\(101\) 1.50000 + 2.59808i 0.149256 + 0.258518i 0.930953 0.365140i \(-0.118979\pi\)
−0.781697 + 0.623658i \(0.785646\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) −9.00000 + 5.19615i −0.878310 + 0.507093i
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −13.5000 7.79423i −1.29307 0.746552i −0.313869 0.949466i \(-0.601625\pi\)
−0.979196 + 0.202915i \(0.934959\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 6.92820i 0.651751i −0.945413 0.325875i \(-0.894341\pi\)
0.945413 0.325875i \(-0.105659\pi\)
\(114\) 0 0
\(115\) 15.5885i 1.45363i
\(116\) 0 0
\(117\) 9.00000 + 5.19615i 0.832050 + 0.480384i
\(118\) 0 0
\(119\) −9.00000 5.19615i −0.825029 0.476331i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) −7.50000 + 4.33013i −0.676252 + 0.390434i
\(124\) 0 0
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 3.50000 + 6.06218i 0.310575 + 0.537931i 0.978487 0.206309i \(-0.0661452\pi\)
−0.667912 + 0.744240i \(0.732812\pi\)
\(128\) 0 0
\(129\) −10.5000 + 6.06218i −0.924473 + 0.533745i
\(130\) 0 0
\(131\) 4.50000 + 2.59808i 0.393167 + 0.226995i 0.683531 0.729921i \(-0.260443\pi\)
−0.290365 + 0.956916i \(0.593777\pi\)
\(132\) 0 0
\(133\) 6.00000 13.8564i 0.520266 1.20150i
\(134\) 0 0
\(135\) −7.50000 + 12.9904i −0.645497 + 1.11803i
\(136\) 0 0
\(137\) −1.50000 2.59808i −0.128154 0.221969i 0.794808 0.606861i \(-0.207572\pi\)
−0.922961 + 0.384893i \(0.874238\pi\)
\(138\) 0 0
\(139\) −16.5000 + 9.52628i −1.39951 + 0.808008i −0.994341 0.106233i \(-0.966121\pi\)
−0.405170 + 0.914241i \(0.632788\pi\)
\(140\) 0 0
\(141\) 1.73205i 0.145865i
\(142\) 0 0
\(143\) 9.00000 + 15.5885i 0.752618 + 1.30357i
\(144\) 0 0
\(145\) 25.9808i 2.15758i
\(146\) 0 0
\(147\) 2.50000 4.33013i 0.206197 0.357143i
\(148\) 0 0
\(149\) 7.50000 12.9904i 0.614424 1.06421i −0.376061 0.926595i \(-0.622722\pi\)
0.990485 0.137619i \(-0.0439449\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −6.00000 + 10.3923i −0.481932 + 0.834730i
\(156\) 0 0
\(157\) −2.50000 + 4.33013i −0.199522 + 0.345582i −0.948373 0.317156i \(-0.897272\pi\)
0.748852 + 0.662738i \(0.230606\pi\)
\(158\) 0 0
\(159\) 1.73205i 0.137361i
\(160\) 0 0
\(161\) 9.00000 + 15.5885i 0.709299 + 1.22854i
\(162\) 0 0
\(163\) 10.3923i 0.813988i −0.913431 0.406994i \(-0.866577\pi\)
0.913431 0.406994i \(-0.133423\pi\)
\(164\) 0 0
\(165\) −9.00000 + 5.19615i −0.700649 + 0.404520i
\(166\) 0 0
\(167\) −10.5000 18.1865i −0.812514 1.40732i −0.911099 0.412188i \(-0.864765\pi\)
0.0985846 0.995129i \(-0.468568\pi\)
\(168\) 0 0
\(169\) 7.00000 12.1244i 0.538462 0.932643i
\(170\) 0 0
\(171\) −1.00000 8.66025i −0.0764719 0.662266i
\(172\) 0 0
\(173\) −7.50000 4.33013i −0.570214 0.329213i 0.187021 0.982356i \(-0.440117\pi\)
−0.757235 + 0.653143i \(0.773450\pi\)
\(174\) 0 0
\(175\) 12.0000 6.92820i 0.907115 0.523723i
\(176\) 0 0
\(177\) 1.50000 + 2.59808i 0.112747 + 0.195283i
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −1.50000 + 0.866025i −0.111494 + 0.0643712i −0.554710 0.832044i \(-0.687171\pi\)
0.443216 + 0.896415i \(0.353837\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −9.00000 5.19615i −0.658145 0.379980i
\(188\) 0 0
\(189\) 17.3205i 1.25988i
\(190\) 0 0
\(191\) 24.2487i 1.75458i −0.479965 0.877288i \(-0.659351\pi\)
0.479965 0.877288i \(-0.340649\pi\)
\(192\) 0 0
\(193\) −10.5000 6.06218i −0.755807 0.436365i 0.0719816 0.997406i \(-0.477068\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 0 0
\(195\) 13.5000 + 7.79423i 0.966755 + 0.558156i
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) 10.5000 6.06218i 0.744325 0.429736i −0.0793146 0.996850i \(-0.525273\pi\)
0.823640 + 0.567113i \(0.191940\pi\)
\(200\) 0 0
\(201\) −5.00000 −0.352673
\(202\) 0 0
\(203\) −15.0000 25.9808i −1.05279 1.82349i
\(204\) 0 0
\(205\) 22.5000 12.9904i 1.57147 0.907288i
\(206\) 0 0
\(207\) 9.00000 + 5.19615i 0.625543 + 0.361158i
\(208\) 0 0
\(209\) 6.00000 13.8564i 0.415029 0.958468i
\(210\) 0 0
\(211\) 11.5000 19.9186i 0.791693 1.37125i −0.133226 0.991086i \(-0.542533\pi\)
0.924918 0.380166i \(-0.124133\pi\)
\(212\) 0 0
\(213\) 4.50000 + 7.79423i 0.308335 + 0.534052i
\(214\) 0 0
\(215\) 31.5000 18.1865i 2.14828 1.24031i
