Properties

Label 672.2.bb.a.271.8
Level $672$
Weight $2$
Character 672.271
Analytic conductor $5.366$
Analytic rank $0$
Dimension $32$
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] = [672,2,Mod(271,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.bb (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: \(5.36594701583\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 271.8
Character \(\chi\) \(=\) 672.271
Dual form 672.2.bb.a.367.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{3} +(2.08776 - 3.61611i) q^{5} +(2.39694 - 1.12013i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{3} +(2.08776 - 3.61611i) q^{5} +(2.39694 - 1.12013i) q^{7} +(0.500000 - 0.866025i) q^{9} +(0.855485 + 1.48174i) q^{11} -1.54062 q^{13} +4.17553i q^{15} +(2.02094 - 1.16679i) q^{17} +(-6.09693 - 3.52006i) q^{19} +(-1.51575 + 2.16853i) q^{21} +(0.406066 + 0.234442i) q^{23} +(-6.21752 - 10.7691i) q^{25} +1.00000i q^{27} +3.33885i q^{29} +(-1.58126 - 2.73883i) q^{31} +(-1.48174 - 0.855485i) q^{33} +(0.953738 - 11.0062i) q^{35} +(7.74648 + 4.47243i) q^{37} +(1.33422 - 0.770311i) q^{39} +5.31411i q^{41} +3.42772 q^{43} +(-2.08776 - 3.61611i) q^{45} +(2.95047 - 5.11037i) q^{47} +(4.49063 - 5.36975i) q^{49} +(-1.16679 + 2.02094i) q^{51} +(-1.35437 + 0.781947i) q^{53} +7.14421 q^{55} +7.04013 q^{57} +(5.26742 - 3.04114i) q^{59} +(4.55959 - 7.89744i) q^{61} +(0.228411 - 2.63587i) q^{63} +(-3.21646 + 5.57107i) q^{65} +(-3.73658 - 6.47195i) q^{67} -0.468884 q^{69} +3.49263i q^{71} +(-12.5811 + 7.26372i) q^{73} +(10.7691 + 6.21752i) q^{75} +(3.71029 + 2.59340i) q^{77} +(1.46108 + 0.843557i) q^{79} +(-0.500000 - 0.866025i) q^{81} +2.72601i q^{83} -9.74391i q^{85} +(-1.66943 - 2.89153i) q^{87} +(1.83829 + 1.06134i) q^{89} +(-3.69278 + 1.72569i) q^{91} +(2.73883 + 1.58126i) q^{93} +(-25.4579 + 14.6981i) q^{95} +1.95202i q^{97} +1.71097 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{9} - 8 q^{11} - 16 q^{25} + 24 q^{35} + 16 q^{43} + 8 q^{49} + 16 q^{57} + 96 q^{59} + 32 q^{67} - 24 q^{73} - 16 q^{81} - 56 q^{91} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.866025 + 0.500000i −0.500000 + 0.288675i
\(4\) 0 0
\(5\) 2.08776 3.61611i 0.933677 1.61718i 0.156700 0.987646i \(-0.449915\pi\)
0.776977 0.629529i \(-0.216752\pi\)
\(6\) 0 0
\(7\) 2.39694 1.12013i 0.905958 0.423368i
\(8\) 0 0
\(9\) 0.500000 0.866025i 0.166667 0.288675i
\(10\) 0 0
\(11\) 0.855485 + 1.48174i 0.257939 + 0.446763i 0.965690 0.259699i \(-0.0836235\pi\)
