Properties

Label 900.2.k.g.307.3
Level $900$
Weight $2$
Character 900.307
Analytic conductor $7.187$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [900,2,Mod(307,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("900.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.k (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 307.3
Root \(0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 900.307
Dual form 900.2.k.g.343.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-1.74238 + 1.74238i) q^{7} -2.82843 q^{8} +O(q^{10})\) \(q+(0.707107 - 1.22474i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(-1.74238 + 1.74238i) q^{7} -2.82843 q^{8} +2.00000i q^{11} +(-4.05317 + 4.05317i) q^{13} +(0.901924 + 3.36603i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(4.24264 + 4.24264i) q^{17} -7.19615 q^{19} +(2.44949 + 1.41421i) q^{22} +(0.378937 + 0.378937i) q^{23} +(2.09808 + 7.83013i) q^{26} +(4.76028 + 1.27551i) q^{28} -7.46410i q^{29} -0.267949i q^{31} +(2.82843 + 4.89898i) q^{32} +(8.19615 - 2.19615i) q^{34} +(-2.07055 - 2.07055i) q^{37} +(-5.08845 + 8.81345i) q^{38} +5.46410 q^{41} +(1.74238 + 1.74238i) q^{43} +(3.46410 - 2.00000i) q^{44} +(0.732051 - 0.196152i) q^{46} +(-9.14162 + 9.14162i) q^{47} +0.928203i q^{49} +(11.0735 + 2.96713i) q^{52} +(1.03528 - 1.03528i) q^{53} +(4.92820 - 4.92820i) q^{56} +(-9.14162 - 5.27792i) q^{58} -4.53590 q^{59} +3.00000 q^{61} +(-0.328169 - 0.189469i) q^{62} +8.00000 q^{64} +(-8.81345 + 8.81345i) q^{67} +(3.10583 - 11.5911i) q^{68} -9.46410i q^{71} +(2.82843 - 2.82843i) q^{73} +(-4.00000 + 1.07180i) q^{74} +(7.19615 + 12.4641i) q^{76} +(-3.48477 - 3.48477i) q^{77} -0.535898 q^{79} +(3.86370 - 6.69213i) q^{82} +(0.656339 + 0.656339i) q^{83} +(3.36603 - 0.901924i) q^{86} -5.65685i q^{88} +6.92820i q^{89} -14.1244i q^{91} +(0.277401 - 1.03528i) q^{92} +(4.73205 + 17.6603i) q^{94} +(4.43211 + 4.43211i) q^{97} +(1.13681 + 0.656339i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 28 q^{14} - 16 q^{16} - 16 q^{19} - 4 q^{26} + 24 q^{34} + 16 q^{41} - 8 q^{46} - 16 q^{56} - 64 q^{59} + 24 q^{61} + 64 q^{64} - 32 q^{74} + 16 q^{76} - 32 q^{79} + 20 q^{86} + 24 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 1.22474i 0.500000 0.866025i
\(3\) 0 0
\(4\) −1.00000 1.73205i −0.500000 0.866025i
\(5\) 0 0
\(6\) 0 0
\(7\) −1.74238 + 1.74238i −0.658559 + 0.658559i −0.955039 0.296480i \(-0.904187\pi\)
0.296480 + 0.955039i \(0.404187\pi\)
\(8\) −2.82843 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000i 0.603023i 0.953463 + 0.301511i \(0.0974911\pi\)
−0.953463 + 0.301511i \(0.902509\pi\)
\(12\) 0 0
\(13\) −4.05317 + 4.05317i −1.12415 + 1.12415i −0.133037 + 0.991111i \(0.542473\pi\)
−0.991111 + 0.133037i \(0.957527\pi\)
\(14\) 0.901924 + 3.36603i 0.241049 + 0.899608i
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 4.24264 + 4.24264i 1.02899 + 1.02899i 0.999567 + 0.0294245i \(0.00936746\pi\)
0.0294245 + 0.999567i \(0.490633\pi\)
\(18\) 0 0
\(19\) −7.19615 −1.65091 −0.825455 0.564467i \(-0.809082\pi\)
−0.825455 + 0.564467i \(0.809082\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.44949 + 1.41421i 0.522233 + 0.301511i
\(23\) 0.378937 + 0.378937i 0.0790139 + 0.0790139i 0.745509 0.666495i \(-0.232206\pi\)
−0.666495 + 0.745509i \(0.732206\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.09808 + 7.83013i 0.411467 + 1.53561i
\(27\) 0 0
\(28\) 4.76028 + 1.27551i 0.899608 + 0.241049i
\(29\) 7.46410i 1.38605i −0.720914 0.693024i \(-0.756278\pi\)
0.720914 0.693024i \(-0.243722\pi\)
\(30\) 0 0
\(31\) 0.267949i 0.0481251i −0.999710 0.0240625i \(-0.992340\pi\)
0.999710 0.0240625i \(-0.00766009\pi\)
\(32\) 2.82843 + 4.89898i 0.500000 + 0.866025i
\(33\) 0 0
\(34\) 8.19615 2.19615i 1.40563 0.376637i
\(35\) 0 0
\(36\) 0 0
\(37\) −2.07055 2.07055i −0.340397 0.340397i 0.516120 0.856516i \(-0.327376\pi\)
−0.856516 + 0.516120i \(0.827376\pi\)
\(38\) −5.08845 + 8.81345i −0.825455 + 1.42973i
\(39\) 0 0
\(40\) 0 0
\(41\) 5.46410 0.853349 0.426675 0.904405i \(-0.359685\pi\)
0.426675 + 0.904405i \(0.359685\pi\)
\(42\) 0 0
\(43\) 1.74238 + 1.74238i 0.265711 + 0.265711i 0.827369 0.561658i \(-0.189836\pi\)
−0.561658 + 0.827369i \(0.689836\pi\)
\(44\) 3.46410 2.00000i 0.522233 0.301511i
\(45\) 0 0
\(46\) 0.732051 0.196152i 0.107935 0.0289211i
\(47\) −9.14162 + 9.14162i −1.33344 + 1.33344i −0.431173 + 0.902269i \(0.641900\pi\)
−0.902269 + 0.431173i \(0.858100\pi\)
\(48\) 0 0
\(49\) 0.928203i 0.132600i
\(50\) 0 0
\(51\) 0 0
\(52\) 11.0735 + 2.96713i 1.53561 + 0.411467i
\(53\) 1.03528 1.03528i 0.142206 0.142206i −0.632420 0.774626i \(-0.717938\pi\)
0.774626 + 0.632420i \(0.217938\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.92820 4.92820i 0.658559 0.658559i
\(57\) 0 0
\(58\) −9.14162 5.27792i −1.20035 0.693024i
\(59\) −4.53590 −0.590524 −0.295262 0.955416i \(-0.595407\pi\)
−0.295262 + 0.955416i \(0.595407\pi\)
\(60\) 0 0
\(61\) 3.00000 0.384111 0.192055 0.981384i \(-0.438485\pi\)
0.192055 + 0.981384i \(0.438485\pi\)
\(62\) −0.328169 0.189469i −0.0416776 0.0240625i
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.81345 + 8.81345i −1.07673 + 1.07673i −0.0799342 + 0.996800i \(0.525471\pi\)
−0.996800 + 0.0799342i \(0.974529\pi\)
\(68\) 3.10583 11.5911i 0.376637 1.40563i
\(69\) 0 0
\(70\) 0 0
\(71\) 9.46410i 1.12318i −0.827415 0.561591i \(-0.810189\pi\)
0.827415 0.561591i \(-0.189811\pi\)
\(72\) 0 0
\(73\) 2.82843 2.82843i 0.331042 0.331042i −0.521940 0.852982i \(-0.674791\pi\)