\(216\) 0 0
\(217\) 13.8564i 0.940634i
\(218\) 0 0
\(219\) −3.50000 6.06218i −0.236508 0.409644i
\(220\) 0 0
\(221\) 15.5885i 1.04859i
\(222\) 0 0
\(223\) −9.50000 + 16.4545i −0.636167 + 1.10187i 0.350100 + 0.936713i \(0.386148\pi\)
−0.986267 + 0.165161i \(0.947186\pi\)
\(224\) 0 0
\(225\) 4.00000 6.92820i 0.266667 0.461880i
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 6.00000 10.3923i 0.394771 0.683763i
\(232\) 0 0
\(233\) −13.5000 + 23.3827i −0.884414 + 1.53185i −0.0380310 + 0.999277i \(0.512109\pi\)
−0.846383 + 0.532574i \(0.821225\pi\)
\(234\) 0 0
\(235\) 5.19615i 0.338960i
\(236\) 0 0
\(237\) −3.50000 6.06218i −0.227349 0.393781i
\(238\) 0 0
\(239\) 17.3205i 1.12037i −0.828367 0.560185i \(-0.810730\pi\)
0.828367 0.560185i \(-0.189270\pi\)
\(240\) 0 0
\(241\) 19.5000 11.2583i 1.25611 0.725213i 0.283790 0.958886i \(-0.408408\pi\)
0.972315 + 0.233674i \(0.0750747\pi\)
\(242\) 0 0
\(243\) −8.00000 13.8564i −0.513200 0.888889i
\(244\) 0 0
\(245\) −7.50000 + 12.9904i −0.479157 + 0.829925i
\(246\) 0 0
\(247\) −22.5000 + 2.59808i −1.43164 + 0.165312i
\(248\) 0 0
\(249\) 3.00000 + 1.73205i 0.190117 + 0.109764i
\(250\) 0 0
\(251\) −4.50000 + 2.59808i −0.284037 + 0.163989i −0.635250 0.772307i \(-0.719103\pi\)
0.351212 + 0.936296i \(0.385770\pi\)
\(252\) 0 0
\(253\) 9.00000 + 15.5885i 0.565825 + 0.980038i
\(254\) 0 0
\(255\) −9.00000 −0.563602
\(256\) 0 0
\(257\) 13.5000 7.79423i 0.842107 0.486191i −0.0158730 0.999874i \(-0.505053\pi\)
0.857980 + 0.513683i \(0.171719\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −15.0000 8.66025i −0.928477 0.536056i
\(262\) 0 0
\(263\) −10.5000 6.06218i −0.647458 0.373810i 0.140024 0.990148i \(-0.455282\pi\)
−0.787482 + 0.616338i \(0.788615\pi\)
\(264\) 0 0
\(265\) 5.19615i 0.319197i
\(266\) 0 0
\(267\) 8.66025i 0.529999i
\(268\) 0 0
\(269\) 22.5000 + 12.9904i 1.37185 + 0.792038i 0.991161 0.132666i \(-0.0423537\pi\)
0.380688 + 0.924703i \(0.375687\pi\)
\(270\) 0 0
\(271\) −10.5000 6.06218i −0.637830 0.368251i 0.145948 0.989292i \(-0.453377\pi\)
−0.783778 + 0.621041i \(0.786710\pi\)
\(272\) 0 0
\(273\) −18.0000 −1.08941
\(274\) 0 0
\(275\) 12.0000 6.92820i 0.723627 0.417786i
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 0 0
\(279\) −4.00000 6.92820i −0.239474 0.414781i
\(280\) 0 0
\(281\) 13.5000 7.79423i 0.805342 0.464965i −0.0399934 0.999200i \(-0.512734\pi\)
0.845336 + 0.534235i \(0.179400\pi\)
\(282\) 0 0
\(283\) 4.50000 + 2.59808i 0.267497 + 0.154440i 0.627750 0.778415i \(-0.283976\pi\)
−0.360252 + 0.932855i \(0.617309\pi\)
\(284\) 0 0
\(285\) −1.50000 12.9904i −0.0888523 0.769484i
\(286\) 0 0
\(287\) −15.0000 + 25.9808i −0.885422 + 1.53360i
\(288\) 0 0
\(289\) 4.00000 + 6.92820i 0.235294 + 0.407541i
\(290\) 0 0
\(291\) −7.50000 + 4.33013i −0.439658 + 0.253837i
\(292\) 0 0
\(293\) 13.8564i 0.809500i −0.914427 0.404750i \(-0.867359\pi\)
0.914427 0.404750i \(-0.132641\pi\)
\(294\) 0 0
\(295\) −4.50000 7.79423i −0.262000 0.453798i
\(296\) 0 0
\(297\) 17.3205i 1.00504i
\(298\) 0 0
\(299\) 13.5000 23.3827i 0.780725 1.35226i
\(300\) 0 0
\(301\) −21.0000 + 36.3731i −1.21042 + 2.09651i
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) 0 0
\(305\) 21.0000 1.20246
\(306\) 0 0
\(307\) −2.50000 + 4.33013i −0.142683 + 0.247133i −0.928506 0.371318i \(-0.878906\pi\)
0.785823 + 0.618451i \(0.212239\pi\)
\(308\) 0 0
\(309\) −8.00000 + 13.8564i −0.455104 + 0.788263i
\(310\) 0 0
\(311\) 3.46410i 0.196431i −0.995165 0.0982156i \(-0.968687\pi\)
0.995165 0.0982156i \(-0.0313135\pi\)
\(312\) 0 0
\(313\) −9.50000 16.4545i −0.536972 0.930062i −0.999065 0.0432311i \(-0.986235\pi\)
0.462093 0.886831i \(-0.347098\pi\)
\(314\) 0 0
\(315\) 20.7846i 1.17108i
\(316\) 0 0
\(317\) 4.50000 2.59808i 0.252745 0.145922i −0.368275 0.929717i \(-0.620052\pi\)
0.621021 + 0.783794i \(0.286718\pi\)
\(318\) 0 0
\(319\) −15.0000 25.9808i −0.839839 1.45464i
\(320\) 0 0
\(321\) 6.00000 10.3923i 0.334887 0.580042i
\(322\) 0 0
\(323\) 10.5000 7.79423i 0.584236 0.433682i
\(324\) 0 0
\(325\) −18.0000 10.3923i −0.998460 0.576461i
\(326\) 0 0
\(327\) 13.5000 7.79423i 0.746552 0.431022i
\(328\) 0 0
\(329\) −3.00000 5.19615i −0.165395 0.286473i
