Properties

Label 1800.2.d.q.1549.2
Level $1800$
Weight $2$
Character 1800.1549
Analytic conductor $14.373$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,2,Mod(1549,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1800.1549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.d (of order \(2\), degree \(1\), 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: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1549.2
Root \(0.264658 + 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 1800.1549
Dual form 1800.2.d.q.1549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.38923 + 0.264658i) q^{2} +(1.85991 - 0.735342i) q^{4} -0.941367i q^{7} +(-2.38923 + 1.51380i) q^{8} +O(q^{10})\) \(q+(-1.38923 + 0.264658i) q^{2} +(1.85991 - 0.735342i) q^{4} -0.941367i q^{7} +(-2.38923 + 1.51380i) q^{8} -4.49828i q^{11} +5.55691 q^{13} +(0.249141 + 1.30777i) q^{14} +(2.91855 - 2.73534i) q^{16} +7.55691i q^{17} -1.05863i q^{19} +(1.19051 + 6.24914i) q^{22} +1.05863i q^{23} +(-7.71982 + 1.47068i) q^{26} +(-0.692226 - 1.75086i) q^{28} -2.00000i q^{29} +3.55691 q^{31} +(-3.33060 + 4.57243i) q^{32} +(-2.00000 - 10.4983i) q^{34} -7.43965 q^{37} +(0.280176 + 1.47068i) q^{38} +3.88273 q^{41} +1.88273 q^{43} +(-3.30777 - 8.36641i) q^{44} +(-0.280176 - 1.47068i) q^{46} -10.0552i q^{47} +6.11383 q^{49} +(10.3354 - 4.08623i) q^{52} +2.00000 q^{53} +(1.42504 + 2.24914i) q^{56} +(0.529317 + 2.77846i) q^{58} -8.49828i q^{59} +8.99656i q^{61} +(-4.94137 + 0.941367i) q^{62} +(3.41683 - 7.23362i) q^{64} -4.00000 q^{67} +(5.55691 + 14.0552i) q^{68} +12.9966 q^{71} -6.00000i q^{73} +(10.3354 - 1.96896i) q^{74} +(-0.778457 - 1.96896i) q^{76} -4.23453 q^{77} -11.5569 q^{79} +(-5.39400 + 1.02760i) q^{82} +5.88273 q^{83} +(-2.61555 + 0.498281i) q^{86} +(6.80949 + 10.7474i) q^{88} -4.11727 q^{89} -5.23109i q^{91} +(0.778457 + 1.96896i) q^{92} +(2.66119 + 13.9690i) q^{94} -17.1138i q^{97} +(-8.49351 + 1.61808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{4} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{4} - 6 q^{8} - 16 q^{14} + 10 q^{16} - 12 q^{22} - 28 q^{26} - 20 q^{28} - 12 q^{31} - 10 q^{32} - 12 q^{34} - 8 q^{37} + 20 q^{38} + 20 q^{41} + 8 q^{43} - 4 q^{44} - 20 q^{46} - 30 q^{49} + 12 q^{52} + 12 q^{53} - 4 q^{56} + 4 q^{58} - 28 q^{62} - 22 q^{64} - 24 q^{67} + 8 q^{71} + 12 q^{74} + 12 q^{76} - 32 q^{77} - 36 q^{79} + 16 q^{82} + 32 q^{83} + 16 q^{86} + 60 q^{88} - 28 q^{89} - 12 q^{92} - 4 q^{94} - 56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38923 + 0.264658i −0.982333 + 0.187142i
\(3\) 0 0
\(4\) 1.85991 0.735342i 0.929956 0.367671i
\(5\) 0 0
\(6\) 0 0
\(7\) 0.941367i 0.355803i −0.984048 0.177902i \(-0.943069\pi\)
0.984048 0.177902i \(-0.0569309\pi\)
\(8\) −2.38923 + 1.51380i −0.844720 + 0.535209i
\(9\) 0 0
\(10\) 0 0
\(11\) 4.49828i 1.35628i −0.734931 0.678141i \(-0.762786\pi\)
0.734931 0.678141i \(-0.237214\pi\)
\(12\) 0 0
\(13\) 5.55691 1.54121 0.770605 0.637313i \(-0.219954\pi\)
0.770605 + 0.637313i \(0.219954\pi\)
\(14\) 0.249141 + 1.30777i 0.0665856 + 0.349517i
\(15\) 0 0
\(16\) 2.91855 2.73534i 0.729636 0.683835i
\(17\) 7.55691i 1.83282i 0.400240 + 0.916410i \(0.368927\pi\)
−0.400240 + 0.916410i \(0.631073\pi\)
\(18\) 0 0
\(19\) 1.05863i 0.242867i −0.992600 0.121434i \(-0.961251\pi\)
0.992600 0.121434i \(-0.0387491\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.19051 + 6.24914i 0.253817 + 1.33232i
\(23\) 1.05863i 0.220740i 0.993891 + 0.110370i \(0.0352036\pi\)
−0.993891 + 0.110370i \(0.964796\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −7.71982 + 1.47068i −1.51398 + 0.288425i
\(27\) 0 0
\(28\) −0.692226 1.75086i −0.130818 0.330881i
\(29\) 2.00000i 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) 3.55691 0.638841 0.319420 0.947613i \(-0.396512\pi\)
0.319420 + 0.947613i \(0.396512\pi\)
\(32\) −3.33060 + 4.57243i −0.588772 + 0.808299i
\(33\) 0 0
\(34\) −2.00000 10.4983i −0.342997 1.80044i
\(35\) 0 0
\(36\) 0 0
\(37\) −7.43965 −1.22307 −0.611535 0.791217i \(-0.709448\pi\)
−0.611535 + 0.791217i \(0.709448\pi\)
\(38\) 0.280176 + 1.47068i 0.0454506 + 0.238576i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.88273 0.606381 0.303191 0.952930i \(-0.401948\pi\)
0.303191 + 0.952930i \(0.401948\pi\)
\(42\) 0 0
\(43\) 1.88273 0.287114 0.143557 0.989642i \(-0.454146\pi\)
0.143557 + 0.989642i \(0.454146\pi\)
\(44\) −3.30777 8.36641i −0.498666 1.26128i
\(45\) 0 0
\(46\) −0.280176 1.47068i −0.0413097 0.216840i
\(47\) 10.0552i 1.46670i −0.679851 0.733350i \(-0.737955\pi\)
0.679851 0.733350i \(-0.262045\pi\)
\(48\) 0 0
\(49\) 6.11383 0.873404
\(50\) 0 0
\(51\) 0 0
\(52\) 10.3354 4.08623i 1.43326 0.566658i
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.42504 + 2.24914i 0.190429 + 0.300554i
\(57\) 0 0
\(58\) 0.529317 + 2.77846i 0.0695027 + 0.364829i
\(59\) 8.49828i 1.10638i −0.833054 0.553191i \(-0.813410\pi\)
0.833054 0.553191i \(-0.186590\pi\)
\(60\) 0 0
\(61\) 8.99656i 1.15189i 0.817488 + 0.575946i \(0.195366\pi\)
−0.817488 + 0.575946i \(0.804634\pi\)
\(62\) −4.94137 + 0.941367i −0.627554 + 0.119554i
\(63\) 0 0
\(64\) 3.41683 7.23362i 0.427103 0.904203i
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 5.55691 + 14.0552i 0.673875 + 1.70444i
\(69\) 0 0
\(70\) 0 0
\(71\) 12.9966 1.54241 0.771204 0.636588i \(-0.219655\pi\)