0.852982 + 0.521940i \(0.174791\pi\)
\(74\) −4.00000 + 1.07180i −0.464991 + 0.124594i
\(75\) 0 0
\(76\) 7.19615 + 12.4641i 0.825455 + 1.42973i
\(77\) −3.48477 3.48477i −0.397126 0.397126i
\(78\) 0 0
\(79\) −0.535898 −0.0602933 −0.0301466 0.999545i \(-0.509597\pi\)
−0.0301466 + 0.999545i \(0.509597\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.86370 6.69213i 0.426675 0.739022i
\(83\) 0.656339 + 0.656339i 0.0720425 + 0.0720425i 0.742210 0.670167i \(-0.233778\pi\)
−0.670167 + 0.742210i \(0.733778\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.36603 0.901924i 0.362968 0.0972569i
\(87\) 0 0
\(88\) 5.65685i 0.603023i
\(89\) 6.92820i 0.734388i 0.930144 + 0.367194i \(0.119682\pi\)
−0.930144 + 0.367194i \(0.880318\pi\)
\(90\) 0 0
\(91\) 14.1244i 1.48063i
\(92\) 0.277401 1.03528i 0.0289211 0.107935i
\(93\) 0 0
\(94\) 4.73205 + 17.6603i 0.488074 + 1.82152i
\(95\) 0 0
\(96\) 0 0
\(97\) 4.43211 + 4.43211i 0.450013 + 0.450013i 0.895359 0.445346i \(-0.146919\pi\)
−0.445346 + 0.895359i \(0.646919\pi\)
\(98\) 1.13681 + 0.656339i 0.114835 + 0.0663002i
\(99\) 0 0
\(100\) 0 0
\(101\) 1.46410 0.145684 0.0728418 0.997344i \(-0.476793\pi\)
0.0728418 + 0.997344i \(0.476793\pi\)
\(102\) 0 0
\(103\) −4.89898 4.89898i −0.482711 0.482711i 0.423286 0.905996i \(-0.360877\pi\)
−0.905996 + 0.423286i \(0.860877\pi\)
\(104\) 11.4641 11.4641i 1.12415 1.12415i
\(105\) 0 0
\(106\) −0.535898 2.00000i −0.0520511 0.194257i
\(107\) 4.62158 4.62158i 0.446785 0.446785i −0.447499 0.894284i \(-0.647685\pi\)
0.894284 + 0.447499i \(0.147685\pi\)
\(108\) 0 0
\(109\) 7.00000i 0.670478i −0.942133 0.335239i \(-0.891183\pi\)
0.942133 0.335239i \(-0.108817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.55103 9.52056i −0.241049 0.899608i
\(113\) −9.79796 + 9.79796i −0.921714 + 0.921714i −0.997151 0.0754362i \(-0.975965\pi\)
0.0754362 + 0.997151i \(0.475965\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −12.9282 + 7.46410i −1.20035 + 0.693024i
\(117\) 0 0
\(118\) −3.20736 + 5.55532i −0.295262 + 0.511409i
\(119\) −14.7846 −1.35530
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 2.12132 3.67423i 0.192055 0.332650i
\(123\) 0 0
\(124\) −0.464102 + 0.267949i −0.0416776 + 0.0240625i
\(125\) 0 0
\(126\) 0 0
\(127\) −2.07055 + 2.07055i −0.183732 + 0.183732i −0.792980 0.609248i \(-0.791471\pi\)
0.609248 + 0.792980i \(0.291471\pi\)
\(128\) 5.65685 9.79796i 0.500000 0.866025i
\(129\) 0 0
\(130\) 0 0
\(131\) 9.46410i 0.826882i 0.910531 + 0.413441i \(0.135673\pi\)
−0.910531 + 0.413441i \(0.864327\pi\)
\(132\) 0 0
\(133\) 12.5385 12.5385i 1.08722 1.08722i
\(134\) 4.56218 + 17.0263i 0.394112 + 1.47085i
\(135\) 0 0
\(136\) −12.0000 12.0000i −1.02899 1.02899i
\(137\) −6.03579 6.03579i −0.515672 0.515672i 0.400586 0.916259i \(-0.368806\pi\)
−0.916259 + 0.400586i \(0.868806\pi\)
\(138\) 0 0
\(139\) −19.4641 −1.65092 −0.825462 0.564458i \(-0.809085\pi\)
−0.825462 + 0.564458i \(0.809085\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −11.5911 6.69213i −0.972704 0.561591i
\(143\) −8.10634 8.10634i −0.677887 0.677887i
\(144\) 0 0
\(145\) 0 0
\(146\) −1.46410 5.46410i −0.121170 0.452212i
\(147\) 0 0
\(148\) −1.51575 + 5.65685i −0.124594 + 0.464991i
\(149\) 15.8564i 1.29901i 0.760358 + 0.649504i \(0.225023\pi\)
−0.760358 + 0.649504i \(0.774977\pi\)
\(150\) 0 0
\(151\) 4.26795i 0.347321i −0.984806 0.173660i \(-0.944440\pi\)
0.984806 0.173660i \(-0.0555595\pi\)
\(152\) 20.3538 1.65091
\(153\) 0 0
\(154\) −6.73205 + 1.80385i −0.542484 + 0.145358i
\(155\) 0 0
\(156\) 0 0
\(157\) −6.88160 6.88160i −0.549211 0.549211i 0.377001 0.926213i \(-0.376955\pi\)
−0.926213 + 0.377001i \(0.876955\pi\)
\(158\) −0.378937 + 0.656339i −0.0301466 + 0.0522155i
\(159\) 0 0
\(160\) 0 0
\(161\) −1.32051 −0.104071
\(162\) 0 0
\(163\) 10.2277 + 10.2277i 0.801092 + 0.801092i 0.983266 0.182174i \(-0.0583134\pi\)
−0.182174 + 0.983266i \(0.558313\pi\)
\(164\) −5.46410 9.46410i −0.426675 0.739022i
\(165\) 0 0
\(166\) 1.26795 0.339746i 0.0984119 0.0263694i
\(167\) 3.10583 3.10583i 0.240336 0.240336i −0.576653 0.816989i \(-0.695642\pi\)
0.816989 + 0.576653i \(0.195642\pi\)
\(168\) 0 0
\(169\) 19.8564i 1.52742i
\(170\) 0 0
\(171\) 0 0
\(172\) 1.27551 4.76028i 0.0972569 0.362968i
\(173\) 6.31319 6.31319i 0.479983 0.479983i −0.425143 0.905126i \(-0.639776\pi\)
0.905126 + 0.425143i \(0.139776\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −6.92820 4.00000i −0.522233 0.301511i
\(177\) 0 0
\(178\) 8.48528 + 4.89898i 0.635999 + 0.367194i
\(179\) 25.3205 1.89254 0.946272 0.323372i \(-0.104817\pi\)
0.946272 + 0.323372i \(0.104817\pi\)
\(180\) 0 0
\(181\) −13.9282 −1.03528 −0.517638 0.855600i \(-0.673188\pi\)
−0.517638 + 0.855600i \(0.673188\pi\)
\(182\) −17.2987 9.98743i −1.28227 0.740317i
\(183\) 0 0
\(184\) −1.07180 1.07180i −0.0790139 0.0790139i
\(185\) 0 0
\(186\) 0 0
\(187\) −8.48528 + 8.48528i −0.620505 + 0.620505i
\(188\) 24.9754 + 6.69213i 1.82152 + 0.488074i
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9282i 1.22488i −0.790516 0.612441i \(-0.790188\pi\)
0.790516 0.612441i \(-0.209812\pi\)
\(192\) 0 0
\(193\) −9.71003 + 9.71003i −0.698943 + 0.698943i −0.964183 0.265240i \(-0.914549\pi\)
0.265240 + 0.964183i \(0.414549\pi\)
\(194\) 8.56218 2.29423i 0.614729 0.164716i
\(195\) 0 0
\(196\) 1.60770 0.928203i 0.114835 0.0663002i
\(197\) −10.1769 10.1769i −0.725074 0.725074i 0.244560 0.969634i \(-0.421356\pi\)
−0.969634 + 0.244560i \(0.921356\pi\)
\(198\) 0 0
\(199\) 14.1244 1.00125 0.500625 0.865665i \(-0.333104\pi\)