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 15.0000 0.819538
\(336\) 0 0
\(337\) −28.5000 16.4545i −1.55249 0.896333i −0.997938 0.0641864i \(-0.979555\pi\)
−0.554556 0.832146i \(-0.687112\pi\)
\(338\) 0 0
\(339\) 6.00000 + 3.46410i 0.325875 + 0.188144i
\(340\) 0 0
\(341\) 13.8564i 0.750366i
\(342\) 0 0
\(343\) 6.92820i 0.374088i
\(344\) 0 0
\(345\) 13.5000 + 7.79423i 0.726816 + 0.419627i
\(346\) 0 0
\(347\) −13.5000 7.79423i −0.724718 0.418416i 0.0917687 0.995780i \(-0.470748\pi\)
−0.816487 + 0.577364i \(0.804081\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −22.5000 + 12.9904i −1.20096 + 0.693375i
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −13.5000 23.3827i −0.716506 1.24102i
\(356\) 0 0
\(357\) 9.00000 5.19615i 0.476331 0.275010i
\(358\) 0 0
\(359\) 13.5000 + 7.79423i 0.712503 + 0.411364i 0.811987 0.583675i \(-0.198386\pi\)
−0.0994843 + 0.995039i \(0.531719\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 0 0
\(363\) 0.500000 0.866025i 0.0262432 0.0454545i
\(364\) 0 0
\(365\) 10.5000 + 18.1865i 0.549595 + 0.951927i
\(366\) 0 0
\(367\) −13.5000 + 7.79423i −0.704694 + 0.406855i −0.809093 0.587680i \(-0.800041\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 0 0
\(369\) 17.3205i 0.901670i
\(370\) 0 0
\(371\) −3.00000 5.19615i −0.155752 0.269771i
\(372\) 0 0
\(373\) 6.92820i 0.358729i 0.983783 + 0.179364i \(0.0574041\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) −1.50000 + 2.59808i −0.0774597 + 0.134164i
\(376\) 0 0
\(377\) −22.5000 + 38.9711i −1.15881 + 2.00712i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 0 0
\(383\) 16.5000 28.5788i 0.843111 1.46031i −0.0441413 0.999025i \(-0.514055\pi\)
0.887252 0.461285i \(-0.152611\pi\)
\(384\) 0 0
\(385\) −18.0000 + 31.1769i −0.917365 + 1.58892i
\(386\) 0 0
\(387\) 24.2487i 1.23263i
\(388\) 0 0
\(389\) 7.50000 + 12.9904i 0.380265 + 0.658638i 0.991100 0.133120i \(-0.0424994\pi\)
−0.610835 + 0.791758i \(0.709166\pi\)
\(390\) 0 0
\(391\) 15.5885i 0.788342i
\(392\) 0 0
\(393\) −4.50000 + 2.59808i −0.226995 + 0.131056i
\(394\) 0 0
\(395\) 10.5000 + 18.1865i 0.528312 + 0.915064i
\(396\) 0 0
\(397\) 5.50000 9.52628i 0.276037 0.478110i −0.694359 0.719629i \(-0.744312\pi\)
0.970396 + 0.241518i \(0.0776454\pi\)
\(398\) 0 0
\(399\) 9.00000 + 12.1244i 0.450564 + 0.606977i
\(400\) 0 0
\(401\) −4.50000 2.59808i −0.224719 0.129742i 0.383414 0.923576i \(-0.374748\pi\)
−0.608134 + 0.793835i \(0.708081\pi\)
\(402\) 0 0
\(403\) −18.0000 + 10.3923i −0.896644 + 0.517678i
\(404\) 0 0
\(405\) 1.50000 + 2.59808i 0.0745356 + 0.129099i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.50000 0.866025i 0.0741702 0.0428222i −0.462456 0.886642i \(-0.653032\pi\)
0.536626 + 0.843820i \(0.319698\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0 0
\(413\) 9.00000 + 5.19615i 0.442861 + 0.255686i
\(414\) 0 0
\(415\) −9.00000 5.19615i −0.441793 0.255069i
\(416\) 0 0
\(417\) 19.0526i 0.933008i
\(418\) 0 0
\(419\) 24.2487i 1.18463i −0.805708 0.592314i \(-0.798215\pi\)
0.805708 0.592314i \(-0.201785\pi\)
\(420\) 0 0
\(421\) −19.5000 11.2583i −0.950372 0.548697i −0.0571754 0.998364i \(-0.518209\pi\)
−0.893196 + 0.449667i \(0.851543\pi\)
\(422\) 0 0
\(423\) −3.00000 1.73205i −0.145865 0.0842152i
\(424\) 0 0
\(425\) 12.0000 0.582086
\(426\) 0 0
\(427\) −21.0000 + 12.1244i −1.01626 + 0.586739i
\(428\) 0 0
\(429\) −18.0000 −0.869048
\(430\) 0 0
\(431\) 7.50000 + 12.9904i 0.361262 + 0.625725i 0.988169 0.153370i \(-0.0490126\pi\)
−0.626907 + 0.779094i \(0.715679\pi\)
\(432\) 0 0
\(433\) 1.50000 0.866025i 0.0720854 0.0416185i −0.463524 0.886084i \(-0.653415\pi\)
0.535609 + 0.844466i \(0.320082\pi\)
\(434\) 0 0
\(435\) −22.5000 12.9904i −1.07879 0.622841i
\(436\) 0 0
\(437\) −22.5000 + 2.59808i −1.07632 + 0.124283i
\(438\) 0 0
\(439\) 8.50000 14.7224i 0.405683 0.702663i −0.588718 0.808339i \(-0.700367\pi\)
0.994401 + 0.105675i \(0.0337004\pi\)
\(440\) 0 0
\(441\) −5.00000 8.66025i −0.238095 0.412393i
\(442\) 0 0
\(443\) 19.5000 11.2583i 0.926473 0.534899i 0.0407786 0.999168i \(-0.487016\pi\)
0.885694 + 0.464269i \(0.153683\pi\)
\(444\) 0 0
\(445\) 25.9808i 1.23161i