−0.707751 + 0.706462i \(0.750290\pi\)
\(12\) 0 0
\(13\) −1.54062 −0.427292 −0.213646 0.976911i \(-0.568534\pi\)
−0.213646 + 0.976911i \(0.568534\pi\)
\(14\) 0 0
\(15\) 4.17553i 1.07812i
\(16\) 0 0
\(17\) 2.02094 1.16679i 0.490149 0.282988i −0.234487 0.972119i \(-0.575341\pi\)
0.724636 + 0.689132i \(0.242008\pi\)
\(18\) 0 0
\(19\) −6.09693 3.52006i −1.39873 0.807558i −0.404472 0.914551i \(-0.632545\pi\)
−0.994260 + 0.106992i \(0.965878\pi\)
\(20\) 0 0
\(21\) −1.51575 + 2.16853i −0.330763 + 0.473212i
\(22\) 0 0
\(23\) 0.406066 + 0.234442i 0.0846705 + 0.0488846i 0.541737 0.840548i \(-0.317767\pi\)
−0.457067 + 0.889432i \(0.651100\pi\)
\(24\) 0 0
\(25\) −6.21752 10.7691i −1.24350 2.15381i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 3.33885i 0.620009i 0.950735 + 0.310005i \(0.100331\pi\)
−0.950735 + 0.310005i \(0.899669\pi\)
\(30\) 0 0
\(31\) −1.58126 2.73883i −0.284003 0.491908i 0.688364 0.725366i \(-0.258329\pi\)
−0.972367 + 0.233458i \(0.924996\pi\)
\(32\) 0 0
\(33\) −1.48174 0.855485i −0.257939 0.148921i
\(34\) 0 0
\(35\) 0.953738 11.0062i 0.161211 1.86038i
\(36\) 0 0
\(37\) 7.74648 + 4.47243i 1.27351 + 0.735263i 0.975647 0.219345i \(-0.0703920\pi\)
0.297865 + 0.954608i \(0.403725\pi\)
\(38\) 0 0
\(39\) 1.33422 0.770311i 0.213646 0.123349i
\(40\) 0 0
\(41\) 5.31411i 0.829925i 0.909838 + 0.414963i \(0.136205\pi\)
−0.909838 + 0.414963i \(0.863795\pi\)
\(42\) 0 0
\(43\) 3.42772 0.522722 0.261361 0.965241i \(-0.415829\pi\)
0.261361 + 0.965241i \(0.415829\pi\)
\(44\) 0 0
\(45\) −2.08776 3.61611i −0.311226 0.539058i
\(46\) 0 0
\(47\) 2.95047 5.11037i 0.430371 0.745424i −0.566534 0.824038i \(-0.691716\pi\)
0.996905 + 0.0786139i \(0.0250494\pi\)
\(48\) 0 0
\(49\) 4.49063 5.36975i 0.641519 0.767107i
\(50\) 0 0
\(51\) −1.16679 + 2.02094i −0.163383 + 0.282988i
\(52\) 0 0
\(53\) −1.35437 + 0.781947i −0.186037 + 0.107409i −0.590126 0.807311i \(-0.700922\pi\)
0.404089 + 0.914720i \(0.367589\pi\)
\(54\) 0 0
\(55\) 7.14421 0.963325
\(56\) 0 0
\(57\) 7.04013 0.932488
\(58\) 0 0
\(59\) 5.26742 3.04114i 0.685759 0.395923i −0.116262 0.993219i \(-0.537091\pi\)
0.802021 + 0.597295i \(0.203758\pi\)
\(60\) 0 0
\(61\) 4.55959 7.89744i 0.583795 1.01116i −0.411229 0.911532i \(-0.634900\pi\)
0.995024 0.0996311i \(-0.0317663\pi\)
\(62\) 0 0
\(63\) 0.228411 2.63587i 0.0287771 0.332089i
\(64\) 0 0
\(65\) −3.21646 + 5.57107i −0.398952 + 0.691006i
\(66\) 0 0
\(67\) −3.73658 6.47195i −0.456496 0.790675i 0.542276 0.840200i \(-0.317563\pi\)