0.500625 + 0.865665i \(0.333104\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1.03528 1.79315i 0.0728418 0.126166i
\(203\) 13.0053 + 13.0053i 0.912795 + 0.912795i
\(204\) 0 0
\(205\) 0 0
\(206\) −9.46410 + 2.53590i −0.659395 + 0.176684i
\(207\) 0 0
\(208\) −5.93426 22.1469i −0.411467 1.53561i
\(209\) 14.3923i 0.995537i
\(210\) 0 0
\(211\) 7.19615i 0.495404i 0.968836 + 0.247702i \(0.0796753\pi\)
−0.968836 + 0.247702i \(0.920325\pi\)
\(212\) −2.82843 0.757875i −0.194257 0.0520511i
\(213\) 0 0
\(214\) −2.39230 8.92820i −0.163535 0.610319i
\(215\) 0 0
\(216\) 0 0
\(217\) 0.466870 + 0.466870i 0.0316932 + 0.0316932i
\(218\) −8.57321 4.94975i −0.580651 0.335239i
\(219\) 0 0
\(220\) 0 0
\(221\) −34.3923 −2.31348
\(222\) 0 0
\(223\) 8.05558 + 8.05558i 0.539441 + 0.539441i 0.923365 0.383924i \(-0.125427\pi\)
−0.383924 + 0.923365i \(0.625427\pi\)
\(224\) −13.4641 3.60770i −0.899608 0.241049i
\(225\) 0 0
\(226\) 5.07180 + 18.9282i 0.337371 + 1.25909i
\(227\) 8.76268 8.76268i 0.581600 0.581600i −0.353743 0.935343i \(-0.615091\pi\)
0.935343 + 0.353743i \(0.115091\pi\)
\(228\) 0 0
\(229\) 24.8564i 1.64256i 0.570527 + 0.821279i \(0.306739\pi\)
−0.570527 + 0.821279i \(0.693261\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 21.1117i 1.38605i
\(233\) 8.76268 8.76268i 0.574062 0.574062i −0.359199 0.933261i \(-0.616950\pi\)
0.933261 + 0.359199i \(0.116950\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.53590 + 7.85641i 0.295262 + 0.511409i
\(237\) 0 0
\(238\) −10.4543 + 18.1074i −0.677651 + 1.17373i
\(239\) −17.4641 −1.12966 −0.564829 0.825208i \(-0.691058\pi\)
−0.564829 + 0.825208i \(0.691058\pi\)
\(240\) 0 0
\(241\) −14.8564 −0.956985 −0.478493 0.878092i \(-0.658817\pi\)
−0.478493 + 0.878092i \(0.658817\pi\)
\(242\) 4.94975 8.57321i 0.318182 0.551107i
\(243\) 0 0
\(244\) −3.00000 5.19615i −0.192055 0.332650i
\(245\) 0 0
\(246\) 0 0
\(247\) 29.1672 29.1672i 1.85587 1.85587i
\(248\) 0.757875i 0.0481251i
\(249\) 0 0
\(250\) 0 0
\(251\) 2.14359i 0.135302i −0.997709 0.0676512i \(-0.978449\pi\)
0.997709 0.0676512i \(-0.0215505\pi\)
\(252\) 0 0
\(253\) −0.757875 + 0.757875i −0.0476472 + 0.0476472i
\(254\) 1.07180 + 4.00000i 0.0672505 + 0.250982i
\(255\) 0 0
\(256\) −8.00000 13.8564i −0.500000 0.866025i
\(257\) 9.04008 + 9.04008i 0.563905 + 0.563905i 0.930414 0.366509i \(-0.119447\pi\)
−0.366509 + 0.930414i \(0.619447\pi\)
\(258\) 0 0
\(259\) 7.21539 0.448343
\(260\) 0 0
\(261\) 0 0
\(262\) 11.5911 + 6.69213i 0.716101 + 0.413441i
\(263\) 15.1774 + 15.1774i 0.935879 + 0.935879i 0.998065 0.0621853i \(-0.0198070\pi\)
−0.0621853 + 0.998065i \(0.519807\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.49038 24.2224i −0.397951 1.48517i
\(267\) 0 0
\(268\) 24.0788 + 6.45189i 1.47085 + 0.394112i
\(269\) 1.60770i 0.0980229i 0.998798 + 0.0490115i \(0.0156071\pi\)
−0.998798 + 0.0490115i \(0.984393\pi\)
\(270\) 0 0
\(271\) 24.2487i 1.47300i 0.676435 + 0.736502i \(0.263524\pi\)
−0.676435 + 0.736502i \(0.736476\pi\)
\(272\) −23.1822 + 6.21166i −1.40563 + 0.376637i
\(273\) 0 0
\(274\) −11.6603 + 3.12436i −0.704422 + 0.188749i
\(275\) 0 0
\(276\) 0 0
\(277\) −0.845807 0.845807i −0.0508196 0.0508196i 0.681240 0.732060i \(-0.261441\pi\)
−0.732060 + 0.681240i \(0.761441\pi\)
\(278\) −13.7632 + 23.8386i −0.825462 + 1.42974i
\(279\) 0 0
\(280\) 0 0
\(281\) 14.3923 0.858573 0.429286 0.903168i \(-0.358765\pi\)
0.429286 + 0.903168i \(0.358765\pi\)
\(282\) 0 0
\(283\) −15.1266 15.1266i −0.899186 0.899186i 0.0961785 0.995364i \(-0.469338\pi\)
−0.995364 + 0.0961785i \(0.969338\pi\)
\(284\) −16.3923 + 9.46410i −0.972704 + 0.561591i
\(285\) 0 0
\(286\) −15.6603 + 4.19615i −0.926010 + 0.248124i
\(287\) −9.52056 + 9.52056i −0.561981 + 0.561981i
\(288\) 0 0
\(289\) 19.0000i 1.11765i
\(290\) 0 0
\(291\) 0 0
\(292\) −7.72741 2.07055i −0.452212 0.121170i
\(293\) 8.86422 8.86422i 0.517853 0.517853i −0.399068 0.916921i \(-0.630666\pi\)
0.916921 + 0.399068i \(0.130666\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 5.85641 + 5.85641i 0.340397 + 0.340397i
\(297\) 0 0
\(298\) 19.4201 + 11.2122i 1.12497 + 0.649504i
\(299\) −3.07180 −0.177647
\(300\) 0 0
\(301\) −6.07180 −0.349973
\(302\) −5.22715 3.01790i −0.300789 0.173660i
\(303\) 0 0
\(304\) 14.3923 24.9282i 0.825455 1.42973i
\(305\) 0 0
\(306\) 0 0
\(307\) 9.46979 9.46979i 0.540469 0.540469i −0.383197 0.923667i \(-0.625177\pi\)
0.923667 + 0.383197i \(0.125177\pi\)
\(308\) −2.55103 + 9.52056i −0.145358 + 0.542484i
\(309\) 0 0
\(310\) 0 0
\(311\) 27.4641i 1.55735i 0.627430 + 0.778673i \(0.284107\pi\)
−0.627430 + 0.778673i \(0.715893\pi\)
\(312\) 0 0
\(313\) −5.74479 + 5.74479i −0.324715 + 0.324715i −0.850572 0.525858i \(-0.823744\pi\)
0.525858 + 0.850572i \(0.323744\pi\)
\(314\) −13.2942 + 3.56218i −0.750237 + 0.201025i
\(315\) 0 0
\(316\) 0.535898 + 0.928203i 0.0301466 + 0.0522155i
\(317\) −4.89898 4.89898i −0.275154 0.275154i 0.556017 0.831171i \(-0.312329\pi\)
−0.831171 + 0.556017i \(0.812329\pi\)
\(318\) 0 0
\(319\) 14.9282 0.835819
\(320\) 0 0
\(321\) 0 0
\(322\) −0.933740 + 1.61729i −0.0520353 + 0.0901278i
\(323\) −30.5307 30.5307i −1.69877 1.69877i
\(324\) 0 0
\(325\) 0 0
\(326\) 19.7583 5.29423i 1.09431 0.293220i
\(327\) 0 0
\(328\) −15.4548 −0.853349
\(329\) 31.8564i 1.75630i
\(330\) 0 0
\(331\) 2.39230i 0.131493i −0.997836 0.0657465i \(-0.979057\pi\)
0.997836 0.0657465i \(-0.0209429\pi\)
\(332\) 0.480473 1.79315i 0.0263694 0.0984119i
\(333\) 0 0
\(334\) −1.60770 6.00000i −0.0879692 0.328305i
\(335\) 0 0