\(446\) 0 0
\(447\) 7.50000 + 12.9904i 0.354738 + 0.614424i
\(448\) 0 0
\(449\) 6.92820i 0.326962i 0.986546 + 0.163481i \(0.0522723\pi\)
−0.986546 + 0.163481i \(0.947728\pi\)
\(450\) 0 0
\(451\) −15.0000 + 25.9808i −0.706322 + 1.22339i
\(452\) 0 0
\(453\) 4.00000 6.92820i 0.187936 0.325515i
\(454\) 0 0
\(455\) 54.0000 2.53156
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 7.50000 12.9904i 0.350070 0.606339i
\(460\) 0 0
\(461\) 7.50000 12.9904i 0.349310 0.605022i −0.636817 0.771015i \(-0.719749\pi\)
0.986127 + 0.165992i \(0.0530827\pi\)
\(462\) 0 0
\(463\) 10.3923i 0.482971i 0.970404 + 0.241486i \(0.0776347\pi\)
−0.970404 + 0.241486i \(0.922365\pi\)
\(464\) 0 0
\(465\) −6.00000 10.3923i −0.278243 0.481932i
\(466\) 0 0
\(467\) 17.3205i 0.801498i 0.916188 + 0.400749i \(0.131250\pi\)
−0.916188 + 0.400749i \(0.868750\pi\)
\(468\) 0 0
\(469\) −15.0000 + 8.66025i −0.692636 + 0.399893i
\(470\) 0 0
\(471\) −2.50000 4.33013i −0.115194 0.199522i
\(472\) 0 0
\(473\) −21.0000 + 36.3731i −0.965581 + 1.67244i
\(474\) 0 0
\(475\) 2.00000 + 17.3205i 0.0917663 + 0.794719i
\(476\) 0 0
\(477\) −3.00000 1.73205i −0.137361 0.0793052i
\(478\) 0 0
\(479\) −13.5000 + 7.79423i −0.616831 + 0.356127i −0.775634 0.631183i \(-0.782570\pi\)
0.158803 + 0.987310i \(0.449236\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −18.0000 −0.819028
\(484\) 0 0
\(485\) 22.5000 12.9904i 1.02167 0.589863i
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) 9.00000 + 5.19615i 0.406994 + 0.234978i
\(490\) 0 0
\(491\) 16.5000 + 9.52628i 0.744635 + 0.429915i 0.823752 0.566950i \(-0.191877\pi\)
−0.0791174 + 0.996865i \(0.525210\pi\)
\(492\) 0 0
\(493\) 25.9808i 1.17011i
\(494\) 0 0
\(495\) 20.7846i 0.934199i
\(496\) 0 0
\(497\) 27.0000 + 15.5885i 1.21112 + 0.699238i
\(498\) 0 0
\(499\) 10.5000 + 6.06218i 0.470045 + 0.271380i 0.716258 0.697835i \(-0.245853\pi\)
−0.246214 + 0.969216i \(0.579187\pi\)
\(500\) 0 0
\(501\) 21.0000 0.938211
\(502\) 0 0
\(503\) −7.50000 + 4.33013i −0.334408 + 0.193071i −0.657797 0.753196i \(-0.728511\pi\)
0.323388 + 0.946266i \(0.395178\pi\)
\(504\) 0 0
\(505\) 9.00000 0.400495
\(506\) 0 0
\(507\) 7.00000 + 12.1244i 0.310881 + 0.538462i
\(508\) 0 0
\(509\) 16.5000 9.52628i 0.731350 0.422245i −0.0875661 0.996159i \(-0.527909\pi\)
0.818916 + 0.573914i \(0.194576\pi\)
\(510\) 0 0
\(511\) −21.0000 12.1244i −0.928985 0.536350i
\(512\) 0 0
\(513\) 20.0000 + 8.66025i 0.883022 + 0.382360i
\(514\) 0 0
\(515\) 24.0000 41.5692i 1.05757 1.83176i
\(516\) 0 0
\(517\) −3.00000 5.19615i −0.131940 0.228527i
\(518\) 0 0
\(519\) 7.50000 4.33013i 0.329213 0.190071i
\(520\) 0 0
\(521\) 41.5692i 1.82118i −0.413310 0.910590i \(-0.635627\pi\)
0.413310 0.910590i \(-0.364373\pi\)
\(522\) 0 0
\(523\) 2.50000 + 4.33013i 0.109317 + 0.189343i 0.915494 0.402332i \(-0.131800\pi\)
−0.806177 + 0.591675i \(0.798467\pi\)
\(524\) 0 0
\(525\) 13.8564i 0.604743i
\(526\) 0 0
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 0 0
\(529\) 2.00000 3.46410i 0.0869565 0.150613i
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 0 0
\(533\) 45.0000 1.94917
\(534\) 0 0
\(535\) −18.0000 + 31.1769i −0.778208 + 1.34790i
\(536\) 0 0
\(537\) −6.00000 + 10.3923i −0.258919 + 0.448461i
\(538\) 0 0
\(539\) 17.3205i 0.746047i
\(540\) 0 0
\(541\) −2.50000 4.33013i −0.107483 0.186167i 0.807267 0.590187i \(-0.200946\pi\)
−0.914750 + 0.404020i \(0.867613\pi\)
\(542\) 0 0
\(543\) 1.73205i 0.0743294i
\(544\) 0 0
\(545\) −40.5000 + 23.3827i −1.73483 + 1.00160i
\(546\) 0 0
\(547\) 0.500000 + 0.866025i 0.0213785 + 0.0370286i 0.876517 0.481371i \(-0.159861\pi\)
−0.855138 + 0.518400i \(0.826528\pi\)
\(548\) 0 0
\(549\) −7.00000 + 12.1244i −0.298753 + 0.517455i
\(550\) 0 0
\(551\) 37.5000 4.33013i 1.59755 0.184470i
\(552\) 0 0
\(553\) −21.0000 12.1244i −0.893011 0.515580i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.50000 7.79423i −0.190671 0.330252i 0.754802 0.655953i \(-0.227733\pi\)
−0.945473 + 0.325701i \(0.894400\pi\)
\(558\) 0 0
\(559\) 63.0000 2.66462
\(560\) 0 0
\(561\) 9.00000 5.19615i 0.379980 0.219382i
\(562\) 0 0
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) 0 0
\(565\) −18.0000 10.3923i −0.757266 0.437208i