0.771204 + 0.636588i \(0.219655\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 10.3354 1.96896i 1.20146 0.228887i
\(75\) 0 0
\(76\) −0.778457 1.96896i −0.0892952 0.225856i
\(77\) −4.23453 −0.482570
\(78\) 0 0
\(79\) −11.5569 −1.30025 −0.650127 0.759825i \(-0.725284\pi\)
−0.650127 + 0.759825i \(0.725284\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −5.39400 + 1.02760i −0.595668 + 0.113479i
\(83\) 5.88273 0.645714 0.322857 0.946448i \(-0.395357\pi\)
0.322857 + 0.946448i \(0.395357\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.61555 + 0.498281i −0.282042 + 0.0537310i
\(87\) 0 0
\(88\) 6.80949 + 10.7474i 0.725894 + 1.14568i
\(89\) −4.11727 −0.436429 −0.218215 0.975901i \(-0.570023\pi\)
−0.218215 + 0.975901i \(0.570023\pi\)
\(90\) 0 0
\(91\) 5.23109i 0.548368i
\(92\) 0.778457 + 1.96896i 0.0811598 + 0.205279i
\(93\) 0 0
\(94\) 2.66119 + 13.9690i 0.274481 + 1.44079i
\(95\) 0 0
\(96\) 0 0
\(97\) 17.1138i 1.73765i −0.495123 0.868823i \(-0.664877\pi\)
0.495123 0.868823i \(-0.335123\pi\)
\(98\) −8.49351 + 1.61808i −0.857974 + 0.163450i
\(99\) 0 0
\(100\) 0 0
\(101\) 2.00000i 0.199007i −0.995037 0.0995037i \(-0.968274\pi\)
0.995037 0.0995037i \(-0.0317255\pi\)
\(102\) 0 0
\(103\) 10.1725i 1.00232i 0.865354 + 0.501161i \(0.167094\pi\)
−0.865354 + 0.501161i \(0.832906\pi\)
\(104\) −13.2767 + 8.41205i −1.30189 + 0.824869i
\(105\) 0 0
\(106\) −2.77846 + 0.529317i −0.269868 + 0.0514118i
\(107\) 17.2311 1.66579 0.832896 0.553429i \(-0.186681\pi\)
0.832896 + 0.553429i \(0.186681\pi\)
\(108\) 0 0
\(109\) 1.88273i 0.180333i −0.995927 0.0901666i \(-0.971260\pi\)
0.995927 0.0901666i \(-0.0287399\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.57496 2.74742i −0.243311 0.259607i
\(113\) 15.3224i 1.44141i −0.693243 0.720704i \(-0.743819\pi\)
0.693243 0.720704i \(-0.256181\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.47068 3.71982i −0.136550 0.345377i
\(117\) 0 0
\(118\) 2.24914 + 11.8061i 0.207050 + 1.08684i
\(119\) 7.11383 0.652124
\(120\) 0 0
\(121\) −9.23453 −0.839503
\(122\) −2.38101 12.4983i −0.215567 1.13154i
\(123\) 0 0
\(124\) 6.61555 2.61555i 0.594094 0.234883i
\(125\) 0 0
\(126\) 0 0
\(127\) 18.1725i 1.61255i 0.591544 + 0.806273i \(0.298519\pi\)
−0.591544 + 0.806273i \(0.701481\pi\)
\(128\) −2.83231 + 10.9534i −0.250344 + 0.968157i
\(129\) 0 0
\(130\) 0 0
\(131\) 6.38101i 0.557512i 0.960362 + 0.278756i \(0.0899220\pi\)
−0.960362 + 0.278756i \(0.910078\pi\)
\(132\) 0 0
\(133\) −0.996562 −0.0864129
\(134\) 5.55691 1.05863i 0.480044 0.0914520i
\(135\) 0 0
\(136\) −11.4396 18.0552i −0.980942 1.54822i
\(137\) 4.44309i 0.379598i −0.981823 0.189799i \(-0.939216\pi\)
0.981823 0.189799i \(-0.0607837\pi\)
\(138\) 0 0
\(139\) 20.1725i 1.71101i −0.517798 0.855503i \(-0.673248\pi\)
0.517798 0.855503i \(-0.326752\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −18.0552 + 3.43965i −1.51516 + 0.288649i
\(143\) 24.9966i 2.09032i
\(144\) 0 0
\(145\) 0 0
\(146\) 1.58795 + 8.33537i 0.131420 + 0.689840i
\(147\) 0 0
\(148\) −13.8371 + 5.47068i −1.13740 + 0.449687i
\(149\) 2.00000i 0.163846i −0.996639 0.0819232i \(-0.973894\pi\)
0.996639 0.0819232i \(-0.0261062\pi\)
\(150\) 0 0
\(151\) 9.67418 0.787274 0.393637 0.919266i \(-0.371217\pi\)
0.393637 + 0.919266i \(0.371217\pi\)
\(152\) 1.60256 + 2.52932i 0.129985 + 0.205155i
\(153\) 0 0
\(154\) 5.88273 1.12070i 0.474044 0.0903089i
\(155\) 0 0
\(156\) 0 0
\(157\) 4.32582 0.345238 0.172619 0.984989i \(-0.444777\pi\)
0.172619 + 0.984989i \(0.444777\pi\)
\(158\) 16.0552 3.05863i 1.27728 0.243332i
\(159\) 0 0
\(160\) 0 0
\(161\) 0.996562 0.0785401
\(162\) 0 0
\(163\) 6.11727 0.479141 0.239571 0.970879i \(-0.422993\pi\)
0.239571 + 0.970879i \(0.422993\pi\)
\(164\) 7.22154 2.85514i 0.563908 0.222949i
\(165\) 0 0
\(166\) −8.17246 + 1.55691i −0.634306 + 0.120840i
\(167\) 6.05520i 0.468565i −0.972169 0.234283i \(-0.924726\pi\)
0.972169 0.234283i \(-0.0752741\pi\)
\(168\) 0 0
\(169\) 17.8793 1.37533
\(170\) 0 0
\(171\) 0 0
\(172\) 3.50172 1.38445i 0.267004 0.105564i
\(173\) 16.8793 1.28331 0.641655 0.766994i \(-0.278248\pi\)
0.641655 + 0.766994i \(0.278248\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −12.3043 13.1284i −0.927474 0.989593i
\(177\) 0 0
\(178\) 5.71982 1.08967i 0.428719 0.0816741i
\(179\) 10.6155i 0.793443i 0.917939 + 0.396722i \(0.129852\pi\)
−0.917939 + 0.396722i \(0.870148\pi\)
\(180\) 0 0
\(181\) 14.1173i 1.04933i 0.851309 + 0.524664i \(0.175809\pi\)
−0.851309 + 0.524664i \(0.824191\pi\)
\(182\) 1.38445 + 7.26719i 0.102622 + 0.538680i
\(183\) 0 0
\(184\) −1.60256 2.52932i −0.118142 0.186464i
\(185\) 0 0
\(186\) 0 0
\(187\) 33.9931 2.48582
\(188\) −7.39400 18.7018i −0.539263 1.36397i
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 4.87930i 0.351219i −0.984460 0.175610i \(-0.943810\pi\)
0.984460 0.175610i \(-0.0561897\pi\)
\(194\) 4.52932 + 23.7750i 0.325186 + 1.70695i
\(195\) 0 0
\(196\) 11.3712 4.49575i 0.812227 0.321125i
\(197\) 2.88617 0.205631 0.102816 0.994700i \(-0.467215\pi\)
0.102816 + 0.994700i \(0.467215\pi\)
\(198\) 0 0
\(199\) 17.6742 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.529317 + 2.77846i 0.0372426 + 0.195492i
\(203\) −1.88273 −0.132142
\(204\) 0 0
\(205\) 0 0
\(206\) −2.69223 14.1319i −0.187576 0.984614i
\(207\) 0 0
\(208\) 16.2181 15.2001i 1.12452 1.05393i
\(209\) −4.76203 −0.329396