\(336\) 0 0
\(337\) −8.01841 8.01841i −0.436791 0.436791i 0.454140 0.890930i \(-0.349947\pi\)
−0.890930 + 0.454140i \(0.849947\pi\)
\(338\) −24.3190 14.0406i −1.32278 0.763708i
\(339\) 0 0
\(340\) 0 0
\(341\) 0.535898 0.0290205
\(342\) 0 0
\(343\) −13.8140 13.8140i −0.745884 0.745884i
\(344\) −4.92820 4.92820i −0.265711 0.265711i
\(345\) 0 0
\(346\) −3.26795 12.1962i −0.175686 0.655669i
\(347\) 7.82894 7.82894i 0.420280 0.420280i −0.465020 0.885300i \(-0.653953\pi\)
0.885300 + 0.465020i \(0.153953\pi\)
\(348\) 0 0
\(349\) 3.85641i 0.206429i −0.994659 0.103214i \(-0.967087\pi\)
0.994659 0.103214i \(-0.0329128\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −9.79796 + 5.65685i −0.522233 + 0.301511i
\(353\) −11.2122 + 11.2122i −0.596764 + 0.596764i −0.939450 0.342686i \(-0.888663\pi\)
0.342686 + 0.939450i \(0.388663\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 12.0000 6.92820i 0.635999 0.367194i
\(357\) 0 0
\(358\) 17.9043 31.0112i 0.946272 1.63899i
\(359\) 24.3923 1.28738 0.643688 0.765288i \(-0.277403\pi\)
0.643688 + 0.765288i \(0.277403\pi\)
\(360\) 0 0
\(361\) 32.7846 1.72551
\(362\) −9.84873 + 17.0585i −0.517638 + 0.896575i
\(363\) 0 0
\(364\) −24.4641 + 14.1244i −1.28227 + 0.740317i
\(365\) 0 0
\(366\) 0 0
\(367\) 7.50077 7.50077i 0.391537 0.391537i −0.483698 0.875235i \(-0.660707\pi\)
0.875235 + 0.483698i \(0.160707\pi\)
\(368\) −2.07055 + 0.554803i −0.107935 + 0.0289211i
\(369\) 0 0
\(370\) 0 0
\(371\) 3.60770i 0.187302i
\(372\) 0 0
\(373\) −13.6753 + 13.6753i −0.708078 + 0.708078i −0.966131 0.258052i \(-0.916919\pi\)
0.258052 + 0.966131i \(0.416919\pi\)
\(374\) 4.39230 + 16.3923i 0.227121 + 0.847626i
\(375\) 0 0
\(376\) 25.8564 25.8564i 1.33344 1.33344i
\(377\) 30.2533 + 30.2533i 1.55812 + 1.55812i
\(378\) 0 0
\(379\) 15.7321 0.808101 0.404051 0.914737i \(-0.367602\pi\)
0.404051 + 0.914737i \(0.367602\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −20.7327 11.9700i −1.06078 0.612441i
\(383\) 11.8685 + 11.8685i 0.606453 + 0.606453i 0.942017 0.335565i \(-0.108927\pi\)
−0.335565 + 0.942017i \(0.608927\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.02628 + 18.7583i 0.255831 + 0.954774i
\(387\) 0 0
\(388\) 3.24453 12.1087i 0.164716 0.614729i
\(389\) 14.5359i 0.736999i 0.929628 + 0.368500i \(0.120128\pi\)
−0.929628 + 0.368500i \(0.879872\pi\)
\(390\) 0 0
\(391\) 3.21539i 0.162609i
\(392\) 2.62536i 0.132600i
\(393\) 0 0
\(394\) −19.6603 + 5.26795i −0.990469 + 0.265395i
\(395\) 0 0
\(396\) 0 0
\(397\) 13.8511 + 13.8511i 0.695168 + 0.695168i 0.963364 0.268196i \(-0.0864275\pi\)
−0.268196 + 0.963364i \(0.586427\pi\)
\(398\) 9.98743 17.2987i 0.500625 0.867107i
\(399\) 0 0
\(400\) 0 0
\(401\) 33.1769 1.65678 0.828388 0.560155i \(-0.189258\pi\)
0.828388 + 0.560155i \(0.189258\pi\)
\(402\) 0 0
\(403\) 1.08604 + 1.08604i 0.0540997 + 0.0540997i
\(404\) −1.46410 2.53590i −0.0728418 0.126166i
\(405\) 0 0
\(406\) 25.1244 6.73205i 1.24690 0.334106i
\(407\) 4.14110 4.14110i 0.205267 0.205267i
\(408\) 0 0
\(409\) 27.7846i 1.37386i 0.726723 + 0.686930i \(0.241042\pi\)
−0.726723 + 0.686930i \(0.758958\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3.58630 + 13.3843i −0.176684 + 0.659395i
\(413\) 7.90327 7.90327i 0.388895 0.388895i
\(414\) 0 0
\(415\) 0 0
\(416\) −31.3205 8.39230i −1.53561 0.411467i
\(417\) 0 0
\(418\) −17.6269 10.1769i −0.862160 0.497768i
\(419\) 26.5359 1.29636 0.648182 0.761486i \(-0.275530\pi\)
0.648182 + 0.761486i \(0.275530\pi\)
\(420\) 0 0
\(421\) 6.78461 0.330662 0.165331 0.986238i \(-0.447131\pi\)
0.165331 + 0.986238i \(0.447131\pi\)
\(422\) 8.81345 + 5.08845i 0.429032 + 0.247702i
\(423\) 0 0
\(424\) −2.92820 + 2.92820i −0.142206 + 0.142206i
\(425\) 0 0
\(426\) 0 0
\(427\) −5.22715 + 5.22715i −0.252959 + 0.252959i
\(428\) −12.6264 3.38323i −0.610319 0.163535i
\(429\) 0 0
\(430\) 0 0
\(431\) 25.7128i 1.23854i −0.785177 0.619271i \(-0.787428\pi\)
0.785177 0.619271i \(-0.212572\pi\)
\(432\) 0 0
\(433\) 3.11943 3.11943i 0.149910 0.149910i −0.628168 0.778078i \(-0.716195\pi\)
0.778078 + 0.628168i \(0.216195\pi\)
\(434\) 0.901924 0.241670i 0.0432937 0.0116005i
\(435\) 0 0
\(436\) −12.1244 + 7.00000i −0.580651 + 0.335239i
\(437\) −2.72689 2.72689i −0.130445 0.130445i
\(438\) 0 0
\(439\) 27.9808 1.33545 0.667724 0.744409i \(-0.267268\pi\)
0.667724 + 0.744409i \(0.267268\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −24.3190 + 42.1218i −1.15674 + 2.00353i
\(443\) −9.04008 9.04008i −0.429507 0.429507i 0.458953 0.888460i \(-0.348225\pi\)
−0.888460 + 0.458953i \(0.848225\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 15.5622 4.16987i 0.736890 0.197449i
\(447\) 0 0
\(448\) −13.9391 + 13.9391i −0.658559 + 0.658559i
\(449\) 33.8564i 1.59778i 0.601475 + 0.798891i \(0.294580\pi\)
−0.601475 + 0.798891i \(0.705420\pi\)
\(450\) 0 0
\(451\) 10.9282i 0.514589i
\(452\) 26.7685 + 7.17260i 1.25909 + 0.337371i
\(453\) 0 0
\(454\) −4.53590 16.9282i −0.212880 0.794480i
\(455\) 0 0
\(456\) 0 0
\(457\) 25.2528 + 25.2528i 1.18127 + 1.18127i 0.979414 + 0.201861i \(0.0646988\pi\)
0.201861 + 0.979414i \(0.435301\pi\)
\(458\) 30.4428 + 17.5761i 1.42250 + 0.821279i
\(459\) 0 0
\(460\) 0 0
\(461\) −15.6077 −0.726923 −0.363461 0.931609i \(-0.618405\pi\)
−0.363461 + 0.931609i \(0.618405\pi\)
\(462\) 0 0
\(463\) 9.24316 + 9.24316i 0.429566 + 0.429566i 0.888480 0.458915i \(-0.151762\pi\)
−0.458915 + 0.888480i \(0.651762\pi\)
\(464\) 25.8564 + 14.9282i 1.20035 + 0.693024i
\(465\) 0 0
\(466\) −4.53590 16.9282i −0.210121 0.784184i