\(566\) 0 0
\(567\) −3.00000 1.73205i −0.125988 0.0727393i
\(568\) 0 0
\(569\) 20.7846i 0.871336i −0.900107 0.435668i \(-0.856512\pi\)
0.900107 0.435668i \(-0.143488\pi\)
\(570\) 0 0
\(571\) 10.3923i 0.434904i −0.976071 0.217452i \(-0.930225\pi\)
0.976071 0.217452i \(-0.0697746\pi\)
\(572\) 0 0
\(573\) 21.0000 + 12.1244i 0.877288 + 0.506502i
\(574\) 0 0
\(575\) −18.0000 10.3923i −0.750652 0.433389i
\(576\) 0 0
\(577\) 46.0000 1.91501 0.957503 0.288425i \(-0.0931316\pi\)
0.957503 + 0.288425i \(0.0931316\pi\)
\(578\) 0 0
\(579\) 10.5000 6.06218i 0.436365 0.251936i
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −3.00000 5.19615i −0.124247 0.215203i
\(584\) 0 0
\(585\) 27.0000 15.5885i 1.11631 0.644503i
\(586\) 0 0
\(587\) 4.50000 + 2.59808i 0.185735 + 0.107234i 0.589984 0.807415i \(-0.299134\pi\)
−0.404249 + 0.914649i \(0.632467\pi\)
\(588\) 0 0
\(589\) 16.0000 + 6.92820i 0.659269 + 0.285472i
\(590\) 0 0
\(591\) 9.00000 15.5885i 0.370211 0.641223i
\(592\) 0 0
\(593\) −1.50000 2.59808i −0.0615976 0.106690i 0.833582 0.552396i \(-0.186286\pi\)
−0.895180 + 0.445705i \(0.852953\pi\)
\(594\) 0 0
\(595\) −27.0000 + 15.5885i −1.10689 + 0.639064i
\(596\) 0 0
\(597\) 12.1244i 0.496217i
\(598\) 0 0
\(599\) −4.50000 7.79423i −0.183865 0.318464i 0.759328 0.650708i \(-0.225528\pi\)
−0.943193 + 0.332244i \(0.892194\pi\)
\(600\) 0 0
\(601\) 6.92820i 0.282607i 0.989966 + 0.141304i \(0.0451294\pi\)
−0.989966 + 0.141304i \(0.954871\pi\)
\(602\) 0 0
\(603\) −5.00000 + 8.66025i −0.203616 + 0.352673i
\(604\) 0 0
\(605\) −1.50000 + 2.59808i −0.0609837 + 0.105627i
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 0 0
\(609\) 30.0000 1.21566
\(610\) 0 0
\(611\) −4.50000 + 7.79423i −0.182051 + 0.315321i
\(612\) 0 0
\(613\) −8.50000 + 14.7224i −0.343312 + 0.594633i −0.985046 0.172294i \(-0.944882\pi\)
0.641734 + 0.766927i \(0.278215\pi\)
\(614\) 0 0
\(615\) 25.9808i 1.04765i
\(616\) 0 0
\(617\) −13.5000 23.3827i −0.543490 0.941351i −0.998700 0.0509678i \(-0.983769\pi\)
0.455211 0.890384i \(-0.349564\pi\)
\(618\) 0 0
\(619\) 31.1769i 1.25311i 0.779379 + 0.626553i \(0.215535\pi\)
−0.779379 + 0.626553i \(0.784465\pi\)
\(620\) 0 0
\(621\) −22.5000 + 12.9904i −0.902894 + 0.521286i
\(622\) 0 0
\(623\) 15.0000 + 25.9808i 0.600962 + 1.04090i
\(624\) 0 0
\(625\) 14.5000 25.1147i 0.580000 1.00459i
\(626\) 0 0
\(627\) 9.00000 + 12.1244i 0.359425 + 0.484200i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −13.5000 + 7.79423i −0.537427 + 0.310283i −0.744035 0.668140i \(-0.767091\pi\)
0.206609 + 0.978424i \(0.433757\pi\)
\(632\) 0 0
\(633\) 11.5000 + 19.9186i 0.457084 + 0.791693i
\(634\) 0 0
\(635\) 21.0000 0.833360
\(636\) 0 0
\(637\) −22.5000 + 12.9904i −0.891482 + 0.514698i
\(638\) 0 0
\(639\) 18.0000 0.712069
\(640\) 0 0
\(641\) −34.5000 19.9186i −1.36267 0.786737i −0.372690 0.927956i \(-0.621564\pi\)
−0.989978 + 0.141219i \(0.954898\pi\)
\(642\) 0 0
\(643\) −1.50000 0.866025i −0.0591542 0.0341527i 0.470131 0.882597i \(-0.344207\pi\)
−0.529285 + 0.848444i \(0.677540\pi\)
\(644\) 0 0
\(645\) 36.3731i 1.43219i
\(646\) 0 0
\(647\) 3.46410i 0.136188i 0.997679 + 0.0680939i \(0.0216918\pi\)
−0.997679 + 0.0680939i \(0.978308\pi\)
\(648\) 0 0
\(649\) 9.00000 + 5.19615i 0.353281 + 0.203967i
\(650\) 0 0
\(651\) 12.0000 + 6.92820i 0.470317 + 0.271538i
\(652\) 0 0
\(653\) −18.0000 −0.704394 −0.352197 0.935926i \(-0.614565\pi\)
−0.352197 + 0.935926i \(0.614565\pi\)
\(654\) 0 0
\(655\) 13.5000 7.79423i 0.527489 0.304546i
\(656\) 0 0
\(657\) −14.0000 −0.546192
\(658\) 0 0
\(659\) 22.5000 + 38.9711i 0.876476 + 1.51810i 0.855183 + 0.518327i \(0.173445\pi\)
0.0212930 + 0.999773i \(0.493222\pi\)
\(660\) 0 0
\(661\) −19.5000 + 11.2583i −0.758462 + 0.437898i −0.828743 0.559629i \(-0.810944\pi\)
0.0702812 + 0.997527i \(0.477610\pi\)
\(662\) 0 0
\(663\) −13.5000 7.79423i −0.524297 0.302703i
\(664\) 0 0
\(665\) −27.0000 36.3731i −1.04702 1.41049i
\(666\) 0 0
\(667\) −22.5000 + 38.9711i −0.871203 + 1.50897i
\(668\) 0 0
\(669\) −9.50000 16.4545i −0.367291 0.636167i
\(670\) 0 0
\(671\) −21.0000 + 12.1244i −0.810696 + 0.468056i
\(672\) 0 0