\(210\) 0 0
\(211\) 23.9379i 1.64795i −0.566623 0.823977i \(-0.691750\pi\)
0.566623 0.823977i \(-0.308250\pi\)
\(212\) 3.71982 1.47068i 0.255479 0.101007i
\(213\) 0 0
\(214\) −23.9379 + 4.56035i −1.63636 + 0.311739i
\(215\) 0 0
\(216\) 0 0
\(217\) 3.34836i 0.227302i
\(218\) 0.498281 + 2.61555i 0.0337479 + 0.177147i
\(219\) 0 0
\(220\) 0 0
\(221\) 41.9931i 2.82476i
\(222\) 0 0
\(223\) 24.0552i 1.61086i −0.592694 0.805428i \(-0.701936\pi\)
0.592694 0.805428i \(-0.298064\pi\)
\(224\) 4.30434 + 3.13531i 0.287596 + 0.209487i
\(225\) 0 0
\(226\) 4.05520 + 21.2863i 0.269748 + 1.41594i
\(227\) −11.1138 −0.737651 −0.368825 0.929499i \(-0.620240\pi\)
−0.368825 + 0.929499i \(0.620240\pi\)
\(228\) 0 0
\(229\) 17.2311i 1.13866i 0.822108 + 0.569331i \(0.192798\pi\)
−0.822108 + 0.569331i \(0.807202\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.02760 + 4.77846i 0.198772 + 0.313721i
\(233\) 8.44309i 0.553125i 0.960996 + 0.276562i \(0.0891953\pi\)
−0.960996 + 0.276562i \(0.910805\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.24914 15.8061i −0.406784 1.02889i
\(237\) 0 0
\(238\) −9.88273 + 1.88273i −0.640602 + 0.122039i
\(239\) 10.1173 0.654432 0.327216 0.944950i \(-0.393890\pi\)
0.327216 + 0.944950i \(0.393890\pi\)
\(240\) 0 0
\(241\) 16.8793 1.08729 0.543646 0.839315i \(-0.317044\pi\)
0.543646 + 0.839315i \(0.317044\pi\)
\(242\) 12.8289 2.44400i 0.824671 0.157106i
\(243\) 0 0
\(244\) 6.61555 + 16.7328i 0.423517 + 1.07121i
\(245\) 0 0
\(246\) 0 0
\(247\) 5.88273i 0.374309i
\(248\) −8.49828 + 5.38445i −0.539641 + 0.341913i
\(249\) 0 0
\(250\) 0 0
\(251\) 11.8466i 0.747753i −0.927478 0.373877i \(-0.878028\pi\)
0.927478 0.373877i \(-0.121972\pi\)
\(252\) 0 0
\(253\) 4.76203 0.299386
\(254\) −4.80949 25.2457i −0.301774 1.58406i
\(255\) 0 0
\(256\) 1.03581 15.9664i 0.0647382 0.997902i
\(257\) 10.6707i 0.665623i −0.942993 0.332811i \(-0.892003\pi\)
0.942993 0.332811i \(-0.107997\pi\)
\(258\) 0 0
\(259\) 7.00344i 0.435172i
\(260\) 0 0
\(261\) 0 0
\(262\) −1.68879 8.86469i −0.104334 0.547662i
\(263\) 1.94480i 0.119922i 0.998201 + 0.0599609i \(0.0190976\pi\)
−0.998201 + 0.0599609i \(0.980902\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1.38445 0.263748i 0.0848862 0.0161715i
\(267\) 0 0
\(268\) −7.43965 + 2.94137i −0.454449 + 0.179673i
\(269\) 9.76547i 0.595411i 0.954658 + 0.297706i \(0.0962214\pi\)
−0.954658 + 0.297706i \(0.903779\pi\)
\(270\) 0 0
\(271\) 3.44652 0.209361 0.104681 0.994506i \(-0.466618\pi\)
0.104681 + 0.994506i \(0.466618\pi\)
\(272\) 20.6707 + 22.0552i 1.25335 + 1.33729i
\(273\) 0 0
\(274\) 1.17590 + 6.17246i 0.0710387 + 0.372892i
\(275\) 0 0
\(276\) 0 0
\(277\) −18.7880 −1.12886 −0.564431 0.825480i \(-0.690904\pi\)
−0.564431 + 0.825480i \(0.690904\pi\)
\(278\) 5.33881 + 28.0242i 0.320200 + 1.68078i
\(279\) 0 0
\(280\) 0 0
\(281\) 16.8793 1.00693 0.503467 0.864014i \(-0.332057\pi\)
0.503467 + 0.864014i \(0.332057\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 24.1725 9.55691i 1.43437 0.567099i
\(285\) 0 0
\(286\) 6.61555 + 34.7259i 0.391186 + 2.05339i
\(287\) 3.65508i 0.215752i
\(288\) 0 0
\(289\) −40.1070 −2.35923
\(290\) 0 0
\(291\) 0 0
\(292\) −4.41205 11.1595i −0.258196 0.653059i
\(293\) −20.2277 −1.18171 −0.590856 0.806777i \(-0.701210\pi\)
−0.590856 + 0.806777i \(0.701210\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 17.7750 11.2621i 1.03315 0.654598i
\(297\) 0 0
\(298\) 0.529317 + 2.77846i 0.0306625 + 0.160952i
\(299\) 5.88273i 0.340207i
\(300\) 0 0
\(301\) 1.77234i 0.102156i
\(302\) −13.4396 + 2.56035i −0.773365 + 0.147332i
\(303\) 0 0
\(304\) −2.89572 3.08967i −0.166081 0.177205i
\(305\) 0 0
\(306\) 0 0
\(307\) 8.11039 0.462884 0.231442 0.972849i \(-0.425656\pi\)
0.231442 + 0.972849i \(0.425656\pi\)
\(308\) −7.87586 + 3.11383i −0.448769 + 0.177427i
\(309\) 0 0
\(310\) 0 0
\(311\) −31.8759 −1.80751 −0.903757 0.428046i \(-0.859202\pi\)
−0.903757 + 0.428046i \(0.859202\pi\)
\(312\) 0 0
\(313\) 5.11383i 0.289051i −0.989501 0.144525i \(-0.953834\pi\)
0.989501 0.144525i \(-0.0461655\pi\)
\(314\) −6.00955 + 1.14486i −0.339139 + 0.0646084i
\(315\) 0 0
\(316\) −21.4948 + 8.49828i −1.20918 + 0.478066i
\(317\) −24.6448 −1.38419 −0.692094 0.721807i \(-0.743312\pi\)
−0.692094 + 0.721807i \(0.743312\pi\)
\(318\) 0 0
\(319\) −8.99656 −0.503711
\(320\) 0 0
\(321\) 0 0
\(322\) −1.38445 + 0.263748i −0.0771525 + 0.0146981i
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) −8.49828 + 1.61899i −0.470676 + 0.0896673i
\(327\) 0 0
\(328\) −9.27674 + 5.87768i −0.512222 + 0.324540i
\(329\) −9.46563 −0.521857
\(330\) 0 0
\(331\) 11.0518i 0.607460i −0.952758 0.303730i \(-0.901768\pi\)
0.952758 0.303730i \(-0.0982320\pi\)
\(332\) 10.9414 4.32582i 0.600486 0.237410i
\(333\) 0 0
\(334\) 1.60256 + 8.41205i 0.0876881 + 0.460287i
\(335\) 0 0
\(336\) 0 0
\(337\) 19.9931i 1.08909i 0.838730 + 0.544547i \(0.183299\pi\)
−0.838730 + 0.544547i \(0.816701\pi\)
\(338\) −24.8384 + 4.73190i −1.35103 + 0.257382i
\(339\) 0 0
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 12.3449i 0.666563i
\(344\) −4.49828 + 2.85008i −0.242531 + 0.153666i
\(345\) 0 0
\(346\) −23.4492 + 4.46725i −1.26064 + 0.240161i
\(347\) 6.87930 0.369300 0.184650 0.982804i \(-0.440885\pi\)
0.184650 + 0.982804i \(0.440885\pi\)
\(348\) 0 0
\(349\) 4.76203i 0.254906i 0.991845 + 0.127453i \(0.0406801\pi\)