−0.998773 + 0.0495251i \(0.984229\pi\)
\(68\) 0 0
\(69\) −0.468884 −0.0564470
\(70\) 0 0
\(71\) 3.49263i 0.414499i 0.978288 + 0.207249i \(0.0664512\pi\)
−0.978288 + 0.207249i \(0.933549\pi\)
\(72\) 0 0
\(73\) −12.5811 + 7.26372i −1.47251 + 0.850154i −0.999522 0.0309152i \(-0.990158\pi\)
−0.472988 + 0.881069i \(0.656824\pi\)
\(74\) 0 0
\(75\) 10.7691 + 6.21752i 1.24350 + 0.717937i
\(76\) 0 0
\(77\) 3.71029 + 2.59340i 0.422826 + 0.295545i
\(78\) 0 0
\(79\) 1.46108 + 0.843557i 0.164385 + 0.0949075i 0.579935 0.814662i \(-0.303078\pi\)
−0.415551 + 0.909570i \(0.636411\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 2.72601i 0.299219i 0.988745 + 0.149609i \(0.0478016\pi\)
−0.988745 + 0.149609i \(0.952198\pi\)
\(84\) 0 0
\(85\) 9.74391i 1.05688i
\(86\) 0 0
\(87\) −1.66943 2.89153i −0.178981 0.310005i
\(88\) 0 0
\(89\) 1.83829 + 1.06134i 0.194858 + 0.112501i 0.594255 0.804277i \(-0.297447\pi\)
−0.399397 + 0.916778i \(0.630780\pi\)
\(90\) 0 0
\(91\) −3.69278 + 1.72569i −0.387108 + 0.180902i
\(92\) 0 0
\(93\) 2.73883 + 1.58126i 0.284003 + 0.163969i
\(94\) 0 0
\(95\) −25.4579 + 14.6981i −2.61193 + 1.50800i
\(96\) 0 0
\(97\) 1.95202i 0.198198i 0.995078 + 0.0990990i \(0.0315960\pi\)
−0.995078 + 0.0990990i \(0.968404\pi\)
\(98\) 0 0
\(99\) 1.71097 0.171959
\(100\) 0 0
\(101\) 6.89045 + 11.9346i 0.685625 + 1.18754i 0.973240 + 0.229792i \(0.0738044\pi\)
−0.287615 + 0.957746i \(0.592862\pi\)
\(102\) 0 0
\(103\) 3.84129 6.65331i 0.378494 0.655570i −0.612350 0.790587i \(-0.709776\pi\)
0.990843 + 0.135017i \(0.0431089\pi\)
\(104\) 0 0
\(105\) 4.67712 + 10.0085i 0.456440 + 0.976728i
\(106\) 0 0
\(107\) −7.20414 + 12.4779i −0.696450 + 1.20629i 0.273239 + 0.961946i \(0.411905\pi\)
−0.969689 + 0.244341i \(0.921428\pi\)
\(108\) 0 0
\(109\) −2.92380 + 1.68806i −0.280050 + 0.161687i −0.633446 0.773787i \(-0.718360\pi\)
0.353396 + 0.935474i \(0.385027\pi\)
\(110\) 0 0
\(111\) −8.94486 −0.849009
\(112\) 0 0
\(113\) −10.7090 −1.00742 −0.503710 0.863873i \(-0.668032\pi\)
−0.503710 + 0.863873i \(0.668032\pi\)
\(114\) 0 0
\(115\) 1.69554 0.978920i 0.158110 0.0912847i
\(116\) 0 0
\(117\) −0.770311 + 1.33422i −0.0712153 + 0.123349i
\(118\) 0 0
\(119\) 3.53711 5.06042i 0.324246 0.463888i
\(120\) 0 0
\(121\) 4.03629 6.99106i 0.366935 0.635551i
\(122\) 0 0
\(123\) −2.65706 4.60216i −0.239579 0.414963i
\(124\) 0 0
\(125\) −31.0452 −2.77677
\(126\) 0 0
\(127\) 9.49738i 0.842757i 0.906885 + 0.421378i \(0.138454\pi\)