\(673\) 48.4974i 1.86944i −0.355387 0.934719i \(-0.615651\pi\)
0.355387 0.934719i \(-0.384349\pi\)
\(674\) 0 0
\(675\) 10.0000 + 17.3205i 0.384900 + 0.666667i
\(676\) 0 0
\(677\) 41.5692i 1.59763i −0.601574 0.798817i \(-0.705459\pi\)
0.601574 0.798817i \(-0.294541\pi\)
\(678\) 0 0
\(679\) −15.0000 + 25.9808i −0.575647 + 0.997050i
\(680\) 0 0
\(681\) −6.00000 + 10.3923i −0.229920 + 0.398234i
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −9.00000 −0.343872
\(686\) 0 0
\(687\) −11.0000 + 19.0526i −0.419676 + 0.726900i
\(688\) 0 0
\(689\) −4.50000 + 7.79423i −0.171436 + 0.296936i
\(690\) 0 0
\(691\) 10.3923i 0.395342i −0.980268 0.197671i \(-0.936662\pi\)
0.980268 0.197671i \(-0.0633378\pi\)
\(692\) 0 0
\(693\) −12.0000 20.7846i −0.455842 0.789542i
\(694\) 0 0
\(695\) 57.1577i 2.16811i
\(696\) 0 0
\(697\) −22.5000 + 12.9904i −0.852248 + 0.492046i
\(698\) 0 0
\(699\) −13.5000 23.3827i −0.510617 0.884414i
\(700\) 0 0
\(701\) −16.5000 + 28.5788i −0.623196 + 1.07941i 0.365690 + 0.930737i \(0.380833\pi\)
−0.988887 + 0.148671i \(0.952500\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −4.50000 2.59808i −0.169480 0.0978492i
\(706\) 0 0
\(707\) −9.00000 + 5.19615i −0.338480 + 0.195421i
\(708\) 0 0
\(709\) 15.5000 + 26.8468i 0.582115 + 1.00825i 0.995228 + 0.0975728i \(0.0311079\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) −14.0000 −0.525041
\(712\) 0 0
\(713\) −18.0000 + 10.3923i −0.674105 + 0.389195i
\(714\) 0 0
\(715\) 54.0000 2.01949
\(716\) 0 0
\(717\) 15.0000 + 8.66025i 0.560185 + 0.323423i
\(718\) 0 0
\(719\) 1.50000 + 0.866025i 0.0559406 + 0.0322973i 0.527709 0.849425i \(-0.323051\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(720\) 0 0
\(721\) 55.4256i 2.06416i
\(722\) 0 0
\(723\) 22.5167i 0.837404i
\(724\) 0 0
\(725\) 30.0000 + 17.3205i 1.11417 + 0.643268i
\(726\) 0 0
\(727\) 1.50000 + 0.866025i 0.0556319 + 0.0321191i 0.527558 0.849519i \(-0.323108\pi\)
−0.471926 + 0.881638i \(0.656441\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −31.5000 + 18.1865i −1.16507 + 0.672653i
\(732\) 0 0
\(733\) −10.0000 −0.369358 −0.184679 0.982799i \(-0.559125\pi\)
−0.184679 + 0.982799i \(0.559125\pi\)
\(734\) 0 0
\(735\) −7.50000 12.9904i −0.276642 0.479157i
\(736\) 0 0
\(737\) −15.0000 + 8.66025i −0.552532 + 0.319005i
\(738\) 0 0
\(739\) −1.50000 0.866025i −0.0551784 0.0318573i 0.472157 0.881514i \(-0.343476\pi\)
−0.527335 + 0.849657i \(0.676809\pi\)
\(740\) 0 0
\(741\) 9.00000 20.7846i 0.330623 0.763542i
\(742\) 0 0
\(743\) −1.50000 + 2.59808i −0.0550297 + 0.0953142i −0.892228 0.451585i \(-0.850859\pi\)
0.837198 + 0.546899i \(0.184192\pi\)
\(744\) 0 0
\(745\) −22.5000 38.9711i −0.824336 1.42779i
\(746\) 0 0
\(747\) 6.00000 3.46410i 0.219529 0.126745i
\(748\) 0 0
\(749\) 41.5692i 1.51891i
\(750\) 0 0
\(751\) 11.5000 + 19.9186i 0.419641 + 0.726839i 0.995903 0.0904254i \(-0.0288227\pi\)
−0.576262 + 0.817265i \(0.695489\pi\)
\(752\) 0 0
\(753\) 5.19615i 0.189358i
\(754\) 0 0
\(755\) −12.0000 + 20.7846i −0.436725 + 0.756429i
\(756\) 0 0
\(757\) −6.50000 + 11.2583i −0.236247 + 0.409191i −0.959634 0.281251i \(-0.909251\pi\)
0.723388 + 0.690442i \(0.242584\pi\)
\(758\) 0 0
\(759\) −18.0000 −0.653359
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 27.0000 46.7654i 0.977466 1.69302i
\(764\) 0 0
\(765\) −9.00000 + 15.5885i −0.325396 + 0.563602i
\(766\) 0 0
\(767\) 15.5885i 0.562867i
\(768\) 0 0
\(769\) 20.5000 + 35.5070i 0.739249 + 1.28042i 0.952834 + 0.303492i \(0.0981526\pi\)
−0.213585 + 0.976924i \(0.568514\pi\)
\(770\) 0 0
\(771\) 15.5885i 0.561405i
\(772\) 0 0
\(773\) −37.5000 + 21.6506i −1.34878 + 0.778719i −0.988077 0.153961i \(-0.950797\pi\)
−0.360704 + 0.932680i \(0.617464\pi\)
\(774\) 0 0
\(775\) 8.00000 + 13.8564i 0.287368 + 0.497737i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −22.5000 30.3109i −0.806146 1.08600i
\(780\) 0 0
\(781\) 27.0000 + 15.5885i 0.966136 + 0.557799i
\(782\) 0 0
\(783\) 37.5000 21.6506i 1.34014 0.773731i
\(784\) 0 0
\(785\) 7.50000 + 12.9904i 0.267686 + 0.463647i
\(786\) 0 0
\(787\) 28.0000 0.998092 0.499046 0.866575i \(-0.333684\pi\)
0.499046 + 0.866575i \(0.333684\pi\)
\(788\) 0 0
\(789\) 10.5000 6.06218i 0.373810 0.215819i