−0.991845 + 0.127453i \(0.959320\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 20.5681 + 14.9820i 1.09628 + 0.798541i
\(353\) 3.79145i 0.201798i 0.994897 + 0.100899i \(0.0321720\pi\)
−0.994897 + 0.100899i \(0.967828\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −7.65775 + 3.02760i −0.405860 + 0.160462i
\(357\) 0 0
\(358\) −2.80949 14.7474i −0.148486 0.779425i
\(359\) −12.9966 −0.685932 −0.342966 0.939348i \(-0.611432\pi\)
−0.342966 + 0.939348i \(0.611432\pi\)
\(360\) 0 0
\(361\) 17.8793 0.941016
\(362\) −3.73625 19.6121i −0.196373 1.03079i
\(363\) 0 0
\(364\) −3.84664 9.72938i −0.201619 0.509958i
\(365\) 0 0
\(366\) 0 0
\(367\) 22.9345i 1.19717i −0.801059 0.598585i \(-0.795730\pi\)
0.801059 0.598585i \(-0.204270\pi\)
\(368\) 2.89572 + 3.08967i 0.150950 + 0.161060i
\(369\) 0 0
\(370\) 0 0
\(371\) 1.88273i 0.0977467i
\(372\) 0 0
\(373\) −15.4396 −0.799435 −0.399717 0.916638i \(-0.630892\pi\)
−0.399717 + 0.916638i \(0.630892\pi\)
\(374\) −47.2242 + 8.99656i −2.44191 + 0.465201i
\(375\) 0 0
\(376\) 15.2215 + 24.0242i 0.784991 + 1.23895i
\(377\) 11.1138i 0.572391i
\(378\) 0 0
\(379\) 6.28973i 0.323082i 0.986866 + 0.161541i \(0.0516463\pi\)
−0.986866 + 0.161541i \(0.948354\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −11.1138 + 2.11727i −0.568633 + 0.108329i
\(383\) 2.94137i 0.150297i −0.997172 0.0751484i \(-0.976057\pi\)
0.997172 0.0751484i \(-0.0239431\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1.29135 + 6.77846i 0.0657278 + 0.345014i
\(387\) 0 0
\(388\) −12.5845 31.8302i −0.638882 1.61593i
\(389\) 12.2277i 0.619967i 0.950742 + 0.309983i \(0.100324\pi\)
−0.950742 + 0.309983i \(0.899676\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) −14.6073 + 9.25511i −0.737782 + 0.467453i
\(393\) 0 0
\(394\) −4.00955 + 0.763849i −0.201998 + 0.0384822i
\(395\) 0 0
\(396\) 0 0
\(397\) −5.32238 −0.267123 −0.133561 0.991041i \(-0.542641\pi\)
−0.133561 + 0.991041i \(0.542641\pi\)
\(398\) −24.5535 + 4.67762i −1.23075 + 0.234468i
\(399\) 0 0
\(400\) 0 0
\(401\) −6.99656 −0.349392 −0.174696 0.984622i \(-0.555894\pi\)
−0.174696 + 0.984622i \(0.555894\pi\)
\(402\) 0 0
\(403\) 19.7655 0.984588
\(404\) −1.47068 3.71982i −0.0731692 0.185068i
\(405\) 0 0
\(406\) 2.61555 0.498281i 0.129807 0.0247293i
\(407\) 33.4656i 1.65883i
\(408\) 0 0
\(409\) −16.2277 −0.802406 −0.401203 0.915989i \(-0.631408\pi\)
−0.401203 + 0.915989i \(0.631408\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 7.48024 + 18.9199i 0.368525 + 0.932116i
\(413\) −8.00000 −0.393654
\(414\) 0 0
\(415\) 0 0
\(416\) −18.5078 + 25.4086i −0.907421 + 1.24576i
\(417\) 0 0
\(418\) 6.61555 1.26031i 0.323577 0.0616438i
\(419\) 15.6121i 0.762701i −0.924430 0.381351i \(-0.875459\pi\)
0.924430 0.381351i \(-0.124541\pi\)
\(420\) 0 0
\(421\) 33.2311i 1.61958i 0.586717 + 0.809792i \(0.300420\pi\)
−0.586717 + 0.809792i \(0.699580\pi\)
\(422\) 6.33537 + 33.2553i 0.308401 + 1.61884i
\(423\) 0 0
\(424\) −4.77846 + 3.02760i −0.232062 + 0.147033i
\(425\) 0 0
\(426\) 0 0
\(427\) 8.46907 0.409847
\(428\) 32.0483 12.6707i 1.54911 0.612463i
\(429\) 0 0
\(430\) 0 0
\(431\) 12.9966 0.626022 0.313011 0.949749i \(-0.398662\pi\)
0.313011 + 0.949749i \(0.398662\pi\)
\(432\) 0 0
\(433\) 20.2277i 0.972079i 0.873937 + 0.486040i \(0.161559\pi\)
−0.873937 + 0.486040i \(0.838441\pi\)
\(434\) 0.886172 + 4.65164i 0.0425376 + 0.223286i
\(435\) 0 0
\(436\) −1.38445 3.50172i −0.0663033 0.167702i
\(437\) 1.12070 0.0536106
\(438\) 0 0
\(439\) 5.43965 0.259620 0.129810 0.991539i \(-0.458563\pi\)
0.129810 + 0.991539i \(0.458563\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −11.1138 58.3380i −0.528631 2.77486i
\(443\) −15.3484 −0.729223 −0.364611 0.931160i \(-0.618798\pi\)
−0.364611 + 0.931160i \(0.618798\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 6.36641 + 33.4182i 0.301458 + 1.58240i
\(447\) 0 0
\(448\) −6.80949 3.21649i −0.321718 0.151965i
\(449\) 4.22766 0.199515 0.0997577 0.995012i \(-0.468193\pi\)
0.0997577 + 0.995012i \(0.468193\pi\)
\(450\) 0 0
\(451\) 17.4656i 0.822424i
\(452\) −11.2672 28.4983i −0.529964 1.34045i
\(453\) 0 0
\(454\) 15.4396 2.94137i 0.724619 0.138045i
\(455\) 0 0
\(456\) 0 0
\(457\) 2.65164i 0.124038i −0.998075 0.0620192i \(-0.980246\pi\)
0.998075 0.0620192i \(-0.0197540\pi\)
\(458\) −4.56035 23.9379i −0.213091 1.11855i
\(459\) 0 0
\(460\) 0 0
\(461\) 10.2345i 0.476670i 0.971183 + 0.238335i \(0.0766016\pi\)
−0.971183 + 0.238335i \(0.923398\pi\)
\(462\) 0 0
\(463\) 19.0586i 0.885730i 0.896588 + 0.442865i \(0.146038\pi\)
−0.896588 + 0.442865i \(0.853962\pi\)
\(464\) −5.47068 5.83709i −0.253970 0.270980i
\(465\) 0 0
\(466\) −2.23453 11.7294i −0.103513 0.543353i
\(467\) 4.11039 0.190206 0.0951031 0.995467i \(-0.469682\pi\)
0.0951031 + 0.995467i \(0.469682\pi\)
\(468\) 0 0
\(469\) 3.76547i 0.173873i
\(470\) 0 0
\(471\) 0 0
\(472\) 12.8647 + 20.3043i 0.592145 + 0.934583i
\(473\) 8.46907i 0.389408i
\(474\) 0 0
\(475\) 0 0
\(476\) 13.2311 5.23109i 0.606446 0.239767i
\(477\) 0 0
\(478\) −14.0552 + 2.67762i −0.642870 + 0.122471i
\(479\) −25.2311 −1.15284 −0.576419 0.817154i \(-0.695550\pi\)
−0.576419 + 0.817154i \(0.695550\pi\)
\(480\) 0 0
\(481\) −41.3415 −1.88501
\(482\) −23.4492 + 4.46725i −1.06808 + 0.203477i
\(483\) 0 0
\(484\) −17.1754 + 6.79054i −0.780701 + 0.308661i
\(485\) 0 0