−0.906885 + 0.421378i \(0.861546\pi\)
\(128\) 0 0
\(129\) −2.96849 + 1.71386i −0.261361 + 0.150897i
\(130\) 0 0
\(131\) 11.8364 + 6.83375i 1.03415 + 0.597068i 0.918171 0.396184i \(-0.129666\pi\)
0.115981 + 0.993251i \(0.462999\pi\)
\(132\) 0 0
\(133\) −18.5569 1.60805i −1.60909 0.139435i
\(134\) 0 0
\(135\) 3.61611 + 2.08776i 0.311226 + 0.179686i
\(136\) 0 0
\(137\) 5.99460 + 10.3829i 0.512153 + 0.887075i 0.999901 + 0.0140902i \(0.00448520\pi\)
−0.487748 + 0.872985i \(0.662181\pi\)
\(138\) 0 0
\(139\) 6.64909i 0.563968i −0.959419 0.281984i \(-0.909007\pi\)
0.959419 0.281984i \(-0.0909925\pi\)
\(140\) 0 0
\(141\) 5.90095i 0.496950i
\(142\) 0 0
\(143\) −1.31798 2.28281i −0.110215 0.190898i
\(144\) 0 0
\(145\) 12.0737 + 6.97073i 1.00266 + 0.578888i
\(146\) 0 0
\(147\) −1.20413 + 6.89566i −0.0993147 + 0.568744i
\(148\) 0 0
\(149\) 7.03123 + 4.05948i 0.576021 + 0.332566i 0.759550 0.650448i \(-0.225419\pi\)
−0.183530 + 0.983014i \(0.558752\pi\)
\(150\) 0 0
\(151\) 1.07044 0.618020i 0.0871113 0.0502937i −0.455812 0.890076i \(-0.650651\pi\)
0.542923 + 0.839783i \(0.317318\pi\)
\(152\) 0 0
\(153\) 2.33358i 0.188658i
\(154\) 0 0
\(155\) −13.2052 −1.06067
\(156\) 0 0
\(157\) −4.80286 8.31880i −0.383310 0.663913i 0.608223 0.793766i \(-0.291883\pi\)
−0.991533 + 0.129854i \(0.958549\pi\)
\(158\) 0 0
\(159\) 0.781947 1.35437i 0.0620124 0.107409i
\(160\) 0 0
\(161\) 1.23592 + 0.107099i 0.0974041 + 0.00844055i
\(162\) 0 0
\(163\) −9.70461 + 16.8089i −0.760123 + 1.31657i 0.182663 + 0.983176i \(0.441528\pi\)
−0.942787 + 0.333397i \(0.891805\pi\)
\(164\) 0 0
\(165\) −6.18706 + 3.57210i −0.481662 + 0.278088i
\(166\) 0 0
\(167\) 17.7482 1.37340 0.686699 0.726942i \(-0.259059\pi\)
0.686699 + 0.726942i \(0.259059\pi\)
\(168\) 0 0
\(169\) −10.6265 −0.817422
\(170\) 0 0
\(171\) −6.09693 + 3.52006i −0.466244 + 0.269186i
\(172\) 0 0
\(173\) 3.57075 6.18472i 0.271479 0.470216i −0.697762 0.716330i \(-0.745821\pi\)
0.969241 + 0.246114i \(0.0791539\pi\)
\(174\) 0 0
\(175\) −26.9657 18.8484i −2.03842 1.42480i
\(176\) 0 0
\(177\) −3.04114 + 5.26742i −0.228586 + 0.395923i
\(178\) 0 0
\(179\) 11.9581 + 20.7121i 0.893791 + 1.54809i 0.835293 + 0.549805i \(0.185298\pi\)
0.0584980 + 0.998288i \(0.481369\pi\)
\(180\) 0 0
\(181\) 20.5572 1.52800 0.764002 0.645214i \(-0.223232\pi\)
0.764002 + 0.645214i \(0.223232\pi\)
\(182\) 0 0
\(183\) 9.11917i 0.674109i
\(184\) 0 0
\(185\) 32.3456 18.6748i 2.37810 1.37300i
\(186\) 0 0
\(187\) 3.45776 + 1.99634i 0.252857 + 0.145987i