\(790\) 0 0
\(791\) 24.0000 0.853342
\(792\) 0 0
\(793\) 31.5000 + 18.1865i 1.11860 + 0.645823i
\(794\) 0 0
\(795\) −4.50000 2.59808i −0.159599 0.0921443i
\(796\) 0 0
\(797\) 13.8564i 0.490819i 0.969419 + 0.245410i \(0.0789224\pi\)
−0.969419 + 0.245410i \(0.921078\pi\)
\(798\) 0 0
\(799\) 5.19615i 0.183827i
\(800\) 0 0
\(801\) 15.0000 + 8.66025i 0.529999 + 0.305995i
\(802\) 0 0
\(803\) −21.0000 12.1244i −0.741074 0.427859i
\(804\) 0 0
\(805\) 54.0000 1.90325
\(806\) 0 0
\(807\) −22.5000 + 12.9904i −0.792038 + 0.457283i
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 6.50000 + 11.2583i 0.228246 + 0.395333i 0.957288 0.289135i \(-0.0933677\pi\)
−0.729042 + 0.684468i \(0.760034\pi\)
\(812\) 0 0
\(813\) 10.5000 6.06218i 0.368251 0.212610i
\(814\) 0 0
\(815\) −27.0000 15.5885i −0.945769 0.546040i
\(816\) 0 0
\(817\) −31.5000 42.4352i −1.10205 1.48462i
\(818\) 0 0
\(819\) −18.0000 + 31.1769i −0.628971 + 1.08941i
\(820\) 0 0
\(821\) −22.5000 38.9711i −0.785255 1.36010i −0.928846 0.370465i \(-0.879198\pi\)
0.143591 0.989637i \(-0.454135\pi\)
\(822\) 0 0
\(823\) −1.50000 + 0.866025i −0.0522867 + 0.0301877i −0.525915 0.850537i \(-0.676277\pi\)
0.473629 + 0.880725i \(0.342944\pi\)
\(824\) 0 0
\(825\) 13.8564i 0.482418i
\(826\) 0 0
\(827\) −19.5000 33.7750i −0.678081 1.17447i −0.975558 0.219742i \(-0.929478\pi\)
0.297477 0.954729i \(-0.403855\pi\)
\(828\) 0 0
\(829\) 6.92820i 0.240626i 0.992736 + 0.120313i \(0.0383899\pi\)
−0.992736 + 0.120313i \(0.961610\pi\)
\(830\) 0 0
\(831\) 1.00000 1.73205i 0.0346896 0.0600842i
\(832\) 0 0
\(833\) 7.50000 12.9904i 0.259860 0.450090i
\(834\) 0 0
\(835\) −63.0000 −2.18020
\(836\) 0 0
\(837\) 20.0000 0.691301
\(838\) 0 0
\(839\) 10.5000 18.1865i 0.362500 0.627869i −0.625871 0.779926i \(-0.715257\pi\)
0.988372 + 0.152057i \(0.0485899\pi\)
\(840\) 0 0
\(841\) 23.0000 39.8372i 0.793103 1.37370i
\(842\) 0 0
\(843\) 15.5885i 0.536895i
\(844\) 0 0
\(845\) −21.0000 36.3731i −0.722422 1.25127i
\(846\) 0 0
\(847\) 3.46410i 0.119028i
\(848\) 0 0
\(849\) −4.50000 + 2.59808i −0.154440 + 0.0891657i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 17.5000 30.3109i 0.599189 1.03783i −0.393753 0.919216i \(-0.628823\pi\)
0.992941 0.118609i \(-0.0378434\pi\)
\(854\) 0 0
\(855\) −24.0000 10.3923i −0.820783 0.355409i
\(856\) 0 0
\(857\) 37.5000 + 21.6506i 1.28098 + 0.739572i 0.977027 0.213117i \(-0.0683615\pi\)
0.303949 + 0.952688i \(0.401695\pi\)
\(858\) 0 0
\(859\) 49.5000 28.5788i 1.68892 0.975097i 0.733571 0.679613i \(-0.237852\pi\)
0.955348 0.295484i \(-0.0954809\pi\)
\(860\) 0 0
\(861\) −15.0000 25.9808i −0.511199 0.885422i
\(862\) 0 0
\(863\) −12.0000 −0.408485 −0.204242 0.978920i \(-0.565473\pi\)
−0.204242 + 0.978920i \(0.565473\pi\)
\(864\) 0 0
\(865\) −22.5000 + 12.9904i −0.765023 + 0.441686i
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) −21.0000 12.1244i −0.712376 0.411291i
\(870\) 0 0
\(871\) 22.5000 + 12.9904i 0.762383 + 0.440162i
\(872\) 0 0
\(873\) 17.3205i 0.586210i
\(874\) 0 0
\(875\) 10.3923i 0.351324i
\(876\) 0 0
\(877\) 22.5000 + 12.9904i 0.759771 + 0.438654i 0.829214 0.558932i \(-0.188789\pi\)
−0.0694425 + 0.997586i \(0.522122\pi\)
\(878\) 0 0
\(879\) 12.0000 + 6.92820i 0.404750 + 0.233682i
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −4.50000 + 2.59808i −0.151437 + 0.0874322i −0.573804 0.818993i \(-0.694533\pi\)
0.422367 + 0.906425i \(0.361199\pi\)
\(884\) 0 0
\(885\) 9.00000 0.302532
\(886\) 0 0
\(887\) 1.50000 + 2.59808i 0.0503651 + 0.0872349i 0.890109 0.455748i \(-0.150628\pi\)
−0.839744 + 0.542983i \(0.817295\pi\)
\(888\) 0 0
\(889\) −21.0000 + 12.1244i −0.704317 + 0.406638i
\(890\) 0 0
\(891\) −3.00000 1.73205i −0.100504 0.0580259i
\(892\) 0 0
\(893\) 7.50000 0.866025i 0.250978 0.0289804i
\(894\) 0 0
\(895\) 18.0000 31.1769i 0.601674 1.04213i
\(896\) 0 0
\(897\) 13.5000 + 23.3827i 0.450752 + 0.780725i
\(898\) 0 0
\(899\) 30.0000 17.3205i 1.00056 0.577671i
\(900\) 0 0
\(901\) 5.19615i 0.173109i
\(902\) 0 0
\(903\) −21.0000 36.3731i −0.698836 1.21042i
\(904\) 0 0
\(905\) 5.19615i 0.172726i
\(906\) 0 0
\(907\) −6.50000 + 11.2583i −0.215829 + 0.373827i −0.953529 0.301302i \(-0.902579\pi\)