\(486\) 0 0
\(487\) 21.9379i 0.994102i 0.867721 + 0.497051i \(0.165584\pi\)
−0.867721 + 0.497051i \(0.834416\pi\)
\(488\) −13.6190 21.4948i −0.616502 0.973026i
\(489\) 0 0
\(490\) 0 0
\(491\) 7.50172i 0.338548i 0.985569 + 0.169274i \(0.0541423\pi\)
−0.985569 + 0.169274i \(0.945858\pi\)
\(492\) 0 0
\(493\) 15.1138 0.680693
\(494\) 1.55691 + 8.17246i 0.0700489 + 0.367696i
\(495\) 0 0
\(496\) 10.3810 9.72938i 0.466121 0.436862i
\(497\) 12.2345i 0.548794i
\(498\) 0 0
\(499\) 29.1690i 1.30578i 0.757451 + 0.652892i \(0.226445\pi\)
−0.757451 + 0.652892i \(0.773555\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.13531 + 16.4577i 0.139936 + 0.734543i
\(503\) 23.9379i 1.06734i 0.845693 + 0.533670i \(0.179187\pi\)
−0.845693 + 0.533670i \(0.820813\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.61555 + 1.26031i −0.294097 + 0.0560276i
\(507\) 0 0
\(508\) 13.3630 + 33.7992i 0.592886 + 1.49960i
\(509\) 28.6967i 1.27196i 0.771706 + 0.635980i \(0.219404\pi\)
−0.771706 + 0.635980i \(0.780596\pi\)
\(510\) 0 0
\(511\) −5.64820 −0.249862
\(512\) 2.78667 + 22.4552i 0.123155 + 0.992387i
\(513\) 0 0
\(514\) 2.82410 + 14.8241i 0.124566 + 0.653863i
\(515\) 0 0
\(516\) 0 0
\(517\) −45.2311 −1.98926
\(518\) −1.85352 9.72938i −0.0814389 0.427484i
\(519\) 0 0
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) −25.7586 −1.12634 −0.563172 0.826340i \(-0.690419\pi\)
−0.563172 + 0.826340i \(0.690419\pi\)
\(524\) 4.69223 + 11.8681i 0.204981 + 0.518461i
\(525\) 0 0
\(526\) −0.514709 2.70178i −0.0224424 0.117803i
\(527\) 26.8793i 1.17088i
\(528\) 0 0
\(529\) 21.8793 0.951274
\(530\) 0 0
\(531\) 0 0
\(532\) −1.85352 + 0.732814i −0.0803602 + 0.0317715i
\(533\) 21.5760 0.934561
\(534\) 0 0
\(535\) 0 0
\(536\) 9.55691 6.05520i 0.412796 0.261545i
\(537\) 0 0
\(538\) −2.58451 13.5665i −0.111426 0.584892i
\(539\) 27.5017i 1.18458i
\(540\) 0 0
\(541\) 12.3449i 0.530750i 0.964145 + 0.265375i \(0.0854957\pi\)
−0.964145 + 0.265375i \(0.914504\pi\)
\(542\) −4.78801 + 0.912151i −0.205663 + 0.0391802i
\(543\) 0 0
\(544\) −34.5535 25.1690i −1.48147 1.07911i
\(545\) 0 0
\(546\) 0 0
\(547\) −19.8759 −0.849830 −0.424915 0.905233i \(-0.639696\pi\)
−0.424915 + 0.905233i \(0.639696\pi\)
\(548\) −3.26719 8.26375i −0.139567 0.353010i
\(549\) 0 0
\(550\) 0 0
\(551\) −2.11727 −0.0901986
\(552\) 0 0
\(553\) 10.8793i 0.462635i
\(554\) 26.1008 4.97240i 1.10892 0.211257i
\(555\) 0 0
\(556\) −14.8337 37.5190i −0.629087 1.59116i
\(557\) −3.12070 −0.132228 −0.0661142 0.997812i \(-0.521060\pi\)
−0.0661142 + 0.997812i \(0.521060\pi\)
\(558\) 0 0
\(559\) 10.4622 0.442503
\(560\) 0 0
\(561\) 0 0
\(562\) −23.4492 + 4.46725i −0.989145 + 0.188439i
\(563\) −0.651639 −0.0274633 −0.0137317 0.999906i \(-0.504371\pi\)
−0.0137317 + 0.999906i \(0.504371\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 27.7846 5.29317i 1.16787 0.222488i
\(567\) 0 0
\(568\) −31.0518 + 19.6742i −1.30290 + 0.825510i
\(569\) 26.9966 1.13175 0.565877 0.824489i \(-0.308538\pi\)
0.565877 + 0.824489i \(0.308538\pi\)
\(570\) 0 0
\(571\) 14.9414i 0.625277i 0.949872 + 0.312638i \(0.101213\pi\)
−0.949872 + 0.312638i \(0.898787\pi\)
\(572\) −18.3810 46.4914i −0.768549 1.94390i
\(573\) 0 0
\(574\) 0.967346 + 5.07774i 0.0403763 + 0.211941i
\(575\) 0 0
\(576\) 0 0
\(577\) 8.87930i 0.369650i 0.982771 + 0.184825i \(0.0591718\pi\)
−0.982771 + 0.184825i \(0.940828\pi\)
\(578\) 55.7177 10.6146i 2.31755 0.441511i
\(579\) 0 0
\(580\) 0 0
\(581\) 5.53781i 0.229747i
\(582\) 0 0
\(583\) 8.99656i 0.372600i
\(584\) 9.08279 + 14.3354i 0.375849 + 0.593202i
\(585\) 0 0
\(586\) 28.1008 5.35342i 1.16083 0.221148i
\(587\) −1.23109 −0.0508127 −0.0254064 0.999677i \(-0.508088\pi\)
−0.0254064 + 0.999677i \(0.508088\pi\)
\(588\) 0 0
\(589\) 3.76547i 0.155153i
\(590\) 0 0
\(591\) 0 0
\(592\) −21.7129 + 20.3500i −0.892397 + 0.836379i
\(593\) 3.55691i 0.146065i −0.997330 0.0730325i \(-0.976732\pi\)
0.997330 0.0730325i \(-0.0232677\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.47068 3.71982i −0.0602415 0.152370i
\(597\) 0 0
\(598\) −1.55691 8.17246i −0.0636670 0.334197i
\(599\) −19.2242 −0.785480 −0.392740 0.919649i \(-0.628473\pi\)
−0.392740 + 0.919649i \(0.628473\pi\)
\(600\) 0 0
\(601\) −27.7586 −1.13230 −0.566148 0.824303i \(-0.691567\pi\)
−0.566148 + 0.824303i \(0.691567\pi\)
\(602\) 0.469065 + 2.46219i 0.0191177 + 0.100351i
\(603\) 0 0
\(604\) 17.9931 7.11383i 0.732130 0.289458i
\(605\) 0 0
\(606\) 0 0
\(607\) 7.16902i 0.290982i −0.989360 0.145491i \(-0.953524\pi\)
0.989360 0.145491i \(-0.0464761\pi\)
\(608\) 4.84053 + 3.52588i 0.196309 + 0.142993i
\(609\) 0 0
\(610\) 0 0
\(611\) 55.8759i 2.26050i
\(612\) 0 0
\(613\) −9.55691 −0.386000 −0.193000 0.981199i \(-0.561822\pi\)
−0.193000 + 0.981199i \(0.561822\pi\)
\(614\) −11.2672 + 2.14648i −0.454707 + 0.0866250i
\(615\) 0 0
\(616\) 10.1173 6.41023i 0.407636 0.258276i
\(617\) 1.32926i 0.0535139i 0.999642 + 0.0267569i \(0.00851802\pi\)
−0.999642 + 0.0267569i \(0.991482\pi\)
\(618\) 0 0
\(619\) 28.1725i 1.13235i −0.824286 0.566173i \(-0.808423\pi\)
0.824286 0.566173i \(-0.191577\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 44.2829 8.43621i 1.77558 0.338261i
\(623\) 3.87586i 0.155283i
\(624\) 0 0
\(625\) 0 0
\(626\) 1.35342 + 7.10428i 0.0540934 + 0.283944i
\(627\) 0 0
\(628\) 8.04564 3.18096i 0.321056 0.126934i