\(188\) 0 0
\(189\) 1.12013 + 2.39694i 0.0814772 + 0.174352i
\(190\) 0 0
\(191\) 1.74523 + 1.00761i 0.126280 + 0.0729079i 0.561809 0.827267i \(-0.310105\pi\)
−0.435529 + 0.900175i \(0.643439\pi\)
\(192\) 0 0
\(193\) 1.78535 + 3.09232i 0.128512 + 0.222590i 0.923100 0.384559i \(-0.125646\pi\)
−0.794588 + 0.607149i \(0.792313\pi\)
\(194\) 0 0
\(195\) 6.43291i 0.460671i
\(196\) 0 0
\(197\) 17.5393i 1.24962i −0.780775 0.624812i \(-0.785176\pi\)
0.780775 0.624812i \(-0.214824\pi\)
\(198\) 0 0
\(199\) 8.85336 + 15.3345i 0.627598 + 1.08703i 0.988032 + 0.154247i \(0.0492952\pi\)
−0.360434 + 0.932785i \(0.617371\pi\)
\(200\) 0 0
\(201\) 6.47195 + 3.73658i 0.456496 + 0.263558i
\(202\) 0 0
\(203\) 3.73994 + 8.00302i 0.262492 + 0.561702i
\(204\) 0 0
\(205\) 19.2164 + 11.0946i 1.34213 + 0.774882i
\(206\) 0 0
\(207\) 0.406066 0.234442i 0.0282235 0.0162949i
\(208\) 0 0
\(209\) 12.0455i 0.833201i
\(210\) 0 0
\(211\) 4.23050 0.291240 0.145620 0.989341i \(-0.453482\pi\)
0.145620 + 0.989341i \(0.453482\pi\)
\(212\) 0 0
\(213\) −1.74631 3.02471i −0.119656 0.207249i
\(214\) 0 0
\(215\) 7.15626 12.3950i 0.488053 0.845333i
\(216\) 0 0
\(217\) −6.85803 4.79359i −0.465553 0.325410i
\(218\) 0 0
\(219\) 7.26372 12.5811i 0.490837 0.850154i
\(220\) 0 0
\(221\) −3.11350 + 1.79758i −0.209437 + 0.120918i
\(222\) 0 0
\(223\) 1.43532 0.0961162 0.0480581 0.998845i \(-0.484697\pi\)
0.0480581 + 0.998845i \(0.484697\pi\)
\(224\) 0 0
\(225\) −12.4350 −0.829003
\(226\) 0 0
\(227\) −13.8688 + 8.00718i −0.920508 + 0.531455i −0.883797 0.467871i \(-0.845021\pi\)
−0.0367106 + 0.999326i \(0.511688\pi\)
\(228\) 0 0
\(229\) −8.12499 + 14.0729i −0.536914 + 0.929963i 0.462154 + 0.886800i \(0.347077\pi\)
−0.999068 + 0.0431631i \(0.986256\pi\)
\(230\) 0 0
\(231\) −4.50990 0.390805i −0.296730 0.0257131i
\(232\) 0 0
\(233\) −5.93054 + 10.2720i −0.388522 + 0.672941i −0.992251 0.124249i \(-0.960348\pi\)
0.603729 + 0.797190i \(0.293681\pi\)
\(234\) 0 0
\(235\) −12.3198 21.3385i −0.803654 1.39197i
\(236\) 0 0
\(237\) −1.68711 −0.109590
\(238\) 0 0
\(239\) 0.846585i 0.0547610i 0.999625 + 0.0273805i \(0.00871657\pi\)
−0.999625 + 0.0273805i \(0.991283\pi\)
\(240\) 0 0
\(241\) −0.761425 + 0.439609i −0.0490477 + 0.0283177i −0.524323 0.851519i \(-0.675682\pi\)
0.475276 + 0.879837i \(0.342348\pi\)
\(242\) 0 0
\(243\) 0.866025 + 0.500000i 0.0555556 + 0.0320750i
\(244\) 0 0
\(245\) −10.0422 27.4494i −0.641575 1.75368i
\(246\) 0 0
\(247\) 9.39307 + 5.42309i 0.597667 + 0.345063i