0.737700 + 0.675129i \(0.235912\pi\)
\(908\) 0 0
\(909\) −3.00000 + 5.19615i −0.0995037 + 0.172345i
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) −10.5000 + 18.1865i −0.347119 + 0.601228i
\(916\) 0 0
\(917\) −9.00000 + 15.5885i −0.297206 + 0.514776i
\(918\) 0 0
\(919\) 38.1051i 1.25697i −0.777821 0.628486i \(-0.783675\pi\)
0.777821 0.628486i \(-0.216325\pi\)
\(920\) 0 0
\(921\) −2.50000 4.33013i −0.0823778 0.142683i
\(922\) 0 0
\(923\) 46.7654i 1.53930i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 16.0000 + 27.7128i 0.525509 + 0.910208i
\(928\) 0 0
\(929\) −19.5000 + 33.7750i −0.639774 + 1.10812i 0.345708 + 0.938342i \(0.387639\pi\)
−0.985482 + 0.169779i \(0.945695\pi\)
\(930\) 0 0
\(931\) 20.0000 + 8.66025i 0.655474 + 0.283828i
\(932\) 0 0
\(933\) 3.00000 + 1.73205i 0.0982156 + 0.0567048i
\(934\) 0 0
\(935\) −27.0000 + 15.5885i −0.882994 + 0.509797i
\(936\) 0 0
\(937\) 24.5000 + 42.4352i 0.800380 + 1.38630i 0.919366 + 0.393403i \(0.128702\pi\)
−0.118986 + 0.992896i \(0.537964\pi\)
\(938\) 0 0
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) −13.5000 + 7.79423i −0.440087 + 0.254085i −0.703635 0.710562i \(-0.748441\pi\)
0.263547 + 0.964646i \(0.415107\pi\)
\(942\) 0 0
\(943\) 45.0000 1.46540
\(944\) 0 0
\(945\) −45.0000 25.9808i −1.46385 0.845154i
\(946\) 0 0
\(947\) −25.5000 14.7224i −0.828639 0.478415i 0.0247477 0.999694i \(-0.492122\pi\)
−0.853386 + 0.521279i \(0.825455\pi\)
\(948\) 0 0
\(949\) 36.3731i 1.18072i
\(950\) 0 0
\(951\) 5.19615i 0.168497i
\(952\) 0 0
\(953\) −10.5000 6.06218i −0.340128 0.196373i 0.320200 0.947350i \(-0.396250\pi\)
−0.660329 + 0.750977i \(0.729583\pi\)
\(954\) 0 0
\(955\) −63.0000 36.3731i −2.03863 1.17700i
\(956\) 0 0
\(957\) 30.0000 0.969762
\(958\) 0 0
\(959\) 9.00000 5.19615i 0.290625 0.167793i
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −12.0000 20.7846i −0.386695 0.669775i
\(964\) 0 0
\(965\) −31.5000 + 18.1865i −1.01402 + 0.585445i
\(966\) 0 0
\(967\) 43.5000 + 25.1147i 1.39887 + 0.807635i 0.994274 0.106862i \(-0.0340803\pi\)
0.404592 + 0.914497i \(0.367414\pi\)
\(968\) 0 0
\(969\) 1.50000 + 12.9904i 0.0481869 + 0.417311i
\(970\) 0 0
\(971\) 13.5000 23.3827i 0.433236 0.750386i −0.563914 0.825833i \(-0.690705\pi\)
0.997150 + 0.0754473i \(0.0240385\pi\)
\(972\) 0 0
\(973\) −33.0000 57.1577i −1.05793 1.83239i
\(974\) 0 0
\(975\) 18.0000 10.3923i 0.576461 0.332820i
\(976\) 0 0
\(977\) 6.92820i 0.221653i 0.993840 + 0.110826i \(0.0353498\pi\)
−0.993840 + 0.110826i \(0.964650\pi\)
\(978\) 0 0
\(979\) 15.0000 + 25.9808i 0.479402 + 0.830349i
\(980\) 0 0
\(981\) 31.1769i 0.995402i
\(982\) 0 0
\(983\) 4.50000 7.79423i 0.143528 0.248597i −0.785295 0.619122i \(-0.787489\pi\)
0.928823 + 0.370525i \(0.120822\pi\)
\(984\) 0 0
\(985\) −27.0000 + 46.7654i −0.860292 + 1.49007i
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 0 0
\(989\) 63.0000 2.00328
\(990\) 0 0
\(991\) 12.5000 21.6506i 0.397076 0.687755i −0.596288 0.802771i \(-0.703358\pi\)
0.993364 + 0.115015i \(0.0366917\pi\)
\(992\) 0 0
\(993\) 14.0000 24.2487i 0.444277 0.769510i
\(994\) 0 0
\(995\) 36.3731i 1.15310i
\(996\) 0 0
\(997\) 15.5000 + 26.8468i 0.490890 + 0.850246i 0.999945 0.0104877i \(-0.00333839\pi\)
−0.509055 + 0.860734i \(0.670005\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1216.2.n.a.255.1 2
4.3 odd 2 1216.2.n.b.255.1 2
8.3 odd 2 304.2.n.a.255.1 yes 2
8.5 even 2 304.2.n.b.255.1 yes 2
19.12 odd 6 1216.2.n.b.639.1 2
24.5 odd 2 2736.2.bm.h.559.1 2
24.11 even 2 2736.2.bm.g.559.1 2
76.31 even 6 inner 1216.2.n.a.639.1 2
152.69 odd 6 304.2.n.a.31.1 2
152.107 even 6 304.2.n.b.31.1 yes 2
456.107 odd 6 2736.2.bm.h.1855.1 2
456.221 even 6 2736.2.bm.g.1855.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
304.2.n.a.31.1 2 152.69 odd 6
304.2.n.a.255.1 yes 2 8.3 odd 2
304.2.n.b.31.1 yes 2 152.107 even 6
304.2.n.b.255.1 yes 2 8.5 even 2
1216.2.n.a.255.1 2 1.1 even 1 trivial
1216.2.n.a.639.1 2 76.31 even 6 inner
1216.2.n.b.255.1 2 4.3 odd 2
1216.2.n.b.639.1 2 19.12 odd 6
2736.2.bm.g.559.1 2 24.11 even 2
2736.2.bm.g.1855.1 2 456.221 even 6
2736.2.bm.h.559.1 2 24.5 odd 2
2736.2.bm.h.1855.1 2 456.107 odd 6