\(629\) 56.2208i 2.24167i
\(630\) 0 0
\(631\) −23.3224 −0.928449 −0.464225 0.885717i \(-0.653667\pi\)
−0.464225 + 0.885717i \(0.653667\pi\)
\(632\) 27.6121 17.4948i 1.09835 0.695907i
\(633\) 0 0
\(634\) 34.2372 6.52244i 1.35973 0.259039i
\(635\) 0 0
\(636\) 0 0
\(637\) 33.9740 1.34610
\(638\) 12.4983 2.38101i 0.494812 0.0942653i
\(639\) 0 0
\(640\) 0 0
\(641\) −27.1070 −1.07066 −0.535330 0.844643i \(-0.679813\pi\)
−0.535330 + 0.844643i \(0.679813\pi\)
\(642\) 0 0
\(643\) −20.3449 −0.802325 −0.401163 0.916007i \(-0.631394\pi\)
−0.401163 + 0.916007i \(0.631394\pi\)
\(644\) 1.85352 0.732814i 0.0730388 0.0288769i
\(645\) 0 0
\(646\) −11.1138 + 2.11727i −0.437268 + 0.0833027i
\(647\) 37.6965i 1.48200i 0.671503 + 0.741002i \(0.265649\pi\)
−0.671503 + 0.741002i \(0.734351\pi\)
\(648\) 0 0
\(649\) −38.2277 −1.50057
\(650\) 0 0
\(651\) 0 0
\(652\) 11.3776 4.49828i 0.445580 0.176166i
\(653\) −8.64476 −0.338296 −0.169148 0.985591i \(-0.554102\pi\)
−0.169148 + 0.985591i \(0.554102\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 11.3319 10.6206i 0.442438 0.414665i
\(657\) 0 0
\(658\) 13.1499 2.50516i 0.512637 0.0976612i
\(659\) 29.2603i 1.13982i 0.821707 + 0.569910i \(0.193022\pi\)
−0.821707 + 0.569910i \(0.806978\pi\)
\(660\) 0 0
\(661\) 28.7620i 1.11871i 0.828927 + 0.559357i \(0.188952\pi\)
−0.828927 + 0.559357i \(0.811048\pi\)
\(662\) 2.92494 + 15.3534i 0.113681 + 0.596728i
\(663\) 0 0
\(664\) −14.0552 + 8.90528i −0.545447 + 0.345592i
\(665\) 0 0
\(666\) 0 0
\(667\) 2.11727 0.0819809
\(668\) −4.45264 11.2621i −0.172278 0.435745i
\(669\) 0 0
\(670\) 0 0
\(671\) 40.4691 1.56229
\(672\) 0 0
\(673\) 18.0000i 0.693849i −0.937893 0.346925i \(-0.887226\pi\)
0.937893 0.346925i \(-0.112774\pi\)
\(674\) −5.29135 27.7750i −0.203815 1.06985i
\(675\) 0 0
\(676\) 33.2539 13.1474i 1.27900 0.505669i
\(677\) 42.8724 1.64772 0.823860 0.566793i \(-0.191816\pi\)
0.823860 + 0.566793i \(0.191816\pi\)
\(678\) 0 0
\(679\) −16.1104 −0.618260
\(680\) 0 0
\(681\) 0 0
\(682\) 4.23453 + 22.2277i 0.162149 + 0.851141i
\(683\) −26.1173 −0.999349 −0.499675 0.866213i \(-0.666547\pi\)
−0.499675 + 0.866213i \(0.666547\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 3.26719 + 17.1499i 0.124742 + 0.654787i
\(687\) 0 0
\(688\) 5.49484 5.14992i 0.209489 0.196339i
\(689\) 11.1138 0.423403
\(690\) 0 0
\(691\) 5.29317i 0.201362i −0.994919 0.100681i \(-0.967898\pi\)
0.994919 0.100681i \(-0.0321021\pi\)
\(692\) 31.3940 12.4121i 1.19342 0.471835i
\(693\) 0 0
\(694\) −9.55691 + 1.82066i −0.362776 + 0.0691114i
\(695\) 0 0
\(696\) 0 0
\(697\) 29.3415i 1.11139i
\(698\) −1.26031 6.61555i −0.0477035 0.250402i
\(699\) 0 0
\(700\) 0 0
\(701\) 7.99312i 0.301896i −0.988542 0.150948i \(-0.951767\pi\)
0.988542 0.150948i \(-0.0482326\pi\)
\(702\) 0 0
\(703\) 7.87586i 0.297044i
\(704\) −32.5389 15.3698i −1.22635 0.579273i
\(705\) 0 0
\(706\) −1.00344 5.26719i −0.0377649 0.198233i
\(707\) −1.88273 −0.0708075
\(708\) 0 0
\(709\) 28.9966i 1.08899i 0.838764 + 0.544494i \(0.183278\pi\)
−0.838764 + 0.544494i \(0.816722\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.83709 6.23271i 0.368661 0.233581i
\(713\) 3.76547i 0.141018i
\(714\) 0 0
\(715\) 0 0
\(716\) 7.80605 + 19.7440i 0.291726 + 0.737867i
\(717\) 0 0
\(718\) 18.0552 3.43965i 0.673814 0.128367i
\(719\) 26.8793 1.00243 0.501214 0.865323i \(-0.332887\pi\)
0.501214 + 0.865323i \(0.332887\pi\)
\(720\) 0 0
\(721\) 9.57602 0.356630
\(722\) −24.8384 + 4.73190i −0.924391 + 0.176103i
\(723\) 0 0
\(724\) 10.3810 + 26.2569i 0.385807 + 0.975829i
\(725\) 0 0
\(726\) 0 0
\(727\) 41.8138i 1.55079i −0.631478 0.775394i \(-0.717551\pi\)
0.631478 0.775394i \(-0.282449\pi\)
\(728\) 7.91883 + 12.4983i 0.293491 + 0.463217i
\(729\) 0 0
\(730\) 0 0
\(731\) 14.2277i 0.526229i
\(732\) 0 0
\(733\) −30.0844 −1.11119 −0.555597 0.831452i \(-0.687510\pi\)
−0.555597 + 0.831452i \(0.687510\pi\)
\(734\) 6.06980 + 31.8613i 0.224041 + 1.17602i
\(735\) 0 0
\(736\) −4.84053 3.52588i −0.178424 0.129966i
\(737\) 17.9931i 0.662785i
\(738\) 0 0
\(739\) 29.0449i 1.06843i −0.845348 0.534217i \(-0.820607\pi\)
0.845348 0.534217i \(-0.179393\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0.498281 + 2.61555i 0.0182925 + 0.0960198i
\(743\) 43.2863i 1.58802i 0.607905 + 0.794010i \(0.292010\pi\)
−0.607905 + 0.794010i \(0.707990\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 21.4492 4.08623i 0.785311 0.149608i
\(747\) 0 0
\(748\) 63.2242 24.9966i 2.31171 0.913965i
\(749\) 16.2208i 0.592694i
\(750\) 0 0
\(751\) 41.7846 1.52474 0.762370 0.647141i \(-0.224036\pi\)
0.762370 + 0.647141i \(0.224036\pi\)
\(752\) −27.5044 29.3465i −1.00298 1.07016i
\(753\) 0 0
\(754\) 2.94137 + 15.4396i 0.107118 + 0.562279i
\(755\) 0 0
\(756\) 0 0
\(757\) −16.3258 −0.593372 −0.296686 0.954975i \(-0.595881\pi\)
−0.296686 + 0.954975i \(0.595881\pi\)
\(758\) −1.66463 8.73787i −0.0604620 0.317374i
\(759\) 0 0
\(760\) 0 0
\(761\) −50.2208 −1.82050 −0.910251 0.414057i \(-0.864111\pi\)
−0.910251 + 0.414057i \(0.864111\pi\)
\(762\) 0 0
\(763\) −1.77234 −0.0641631
\(764\) 14.8793 5.88273i 0.538314 0.212830i
\(765\) 0 0
\(766\) 0.778457 + 4.08623i 0.0281268 + 0.147642i
\(767\) 47.2242i 1.70517i
\(768\) 0 0
\(769\) 31.3415 1.13020 0.565101 0.825021i \(-0.308837\pi\)
0.565101 + 0.825021i \(0.308837\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.58795 9.07506i −0.129133 0.326619i