\(467\) 2.44949 2.44949i 0.113349 0.113349i −0.648157 0.761506i \(-0.724460\pi\)
0.761506 + 0.648157i \(0.224460\pi\)
\(468\) 0 0
\(469\) 30.7128i 1.41819i
\(470\) 0 0
\(471\) 0 0
\(472\) 12.8295 0.590524
\(473\) −3.48477 + 3.48477i −0.160230 + 0.160230i
\(474\) 0 0
\(475\) 0 0
\(476\) 14.7846 + 25.6077i 0.677651 + 1.17373i
\(477\) 0 0
\(478\) −12.3490 + 21.3891i −0.564829 + 0.978313i
\(479\) 13.3205 0.608630 0.304315 0.952572i \(-0.401573\pi\)
0.304315 + 0.952572i \(0.401573\pi\)
\(480\) 0 0
\(481\) 16.7846 0.765312
\(482\) −10.5051 + 18.1953i −0.478493 + 0.828774i
\(483\) 0 0
\(484\) −7.00000 12.1244i −0.318182 0.551107i
\(485\) 0 0
\(486\) 0 0
\(487\) −15.2282 + 15.2282i −0.690055 + 0.690055i −0.962244 0.272189i \(-0.912252\pi\)
0.272189 + 0.962244i \(0.412252\pi\)
\(488\) −8.48528 −0.384111
\(489\) 0 0
\(490\) 0 0
\(491\) 34.6410i 1.56333i 0.623700 + 0.781664i \(0.285629\pi\)
−0.623700 + 0.781664i \(0.714371\pi\)
\(492\) 0 0
\(493\) 31.6675 31.6675i 1.42623 1.42623i
\(494\) −15.0981 56.3468i −0.679295 2.53516i
\(495\) 0 0
\(496\) 0.928203 + 0.535898i 0.0416776 + 0.0240625i
\(497\) 16.4901 + 16.4901i 0.739682 + 0.739682i
\(498\) 0 0
\(499\) −5.87564 −0.263030 −0.131515 0.991314i \(-0.541984\pi\)
−0.131515 + 0.991314i \(0.541984\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.62536 1.51575i −0.117175 0.0676512i
\(503\) −14.4195 14.4195i −0.642935 0.642935i 0.308341 0.951276i \(-0.400226\pi\)
−0.951276 + 0.308341i \(0.900226\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0.392305 + 1.46410i 0.0174401 + 0.0650873i
\(507\) 0 0
\(508\) 5.65685 + 1.51575i 0.250982 + 0.0672505i
\(509\) 11.0718i 0.490749i −0.969428 0.245374i \(-0.921089\pi\)
0.969428 0.245374i \(-0.0789109\pi\)
\(510\) 0 0
\(511\) 9.85641i 0.436022i
\(512\) −22.6274 −1.00000
\(513\) 0 0
\(514\) 17.4641 4.67949i 0.770309 0.206404i
\(515\) 0 0
\(516\) 0 0
\(517\) −18.2832 18.2832i −0.804096 0.804096i
\(518\) 5.10205 8.83701i 0.224171 0.388276i
\(519\) 0 0
\(520\) 0 0
\(521\) 17.6077 0.771407 0.385704 0.922623i \(-0.373959\pi\)
0.385704 + 0.922623i \(0.373959\pi\)
\(522\) 0 0
\(523\) −13.7124 13.7124i −0.599603 0.599603i 0.340604 0.940207i \(-0.389368\pi\)
−0.940207 + 0.340604i \(0.889368\pi\)
\(524\) 16.3923 9.46410i 0.716101 0.413441i
\(525\) 0 0
\(526\) 29.3205 7.85641i 1.27843 0.342556i
\(527\) 1.13681 1.13681i 0.0495203 0.0495203i
\(528\) 0 0
\(529\) 22.7128i 0.987514i
\(530\) 0 0
\(531\) 0 0
\(532\) −34.2557 9.17878i −1.48517 0.397951i
\(533\) −22.1469 + 22.1469i −0.959291 + 0.959291i
\(534\) 0 0
\(535\) 0 0
\(536\) 24.9282 24.9282i 1.07673 1.07673i
\(537\) 0 0
\(538\) 1.96902 + 1.13681i 0.0848903 + 0.0490115i
\(539\) −1.85641 −0.0799611
\(540\) 0 0
\(541\) −33.7846 −1.45251 −0.726257 0.687423i \(-0.758742\pi\)
−0.726257 + 0.687423i \(0.758742\pi\)
\(542\) 29.6985 + 17.1464i 1.27566 + 0.736502i
\(543\) 0 0
\(544\) −8.78461 + 32.7846i −0.376637 + 1.40563i
\(545\) 0 0
\(546\) 0 0
\(547\) 17.5254 17.5254i 0.749331 0.749331i −0.225023 0.974353i \(-0.572246\pi\)
0.974353 + 0.225023i \(0.0722457\pi\)
\(548\) −4.41851 + 16.4901i −0.188749 + 0.704422i
\(549\) 0 0
\(550\) 0 0
\(551\) 53.7128i 2.28824i
\(552\) 0 0
\(553\) 0.933740 0.933740i 0.0397067 0.0397067i
\(554\) −1.63397 + 0.437822i −0.0694209 + 0.0186013i
\(555\) 0 0
\(556\) 19.4641 + 33.7128i 0.825462 + 1.42974i
\(557\) −25.6317 25.6317i −1.08605 1.08605i −0.995931 0.0901194i \(-0.971275\pi\)
−0.0901194 0.995931i \(-0.528725\pi\)
\(558\) 0 0
\(559\) −14.1244 −0.597397
\(560\) 0 0
\(561\) 0 0
\(562\) 10.1769 17.6269i 0.429286 0.743546i
\(563\) 16.3886 + 16.3886i 0.690695 + 0.690695i 0.962385 0.271690i \(-0.0875824\pi\)
−0.271690 + 0.962385i \(0.587582\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −29.2224 + 7.83013i −1.22831 + 0.329125i
\(567\) 0 0
\(568\) 26.7685i 1.12318i
\(569\) 17.3205i 0.726113i 0.931767 + 0.363057i \(0.118267\pi\)
−0.931767 + 0.363057i \(0.881733\pi\)
\(570\) 0 0
\(571\) 16.8038i 0.703219i −0.936147 0.351610i \(-0.885634\pi\)
0.936147 0.351610i \(-0.114366\pi\)
\(572\) −5.93426 + 22.1469i −0.248124 + 0.926010i
\(573\) 0 0
\(574\) 4.92820 + 18.3923i 0.205699 + 0.767680i
\(575\) 0 0
\(576\) 0 0
\(577\) 10.8468 + 10.8468i 0.451560 + 0.451560i 0.895872 0.444312i \(-0.146552\pi\)
−0.444312 + 0.895872i \(0.646552\pi\)
\(578\) 23.2702 + 13.4350i 0.967911 + 0.558824i
\(579\) 0 0
\(580\) 0 0
\(581\) −2.28719 −0.0948885
\(582\) 0 0
\(583\) 2.07055 + 2.07055i 0.0857535 + 0.0857535i
\(584\) −8.00000 + 8.00000i −0.331042 + 0.331042i
\(585\) 0 0
\(586\) −4.58846 17.1244i −0.189547 0.707401i
\(587\) −13.6617 + 13.6617i −0.563877 + 0.563877i −0.930406 0.366529i \(-0.880546\pi\)
0.366529 + 0.930406i \(0.380546\pi\)
\(588\) 0 0
\(589\) 1.92820i 0.0794502i
\(590\) 0 0
\(591\) 0 0
\(592\) 11.3137 3.03150i 0.464991 0.124594i
\(593\) −22.9048 + 22.9048i −0.940588 + 0.940588i −0.998331 0.0577433i \(-0.981610\pi\)
0.0577433 + 0.998331i \(0.481610\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 27.4641 15.8564i 1.12497 0.649504i
\(597\) 0 0
\(598\) −2.17209 + 3.76217i −0.0888233 + 0.153846i
\(599\) −46.6410 −1.90570 −0.952850 0.303441i \(-0.901864\pi\)
−0.952850 + 0.303441i \(0.901864\pi\)
\(600\) 0 0
\(601\) −19.7846 −0.807031 −0.403516 0.914973i \(-0.632212\pi\)
−0.403516 + 0.914973i \(0.632212\pi\)
\(602\) −4.29341 + 7.43640i −0.174986 + 0.303085i
\(603\) 0 0
\(604\) −7.39230 + 4.26795i −0.300789 + 0.173660i
\(605\) 0 0
\(606\) 0 0
\(607\) 13.9391 13.9391i 0.565769 0.565769i −0.365171 0.930940i \(-0.618990\pi\)