\(773\) −9.11383 −0.327802 −0.163901 0.986477i \(-0.552408\pi\)
−0.163901 + 0.986477i \(0.552408\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 25.9069 + 40.8888i 0.930003 + 1.46782i
\(777\) 0 0
\(778\) −3.23615 16.9870i −0.116022 0.609014i
\(779\) 4.11039i 0.147270i
\(780\) 0 0
\(781\) 58.4622i 2.09194i
\(782\) 11.1138 2.11727i 0.397430 0.0757133i
\(783\) 0 0
\(784\) 17.8435 16.7234i 0.637267 0.597265i
\(785\) 0 0
\(786\) 0 0
\(787\) 36.2208 1.29113 0.645566 0.763705i \(-0.276622\pi\)
0.645566 + 0.763705i \(0.276622\pi\)
\(788\) 5.36802 2.12232i 0.191228 0.0756046i
\(789\) 0 0
\(790\) 0 0
\(791\) −14.4240 −0.512858
\(792\) 0 0
\(793\) 49.9931i 1.77531i
\(794\) 7.39400 1.40861i 0.262403 0.0499898i
\(795\) 0 0
\(796\) 32.8724 12.9966i 1.16513 0.460651i
\(797\) −10.0000 −0.354218 −0.177109 0.984191i \(-0.556675\pi\)
−0.177109 + 0.984191i \(0.556675\pi\)
\(798\) 0 0
\(799\) 75.9862 2.68820
\(800\) 0 0
\(801\) 0 0
\(802\) 9.71982 1.85170i 0.343219 0.0653857i
\(803\) −26.9897 −0.952445
\(804\) 0 0
\(805\) 0 0
\(806\) −27.4588 + 5.23109i −0.967193 + 0.184257i
\(807\) 0 0
\(808\) 3.02760 + 4.77846i 0.106511 + 0.168106i
\(809\) 47.5760 1.67268 0.836342 0.548208i \(-0.184690\pi\)
0.836342 + 0.548208i \(0.184690\pi\)
\(810\) 0 0
\(811\) 20.5174i 0.720463i 0.932863 + 0.360231i \(0.117302\pi\)
−0.932863 + 0.360231i \(0.882698\pi\)
\(812\) −3.50172 + 1.38445i −0.122886 + 0.0485848i
\(813\) 0 0
\(814\) −8.85696 46.4914i −0.310436 1.62952i
\(815\) 0 0
\(816\) 0 0
\(817\) 1.99312i 0.0697306i
\(818\) 22.5439 4.29478i 0.788230 0.150164i
\(819\) 0 0
\(820\) 0 0
\(821\) 44.4622i 1.55174i −0.630892 0.775871i \(-0.717311\pi\)
0.630892 0.775871i \(-0.282689\pi\)
\(822\) 0 0
\(823\) 32.1656i 1.12122i 0.828079 + 0.560611i \(0.189434\pi\)
−0.828079 + 0.560611i \(0.810566\pi\)
\(824\) −15.3991 24.3043i −0.536452 0.846682i
\(825\) 0 0
\(826\) 11.1138 2.11727i 0.386700 0.0736691i
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) 33.8827i 1.17680i −0.808571 0.588398i \(-0.799759\pi\)
0.808571 0.588398i \(-0.200241\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 18.9870 40.1966i 0.658256 1.39357i
\(833\) 46.2017i 1.60079i
\(834\) 0 0
\(835\) 0 0
\(836\) −8.85696 + 3.50172i −0.306324 + 0.121109i
\(837\) 0 0
\(838\) 4.13187 + 21.6888i 0.142733 + 0.749227i
\(839\) 4.52750 0.156307 0.0781533 0.996941i \(-0.475098\pi\)
0.0781533 + 0.996941i \(0.475098\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) −8.79488 46.1656i −0.303092 1.59097i
\(843\) 0 0
\(844\) −17.6026 44.5224i −0.605905 1.53253i
\(845\) 0 0
\(846\) 0 0
\(847\) 8.69308i 0.298698i
\(848\) 5.83709 5.47068i 0.200447 0.187864i
\(849\) 0 0
\(850\) 0 0
\(851\) 7.87586i 0.269981i
\(852\) 0 0
\(853\) 50.4293 1.72667 0.863334 0.504633i \(-0.168372\pi\)
0.863334 + 0.504633i \(0.168372\pi\)
\(854\) −11.7655 + 2.24141i −0.402606 + 0.0766994i
\(855\) 0 0
\(856\) −41.1690 + 26.0844i −1.40713 + 0.891547i
\(857\) 26.4362i 0.903044i 0.892260 + 0.451522i \(0.149119\pi\)
−0.892260 + 0.451522i \(0.850881\pi\)
\(858\) 0 0
\(859\) 0.406994i 0.0138865i −0.999976 0.00694323i \(-0.997790\pi\)
0.999976 0.00694323i \(-0.00221012\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −18.0552 + 3.43965i −0.614962 + 0.117155i
\(863\) 29.9311i 1.01886i 0.860511 + 0.509432i \(0.170145\pi\)
−0.860511 + 0.509432i \(0.829855\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −5.35342 28.1008i −0.181917 0.954905i
\(867\) 0 0
\(868\) −2.46219 6.22766i −0.0835722 0.211380i
\(869\) 51.9862i 1.76351i
\(870\) 0 0
\(871\) −22.2277 −0.753155
\(872\) 2.85008 + 4.49828i 0.0965159 + 0.152331i
\(873\) 0 0
\(874\) −1.55691 + 0.296604i −0.0526634 + 0.0100328i
\(875\) 0 0
\(876\) 0 0
\(877\) −11.2051 −0.378370 −0.189185 0.981941i \(-0.560585\pi\)
−0.189185 + 0.981941i \(0.560585\pi\)
\(878\) −7.55691 + 1.43965i −0.255034 + 0.0485858i
\(879\) 0 0
\(880\) 0 0
\(881\) 48.3380 1.62855 0.814275 0.580479i \(-0.197135\pi\)
0.814275 + 0.580479i \(0.197135\pi\)
\(882\) 0 0
\(883\) 50.5726 1.70190 0.850951 0.525244i \(-0.176026\pi\)
0.850951 + 0.525244i \(0.176026\pi\)
\(884\) 30.8793 + 78.1035i 1.03858 + 2.62691i
\(885\) 0 0
\(886\) 21.3224 4.06207i 0.716339 0.136468i
\(887\) 48.0483i 1.61330i −0.591026 0.806652i \(-0.701277\pi\)
0.591026 0.806652i \(-0.298723\pi\)
\(888\) 0 0
\(889\) 17.1070 0.573749
\(890\) 0 0
\(891\) 0 0
\(892\) −17.6888 44.7405i −0.592264 1.49802i
\(893\) −10.6448 −0.356213
\(894\) 0 0
\(895\) 0 0
\(896\) 10.3112 + 2.66625i 0.344473 + 0.0890731i
\(897\) 0 0
\(898\) −5.87318 + 1.11888i −0.195991 + 0.0373377i
\(899\) 7.11383i 0.237259i
\(900\) 0 0
\(901\) 15.1138i 0.503515i
\(902\) 4.62242 + 24.2637i 0.153910 + 0.807894i
\(903\) 0 0
\(904\) 23.1950 + 36.6087i 0.771454 + 1.21759i
\(905\) 0 0
\(906\) 0 0
\(907\) −6.46219 −0.214573 −0.107287 0.994228i \(-0.534216\pi\)
−0.107287 + 0.994228i \(0.534216\pi\)
\(908\) −20.6707 + 8.17246i −0.685983 + 0.271213i
\(909\) 0 0
\(910\) 0 0
\(911\) −50.3380 −1.66777 −0.833887 0.551935i \(-0.813890\pi\)
−0.833887 + 0.551935i \(0.813890\pi\)
\(912\) 0 0
\(913\) 26.4622i 0.875771i
\(914\) 0.701778 + 3.68373i 0.0232128 + 0.121847i
\(915\) 0 0
\(916\) 12.6707 + 32.0483i 0.418653 + 1.05891i
\(917\) 6.00688 0.198365
\(918\) 0 0
\(919\) −46.4362 −1.53179 −0.765895 0.642966i \(-0.777704\pi\)
−0.765895 + 0.642966i \(0.777704\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −2.70865 14.2181i −0.0892048 0.468248i
\(923\) 72.2208 2.37718
\(924\) 0 0
\(925\) 0 0
\(926\) −5.04403 26.4768i −0.165757 0.870082i
\(927\) 0 0
\(928\) 9.14486 + 6.66119i 0.300195 + 0.218664i
\(929\) −35.9931 −1.18090 −0.590448 0.807076i \(-0.701049\pi\)
−0.590448 + 0.807076i \(0.701049\pi\)
\(930\) 0 0
\(931\) 6.47230i 0.212121i
\(932\) 6.20855 + 15.7034i 0.203368 + 0.514382i
\(933\) 0 0
\(934\) −5.71027 + 1.08785i −0.186846 + 0.0355955i
\(935\) 0 0
\(936\) 0 0
\(937\) 2.70360i 0.0883227i −0.999024 0.0441613i \(-0.985938\pi\)
0.999024 0.0441613i \(-0.0140616\pi\)
\(938\) −0.996562 5.23109i −0.0325389 0.170801i
\(939\) 0 0
\(940\) 0 0
\(941\) 17.7655i 0.579138i 0.957157 + 0.289569i \(0.0935119\pi\)
−0.957157 + 0.289569i \(0.906488\pi\)
\(942\) 0 0
\(943\) 4.11039i 0.133853i
\(944\) −23.2457 24.8026i −0.756583 0.807256i
\(945\) 0 0
\(946\) 2.24141 + 11.7655i 0.0728745 + 0.382528i
\(947\) 26.2277 0.852284 0.426142 0.904656i \(-0.359872\pi\)
0.426142 + 0.904656i \(0.359872\pi\)
\(948\) 0 0
\(949\) 33.3415i 1.08231i
\(950\) 0 0
\(951\) 0 0
\(952\) −16.9966 + 10.7689i −0.550862 + 0.349022i
\(953\) 9.09472i 0.294607i −0.989091 0.147304i \(-0.952941\pi\)
0.989091 0.147304i \(-0.0470594\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 18.8172 7.43965i 0.608593 0.240615i
\(957\) 0 0
\(958\) 35.0518 6.67762i 1.13247 0.215744i
\(959\) −4.18257 −0.135062
\(960\) 0 0
\(961\) −18.3484 −0.591883
\(962\) 57.4328 10.9414i 1.85171 0.352764i
\(963\) 0 0
\(964\) 31.3940 12.4121i 1.01113 0.399765i
\(965\) 0 0
\(966\) 0 0
\(967\) 7.47574i 0.240404i 0.992749 + 0.120202i \(0.0383542\pi\)
−0.992749 + 0.120202i \(0.961646\pi\)
\(968\) 22.0634 13.9792i 0.709145 0.449309i
\(969\) 0 0
\(970\) 0 0
\(971\) 41.0777i 1.31825i 0.752035 + 0.659124i \(0.229073\pi\)
−0.752035 + 0.659124i \(0.770927\pi\)
\(972\) 0 0
\(973\) −18.9897 −0.608781
\(974\) −5.80605 30.4768i −0.186038 0.976540i
\(975\) 0 0
\(976\) 24.6087 + 26.2569i 0.787704 + 0.840462i
\(977\) 4.20855i 0.134644i −0.997731 0.0673218i \(-0.978555\pi\)
0.997731 0.0673218i \(-0.0214454\pi\)
\(978\) 0 0
\(979\) 18.5206i 0.591922i
\(980\) 0 0
\(981\) 0 0
\(982\) −1.98539 10.4216i −0.0633564 0.332567i
\(983\) 8.35504i 0.266484i 0.991084 + 0.133242i \(0.0425388\pi\)
−0.991084 + 0.133242i \(0.957461\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −20.9966 + 4.00000i −0.668667 + 0.127386i
\(987\) 0 0
\(988\) −4.32582 10.9414i −0.137623 0.348091i
\(989\) 1.99312i 0.0633777i
\(990\) 0 0
\(991\) −13.9087 −0.441825 −0.220912 0.975294i \(-0.570903\pi\)
−0.220912 + 0.975294i \(0.570903\pi\)
\(992\) −11.8466 + 16.2637i −0.376131 + 0.516375i
\(993\) 0 0
\(994\) 3.23797 + 16.9966i 0.102702 + 0.539098i
\(995\) 0 0
\(996\) 0 0
\(997\) 34.8984 1.10524 0.552622 0.833432i \(-0.313627\pi\)
0.552622 + 0.833432i \(0.313627\pi\)
\(998\) −7.71982 40.5224i −0.244367 1.28272i
\(999\) 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 1800.2.d.q.1549.2 6
3.2 odd 2 600.2.d.e.349.5 6
4.3 odd 2 7200.2.d.q.2449.4 6
5.2 odd 4 360.2.k.f.181.3 6
5.3 odd 4 1800.2.k.p.901.4 6
5.4 even 2 1800.2.d.r.1549.5 6
8.3 odd 2 7200.2.d.r.2449.4 6
8.5 even 2 1800.2.d.r.1549.6 6
12.11 even 2 2400.2.d.f.49.4 6
15.2 even 4 120.2.k.b.61.4 yes 6
15.8 even 4 600.2.k.c.301.3 6
15.14 odd 2 600.2.d.f.349.2 6
20.3 even 4 7200.2.k.p.3601.4 6
20.7 even 4 1440.2.k.f.721.2 6
20.19 odd 2 7200.2.d.r.2449.3 6
24.5 odd 2 600.2.d.f.349.1 6
24.11 even 2 2400.2.d.e.49.4 6
40.3 even 4 7200.2.k.p.3601.3 6
40.13 odd 4 1800.2.k.p.901.3 6
40.19 odd 2 7200.2.d.q.2449.3 6
40.27 even 4 1440.2.k.f.721.5 6
40.29 even 2 inner 1800.2.d.q.1549.1 6
40.37 odd 4 360.2.k.f.181.4 6
60.23 odd 4 2400.2.k.c.1201.5 6
60.47 odd 4 480.2.k.b.241.2 6
60.59 even 2 2400.2.d.e.49.3 6
120.29 odd 2 600.2.d.e.349.6 6
120.53 even 4 600.2.k.c.301.4 6
120.59 even 2 2400.2.d.f.49.3 6
120.77 even 4 120.2.k.b.61.3 6
120.83 odd 4 2400.2.k.c.1201.2 6
120.107 odd 4 480.2.k.b.241.5 6
240.77 even 4 3840.2.a.bp.1.2 3
240.107 odd 4 3840.2.a.bo.1.2 3
240.197 even 4 3840.2.a.bq.1.2 3
240.227 odd 4 3840.2.a.br.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.k.b.61.3 6 120.77 even 4
120.2.k.b.61.4 yes 6 15.2 even 4
360.2.k.f.181.3 6 5.2 odd 4
360.2.k.f.181.4 6 40.37 odd 4
480.2.k.b.241.2 6 60.47 odd 4
480.2.k.b.241.5 6 120.107 odd 4
600.2.d.e.349.5 6 3.2 odd 2
600.2.d.e.349.6 6 120.29 odd 2
600.2.d.f.349.1 6 24.5 odd 2
600.2.d.f.349.2 6 15.14 odd 2
600.2.k.c.301.3 6 15.8 even 4
600.2.k.c.301.4 6 120.53 even 4
1440.2.k.f.721.2 6 20.7 even 4
1440.2.k.f.721.5 6 40.27 even 4
1800.2.d.q.1549.1 6 40.29 even 2 inner
1800.2.d.q.1549.2 6 1.1 even 1 trivial
1800.2.d.r.1549.5 6 5.4 even 2
1800.2.d.r.1549.6 6 8.5 even 2
1800.2.k.p.901.3 6 40.13 odd 4
1800.2.k.p.901.4 6 5.3 odd 4
2400.2.d.e.49.3 6 60.59 even 2
2400.2.d.e.49.4 6 24.11 even 2
2400.2.d.f.49.3 6 120.59 even 2
2400.2.d.f.49.4 6 12.11 even 2
2400.2.k.c.1201.2 6 120.83 odd 4
2400.2.k.c.1201.5 6 60.23 odd 4
3840.2.a.bo.1.2 3 240.107 odd 4
3840.2.a.bp.1.2 3 240.77 even 4
3840.2.a.bq.1.2 3 240.197 even 4
3840.2.a.br.1.2 3 240.227 odd 4
7200.2.d.q.2449.3 6 40.19 odd 2
7200.2.d.q.2449.4 6 4.3 odd 2
7200.2.d.r.2449.3 6 20.19 odd 2
7200.2.d.r.2449.4 6 8.3 odd 2
7200.2.k.p.3601.3 6 40.3 even 4
7200.2.k.p.3601.4 6 20.3 even 4