0.930940 + 0.365171i \(0.118990\pi\)
\(608\) −20.3538 35.2538i −0.825455 1.42973i
\(609\) 0 0
\(610\) 0 0
\(611\) 74.1051i 2.99797i
\(612\) 0 0
\(613\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) −4.90192 18.2942i −0.197826 0.738295i
\(615\) 0 0
\(616\) 9.85641 + 9.85641i 0.397126 + 0.397126i
\(617\) 10.0754 + 10.0754i 0.405619 + 0.405619i 0.880208 0.474589i \(-0.157403\pi\)
−0.474589 + 0.880208i \(0.657403\pi\)
\(618\) 0 0
\(619\) 35.9808 1.44619 0.723094 0.690749i \(-0.242719\pi\)
0.723094 + 0.690749i \(0.242719\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 33.6365 + 19.4201i 1.34870 + 0.778673i
\(623\) −12.0716 12.0716i −0.483638 0.483638i
\(624\) 0 0
\(625\) 0 0
\(626\) 2.97372 + 11.0981i 0.118854 + 0.443568i
\(627\) 0 0
\(628\) −5.03768 + 18.8009i −0.201025 + 0.750237i
\(629\) 17.5692i 0.700531i
\(630\) 0 0
\(631\) 1.58846i 0.0632355i −0.999500 0.0316177i \(-0.989934\pi\)
0.999500 0.0316177i \(-0.0100659\pi\)
\(632\) 1.51575 0.0602933
\(633\) 0 0
\(634\) −9.46410 + 2.53590i −0.375867 + 0.100713i
\(635\) 0 0
\(636\) 0 0
\(637\) −3.76217 3.76217i −0.149062 0.149062i
\(638\) 10.5558 18.2832i 0.417909 0.723840i
\(639\) 0 0
\(640\) 0 0
\(641\) 3.21539 0.127000 0.0635001 0.997982i \(-0.479774\pi\)
0.0635001 + 0.997982i \(0.479774\pi\)
\(642\) 0 0
\(643\) 12.0716 + 12.0716i 0.476057 + 0.476057i 0.903868 0.427811i \(-0.140715\pi\)
−0.427811 + 0.903868i \(0.640715\pi\)
\(644\) 1.32051 + 2.28719i 0.0520353 + 0.0901278i
\(645\) 0 0
\(646\) −58.9808 + 15.8038i −2.32057 + 0.621794i
\(647\) 3.58630 3.58630i 0.140992 0.140992i −0.633088 0.774080i \(-0.718213\pi\)
0.774080 + 0.633088i \(0.218213\pi\)
\(648\) 0 0
\(649\) 9.07180i 0.356099i
\(650\) 0 0
\(651\) 0 0
\(652\) 7.48717 27.9425i 0.293220 1.09431i
\(653\) 7.34847 7.34847i 0.287568 0.287568i −0.548550 0.836118i \(-0.684820\pi\)
0.836118 + 0.548550i \(0.184820\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −10.9282 + 18.9282i −0.426675 + 0.739022i
\(657\) 0 0
\(658\) −39.0160 22.5259i −1.52100 0.878150i
\(659\) −14.5359 −0.566238 −0.283119 0.959085i \(-0.591369\pi\)
−0.283119 + 0.959085i \(0.591369\pi\)
\(660\) 0 0
\(661\) −7.85641 −0.305579 −0.152789 0.988259i \(-0.548826\pi\)
−0.152789 + 0.988259i \(0.548826\pi\)
\(662\) −2.92996 1.69161i −0.113876 0.0657465i
\(663\) 0 0
\(664\) −1.85641 1.85641i −0.0720425 0.0720425i
\(665\) 0 0
\(666\) 0 0
\(667\) 2.82843 2.82843i 0.109517 0.109517i
\(668\) −8.48528 2.27362i −0.328305 0.0879692i
\(669\) 0 0
\(670\) 0 0
\(671\) 6.00000i 0.231627i
\(672\) 0 0
\(673\) −2.82843 + 2.82843i −0.109028 + 0.109028i −0.759516 0.650488i \(-0.774564\pi\)
0.650488 + 0.759516i \(0.274564\pi\)
\(674\) −15.4904 + 4.15064i −0.596667 + 0.159876i
\(675\) 0 0
\(676\) −34.3923 + 19.8564i −1.32278 + 0.763708i
\(677\) −25.9091 25.9091i −0.995768 0.995768i 0.00422306 0.999991i \(-0.498656\pi\)
−0.999991 + 0.00422306i \(0.998656\pi\)
\(678\) 0 0
\(679\) −15.4449 −0.592719
\(680\) 0 0
\(681\) 0 0
\(682\) 0.378937 0.656339i 0.0145103 0.0251325i
\(683\) 3.58630 + 3.58630i 0.137226 + 0.137226i 0.772383 0.635157i \(-0.219065\pi\)
−0.635157 + 0.772383i \(0.719065\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −26.6865 + 7.15064i −1.01890 + 0.273013i
\(687\) 0 0
\(688\) −9.52056 + 2.55103i −0.362968 + 0.0972569i
\(689\) 8.39230i 0.319721i
\(690\) 0 0
\(691\) 25.3205i 0.963238i 0.876381 + 0.481619i \(0.159951\pi\)
−0.876381 + 0.481619i \(0.840049\pi\)
\(692\) −17.2480 4.62158i −0.655669 0.175686i
\(693\) 0 0
\(694\) −4.05256 15.1244i −0.153833 0.574113i
\(695\) 0 0
\(696\) 0 0
\(697\) 23.1822 + 23.1822i 0.878089 + 0.878089i
\(698\) −4.72311 2.72689i −0.178773 0.103214i
\(699\) 0 0
\(700\) 0 0
\(701\) 15.4641 0.584071 0.292036 0.956407i \(-0.405667\pi\)
0.292036 + 0.956407i \(0.405667\pi\)
\(702\) 0 0
\(703\) 14.9000 + 14.9000i 0.561965 + 0.561965i
\(704\) 16.0000i 0.603023i
\(705\) 0 0
\(706\) 5.80385 + 21.6603i 0.218431 + 0.815194i
\(707\) −2.55103 + 2.55103i −0.0959412 + 0.0959412i
\(708\) 0 0
\(709\) 24.8564i 0.933502i 0.884389 + 0.466751i \(0.154576\pi\)
−0.884389 + 0.466751i \(0.845424\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 19.5959i 0.734388i
\(713\) 0.101536 0.101536i 0.00380255 0.00380255i
\(714\) 0 0
\(715\) 0 0
\(716\) −25.3205 43.8564i −0.946272 1.63899i
\(717\) 0 0
\(718\) 17.2480 29.8744i 0.643688 1.11490i
\(719\) 12.9282 0.482141 0.241070 0.970508i \(-0.422502\pi\)
0.241070 + 0.970508i \(0.422502\pi\)
\(720\) 0 0
\(721\) 17.0718 0.635787
\(722\) 23.1822 40.1528i 0.862753 1.49433i
\(723\) 0 0
\(724\) 13.9282 + 24.1244i 0.517638 + 0.896575i
\(725\) 0 0
\(726\) 0 0
\(727\) −33.3083 + 33.3083i −1.23534 + 1.23534i −0.273453 + 0.961885i \(0.588166\pi\)
−0.961885 + 0.273453i \(0.911834\pi\)
\(728\) 39.9497i 1.48063i
\(729\) 0 0
\(730\) 0 0
\(731\) 14.7846i 0.546829i
\(732\) 0 0
\(733\) −13.9391 + 13.9391i −0.514851 + 0.514851i −0.916009 0.401158i \(-0.868608\pi\)
0.401158 + 0.916009i \(0.368608\pi\)
\(734\) −3.88269 14.4904i −0.143313 0.534850i
\(735\) 0 0
\(736\) −0.784610 + 2.92820i −0.0289211 + 0.107935i
\(737\) −17.6269 17.6269i −0.649295 0.649295i
\(738\) 0 0
\(739\) −14.3923 −0.529429 −0.264715 0.964327i \(-0.585278\pi\)
−0.264715 + 0.964327i \(0.585278\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.41851 + 2.55103i 0.162208 + 0.0936511i
\(743\) 21.8695 + 21.8695i 0.802316 + 0.802316i 0.983457 0.181141i \(-0.0579792\pi\)
−0.181141 + 0.983457i \(0.557979\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.07884 + 26.4186i 0.259175 + 0.967253i
\(747\) 0 0
\(748\) 23.1822 + 6.21166i 0.847626 + 0.227121i
\(749\) 16.1051i 0.588468i
\(750\) 0 0
\(751\) 11.4641i 0.418331i 0.977880 + 0.209166i \(0.0670747\pi\)
−0.977880 + 0.209166i \(0.932925\pi\)
\(752\) −13.3843 49.9507i −0.488074 1.82152i
\(753\) 0 0
\(754\) 58.4449 15.6603i 2.12844 0.570313i
\(755\) 0 0
\(756\) 0 0
\(757\) −20.6448 20.6448i −0.750348 0.750348i 0.224196 0.974544i \(-0.428024\pi\)
−0.974544 + 0.224196i \(0.928024\pi\)
\(758\) 11.1242 19.2677i 0.404051 0.699836i
\(759\) 0 0
\(760\) 0 0
\(761\) −46.6410 −1.69074 −0.845368 0.534185i \(-0.820619\pi\)
−0.845368 + 0.534185i \(0.820619\pi\)
\(762\) 0 0
\(763\) 12.1967 + 12.1967i 0.441549 + 0.441549i
\(764\) −29.3205 + 16.9282i −1.06078 + 0.612441i
\(765\) 0 0
\(766\) 22.9282 6.14359i 0.828430 0.221977i
\(767\) 18.3848 18.3848i 0.663836 0.663836i
\(768\) 0 0
\(769\) 29.9282i 1.07924i −0.841909 0.539619i \(-0.818568\pi\)
0.841909 0.539619i \(-0.181432\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.5283 + 7.10823i 0.954774 + 0.255831i
\(773\) 23.4596 23.4596i 0.843784 0.843784i −0.145565 0.989349i \(-0.546500\pi\)
0.989349 + 0.145565i \(0.0464999\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −12.5359 12.5359i −0.450013 0.450013i
\(777\) 0 0
\(778\) 17.8028 + 10.2784i 0.638260 + 0.368500i
\(779\) −39.3205 −1.40880
\(780\) 0 0
\(781\) 18.9282 0.677304
\(782\) 3.93803 + 2.27362i 0.140824 + 0.0813046i
\(783\) 0 0
\(784\) −3.21539 1.85641i −0.114835 0.0663002i
\(785\) 0 0
\(786\) 0 0
\(787\) −10.0246 + 10.0246i −0.357338 + 0.357338i −0.862831 0.505493i \(-0.831311\pi\)
0.505493 + 0.862831i \(0.331311\pi\)
\(788\) −7.45001 + 27.8038i −0.265395 + 0.990469i
\(789\) 0 0
\(790\) 0 0
\(791\) 34.1436i 1.21401i
\(792\) 0 0
\(793\) −12.1595 + 12.1595i −0.431797 + 0.431797i
\(794\) 26.7583 7.16987i 0.949618 0.254449i
\(795\) 0 0
\(796\) −14.1244 24.4641i −0.500625 0.867107i
\(797\) 9.79796 + 9.79796i 0.347062 + 0.347062i 0.859014 0.511952i \(-0.171078\pi\)
−0.511952 + 0.859014i \(0.671078\pi\)
\(798\) 0 0
\(799\) −77.5692 −2.74420
\(800\) 0 0
\(801\) 0 0
\(802\) 23.4596 40.6333i 0.828388 1.43481i
\(803\) 5.65685 + 5.65685i 0.199626 + 0.199626i
\(804\) 0 0
\(805\) 0 0
\(806\) 2.09808 0.562178i 0.0739016 0.0198019i
\(807\) 0 0
\(808\) −4.14110 −0.145684
\(809\) 49.1769i 1.72897i −0.502660 0.864484i \(-0.667645\pi\)
0.502660 0.864484i \(-0.332355\pi\)
\(810\) 0 0
\(811\) 26.1244i 0.917350i −0.888604 0.458675i \(-0.848324\pi\)
0.888604 0.458675i \(-0.151676\pi\)
\(812\) 9.52056 35.5312i 0.334106 1.24690i
\(813\) 0 0
\(814\) −2.14359 8.00000i −0.0751329 0.280400i
\(815\) 0 0
\(816\) 0 0
\(817\) −12.5385 12.5385i −0.438665 0.438665i
\(818\) 34.0291 + 19.6467i 1.18980 + 0.686930i
\(819\) 0 0
\(820\) 0 0
\(821\) −55.5692 −1.93938 −0.969690 0.244340i \(-0.921429\pi\)
−0.969690 + 0.244340i \(0.921429\pi\)
\(822\) 0 0
\(823\) −33.5114 33.5114i −1.16813 1.16813i −0.982646 0.185488i \(-0.940613\pi\)
−0.185488 0.982646i \(-0.559387\pi\)
\(824\) 13.8564 + 13.8564i 0.482711 + 0.482711i
\(825\) 0 0
\(826\) −4.09103 15.2679i −0.142345 0.531240i
\(827\) −30.0774 + 30.0774i −1.04589 + 1.04589i −0.0469995 + 0.998895i \(0.514966\pi\)
−0.998895 + 0.0469995i \(0.985034\pi\)
\(828\) 0 0
\(829\) 17.7128i 0.615191i 0.951517 + 0.307596i \(0.0995244\pi\)
−0.951517 + 0.307596i \(0.900476\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −32.4254 + 32.4254i −1.12415 + 1.12415i
\(833\) −3.93803 + 3.93803i −0.136445 + 0.136445i
\(834\) 0 0
\(835\) 0 0
\(836\) −24.9282 + 14.3923i −0.862160 + 0.497768i
\(837\) 0 0
\(838\) 18.7637 32.4997i 0.648182 1.12268i
\(839\) 40.6410 1.40308 0.701542 0.712628i \(-0.252495\pi\)
0.701542 + 0.712628i \(0.252495\pi\)
\(840\) 0 0
\(841\) −26.7128 −0.921131
\(842\) 4.79744 8.30942i 0.165331 0.286361i
\(843\) 0 0
\(844\) 12.4641 7.19615i 0.429032 0.247702i
\(845\) 0 0
\(846\) 0 0
\(847\) −12.1967 + 12.1967i −0.419083 + 0.419083i
\(848\) 1.51575 + 5.65685i 0.0520511 + 0.194257i
\(849\) 0 0
\(850\) 0 0
\(851\) 1.56922i 0.0537921i
\(852\) 0 0
\(853\) −11.0227 + 11.0227i −0.377410 + 0.377410i −0.870167 0.492757i \(-0.835989\pi\)
0.492757 + 0.870167i \(0.335989\pi\)
\(854\) 2.70577 + 10.0981i 0.0925896 + 0.345549i
\(855\) 0 0
\(856\) −13.0718 + 13.0718i −0.446785 + 0.446785i
\(857\) 33.7381 + 33.7381i 1.15247 + 1.15247i 0.986056 + 0.166414i \(0.0532188\pi\)
0.166414 + 0.986056i \(0.446781\pi\)
\(858\) 0 0
\(859\) −21.6077 −0.737245 −0.368623 0.929579i \(-0.620170\pi\)
−0.368623 + 0.929579i \(0.620170\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −31.4916 18.1817i −1.07261 0.619271i
\(863\) −27.3233 27.3233i −0.930097 0.930097i 0.0676147 0.997712i \(-0.478461\pi\)
−0.997712 + 0.0676147i \(0.978461\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1.61474 6.02628i −0.0548710 0.204781i
\(867\) 0 0
\(868\) 0.341773 1.27551i 0.0116005 0.0432937i
\(869\) 1.07180i 0.0363582i
\(870\) 0 0
\(871\) 71.4449i 2.42082i
\(872\) 19.7990i 0.670478i
\(873\) 0 0
\(874\) −5.26795 + 1.41154i −0.178191 + 0.0477461i
\(875\) 0 0
\(876\) 0 0
\(877\) −22.5123 22.5123i −0.760186 0.760186i 0.216170 0.976356i \(-0.430643\pi\)
−0.976356 + 0.216170i \(0.930643\pi\)
\(878\) 19.7854 34.2693i 0.667724 1.15653i
\(879\) 0 0
\(880\) 0 0
\(881\) 50.9282 1.71581 0.857907 0.513804i \(-0.171764\pi\)
0.857907 + 0.513804i \(0.171764\pi\)
\(882\) 0 0
\(883\) 21.3383 + 21.3383i 0.718091 + 0.718091i 0.968214 0.250123i \(-0.0804711\pi\)
−0.250123 + 0.968214i \(0.580471\pi\)
\(884\) 34.3923 + 59.5692i 1.15674 + 2.00353i
\(885\) 0 0
\(886\) −17.4641 + 4.67949i −0.586718 + 0.157211i
\(887\) 5.55532 5.55532i 0.186529 0.186529i −0.607665 0.794194i \(-0.707893\pi\)
0.794194 + 0.607665i \(0.207893\pi\)
\(888\) 0 0
\(889\) 7.21539i 0.241996i
\(890\) 0 0
\(891\) 0 0
\(892\) 5.89709 22.0082i 0.197449 0.736890i
\(893\) 65.7845 65.7845i 2.20139 2.20139i
\(894\) 0 0
\(895\) 0 0
\(896\) 7.21539 + 26.9282i 0.241049 + 0.899608i
\(897\) 0 0
\(898\) 41.4655 + 23.9401i 1.38372 + 0.798891i
\(899\) −2.00000 −0.0667037
\(900\) 0 0
\(901\) 8.78461 0.292658
\(902\) 13.3843 + 7.72741i 0.445647 + 0.257294i
\(903\) 0 0
\(904\) 27.7128 27.7128i 0.921714 0.921714i
\(905\) 0 0
\(906\) 0 0
\(907\) 6.96953 6.96953i 0.231420 0.231420i −0.581866 0.813285i \(-0.697677\pi\)
0.813285 + 0.581866i \(0.197677\pi\)
\(908\) −23.9401 6.41473i −0.794480 0.212880i
\(909\) 0 0
\(910\) 0 0
\(911\) 26.3923i 0.874416i −0.899360 0.437208i \(-0.855967\pi\)
0.899360 0.437208i \(-0.144033\pi\)
\(912\) 0 0
\(913\) −1.31268 + 1.31268i −0.0434433 + 0.0434433i
\(914\) 48.7846 13.0718i 1.61365 0.432377i
\(915\) 0 0
\(916\) 43.0526 24.8564i 1.42250 0.821279i
\(917\) −16.4901 16.4901i −0.544551 0.544551i
\(918\) 0 0
\(919\) −9.33975 −0.308090 −0.154045 0.988064i \(-0.549230\pi\)
−0.154045 + 0.988064i \(0.549230\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −11.0363 + 19.1154i −0.363461 + 0.629534i
\(923\) 38.3596 + 38.3596i 1.26262 + 1.26262i
\(924\) 0 0
\(925\) 0 0
\(926\) 17.8564 4.78461i 0.586798 0.157232i
\(927\) 0 0
\(928\) 36.5665 21.1117i 1.20035 0.693024i
\(929\) 7.46410i 0.244889i −0.992475 0.122445i \(-0.960927\pi\)
0.992475 0.122445i \(-0.0390734\pi\)
\(930\) 0 0
\(931\) 6.67949i 0.218912i
\(932\) −23.9401 6.41473i −0.784184 0.210121i
\(933\) 0 0
\(934\) −1.26795 4.73205i −0.0414886 0.154837i
\(935\) 0 0
\(936\) 0 0
\(937\) −37.4123 37.4123i −1.22221 1.22221i −0.966846 0.255360i \(-0.917806\pi\)
−0.255360 0.966846i \(-0.582194\pi\)
\(938\) −37.6154 21.7172i −1.22819 0.709093i
\(939\) 0 0
\(940\) 0 0
\(941\) −24.9282 −0.812636 −0.406318 0.913732i \(-0.633188\pi\)
−0.406318 + 0.913732i \(0.633188\pi\)
\(942\) 0 0
\(943\) 2.07055 + 2.07055i 0.0674265 + 0.0674265i
\(944\) 9.07180 15.7128i 0.295262 0.511409i
\(945\) 0 0
\(946\) 1.80385 + 6.73205i 0.0586481 + 0.218878i
\(947\) 11.4896 11.4896i 0.373361 0.373361i −0.495339 0.868700i \(-0.664956\pi\)
0.868700 + 0.495339i \(0.164956\pi\)
\(948\) 0 0
\(949\) 22.9282i 0.744281i
\(950\) 0 0
\(951\) 0 0
\(952\) 41.8172 1.35530
\(953\) 11.1106 11.1106i 0.359909 0.359909i −0.503870 0.863779i \(-0.668091\pi\)
0.863779 + 0.503870i \(0.168091\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 17.4641 + 30.2487i 0.564829 + 0.978313i
\(957\) 0 0
\(958\) 9.41902 16.3142i 0.304315 0.527089i
\(959\) 21.0333 0.679201
\(960\) 0 0
\(961\) 30.9282 0.997684
\(962\) 11.8685 20.5569i 0.382656 0.662780i
\(963\) 0 0
\(964\) 14.8564 + 25.7321i 0.478493 + 0.828774i
\(965\) 0 0
\(966\) 0 0
\(967\) −33.9411 + 33.9411i −1.09147 + 1.09147i −0.0961015 + 0.995372i \(0.530637\pi\)
−0.995372 + 0.0961015i \(0.969363\pi\)
\(968\) −19.7990 −0.636364
\(969\) 0 0
\(970\) 0 0
\(971\) 19.1769i 0.615416i −0.951481 0.307708i \(-0.900438\pi\)
0.951481 0.307708i \(-0.0995621\pi\)
\(972\) 0 0
\(973\) 33.9139 33.9139i 1.08723 1.08723i
\(974\) 7.88269 + 29.4186i 0.252578 + 0.942632i
\(975\) 0 0
\(976\) −6.00000 + 10.3923i −0.192055 + 0.332650i
\(977\) −27.7023 27.7023i −0.886274 0.886274i 0.107889 0.994163i \(-0.465591\pi\)
−0.994163 + 0.107889i \(0.965591\pi\)
\(978\) 0 0
\(979\) −13.8564 −0.442853
\(980\) 0 0
\(981\) 0 0
\(982\) 42.4264 + 24.4949i 1.35388 + 0.781664i
\(983\) 18.2832 + 18.2832i 0.583145 + 0.583145i 0.935766 0.352621i \(-0.114710\pi\)
−0.352621 + 0.935766i \(0.614710\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −16.3923 61.1769i −0.522037 1.94827i
\(987\) 0 0
\(988\) −79.6864 21.3519i −2.53516 0.679295i
\(989\) 1.32051i 0.0419897i
\(990\) 0 0
\(991\) 57.5885i 1.82936i −0.404181 0.914679i \(-0.632444\pi\)
0.404181 0.914679i \(-0.367556\pi\)
\(992\) 1.31268 0.757875i 0.0416776 0.0240625i
\(993\) 0 0
\(994\) 31.8564 8.53590i 1.01042 0.270742i
\(995\) 0 0
\(996\) 0 0
\(997\) −19.5959 19.5959i −0.620609 0.620609i 0.325078 0.945687i \(-0.394609\pi\)
−0.945687 + 0.325078i \(0.894609\pi\)
\(998\) −4.15471 + 7.19617i −0.131515 + 0.227791i
\(999\) 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 900.2.k.g.307.3 8
3.2 odd 2 300.2.j.a.7.2 8
4.3 odd 2 900.2.k.l.307.2 8
5.2 odd 4 900.2.k.l.343.4 8
5.3 odd 4 900.2.k.l.343.1 8
5.4 even 2 inner 900.2.k.g.307.2 8
12.11 even 2 300.2.j.c.7.3 yes 8
15.2 even 4 300.2.j.c.43.1 yes 8
15.8 even 4 300.2.j.c.43.4 yes 8
15.14 odd 2 300.2.j.a.7.3 yes 8
20.3 even 4 inner 900.2.k.g.343.4 8
20.7 even 4 inner 900.2.k.g.343.1 8
20.19 odd 2 900.2.k.l.307.3 8
60.23 odd 4 300.2.j.a.43.1 yes 8
60.47 odd 4 300.2.j.a.43.4 yes 8
60.59 even 2 300.2.j.c.7.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.j.a.7.2 8 3.2 odd 2
300.2.j.a.7.3 yes 8 15.14 odd 2
300.2.j.a.43.1 yes 8 60.23 odd 4
300.2.j.a.43.4 yes 8 60.47 odd 4
300.2.j.c.7.2 yes 8 60.59 even 2
300.2.j.c.7.3 yes 8 12.11 even 2
300.2.j.c.43.1 yes 8 15.2 even 4
300.2.j.c.43.4 yes 8 15.8 even 4
900.2.k.g.307.2 8 5.4 even 2 inner
900.2.k.g.307.3 8 1.1 even 1 trivial
900.2.k.g.343.1 8 20.7 even 4 inner
900.2.k.g.343.4 8 20.3 even 4 inner
900.2.k.l.307.2 8 4.3 odd 2
900.2.k.l.307.3 8 20.19 odd 2
900.2.k.l.343.1 8 5.3 odd 4
900.2.k.l.343.4 8 5.2 odd 4