\(248\) 0 0
\(249\) −1.36301 2.36080i −0.0863771 0.149609i
\(250\) 0 0
\(251\) 18.1441i 1.14524i −0.819820 0.572622i \(-0.805926\pi\)
0.819820 0.572622i \(-0.194074\pi\)
\(252\) 0 0
\(253\) 0.802247i 0.0504368i
\(254\) 0 0
\(255\) 4.87196 + 8.43848i 0.305094 + 0.528438i
\(256\) 0 0
\(257\) 15.8902 + 9.17421i 0.991203 + 0.572271i 0.905634 0.424061i \(-0.139396\pi\)
0.0855695 + 0.996332i \(0.472729\pi\)
\(258\) 0 0
\(259\) 23.5775 + 2.04311i 1.46504 + 0.126953i
\(260\) 0 0
\(261\) 2.89153 + 1.66943i 0.178981 + 0.103335i
\(262\) 0 0
\(263\) 3.02044 1.74385i 0.186248 0.107531i −0.403977 0.914769i \(-0.632372\pi\)
0.590225 + 0.807239i \(0.299039\pi\)
\(264\) 0 0
\(265\) 6.53008i 0.401140i
\(266\) 0 0
\(267\) −2.12267 −0.129905
\(268\) 0 0
\(269\) 7.35605 + 12.7410i 0.448506 + 0.776835i 0.998289 0.0584722i \(-0.0186229\pi\)
−0.549783 + 0.835308i \(0.685290\pi\)
\(270\) 0 0
\(271\) −9.95139 + 17.2363i −0.604504 + 1.04703i 0.387626 + 0.921817i \(0.373295\pi\)
−0.992130 + 0.125215i \(0.960038\pi\)
\(272\) 0 0
\(273\) 2.33519 3.34088i 0.141332 0.202199i
\(274\) 0 0
\(275\) 10.6380 18.4255i 0.641495 1.11110i
\(276\) 0 0
\(277\) −22.9034 + 13.2233i −1.37613 + 0.794510i −0.991691 0.128640i \(-0.958939\pi\)
−0.384440 + 0.923150i \(0.625605\pi\)
\(278\) 0 0
\(279\) −3.16253 −0.189336
\(280\) 0 0
\(281\) −20.2837 −1.21003 −0.605013 0.796216i \(-0.706832\pi\)
−0.605013 + 0.796216i \(0.706832\pi\)
\(282\) 0 0
\(283\) −5.70426 + 3.29336i −0.339083 + 0.195770i −0.659866 0.751383i \(-0.729387\pi\)
0.320783 + 0.947153i \(0.396054\pi\)
\(284\) 0 0
\(285\) 14.6981 25.4579i 0.870642 1.50800i
\(286\) 0 0
\(287\) 5.95248 + 12.7376i 0.351364 + 0.751877i
\(288\) 0 0
\(289\) −5.77721 + 10.0064i −0.339836 + 0.588613i
\(290\) 0 0
\(291\) −0.976012 1.69050i −0.0572148 0.0990990i
\(292\) 0 0
\(293\) 12.9438 0.756187 0.378094 0.925767i \(-0.376580\pi\)
0.378094 + 0.925767i \(0.376580\pi\)
\(294\) 0 0
\(295\) 25.3968i 1.47866i
\(296\) 0 0
\(297\) −1.48174 + 0.855485i −0.0859795 + 0.0496403i
\(298\) 0 0
\(299\) −0.625594 0.361187i −0.0361790 0.0208880i
\(300\) 0 0
\(301\) 8.21603 3.83948i 0.473564 0.221304i
\(302\) 0 0
\(303\) −11.9346 6.89045i −0.685625 0.395846i
\(304\) 0 0
\(305\) −19.0387 32.9760i −1.09015 1.88820i
\(306\) 0 0
\(307\) 11.8773i 0.677871i −0.940810 0.338936i \(-0.889933\pi\)
0.940810 0.338936i \(-0.110067\pi\)
\(308\) 0 0
\(309\) 7.68258i 0.437047i
\(310\) 0 0
\(311\) −5.91849 10.2511i −0.335607 0.581288i 0.647994 0.761645i \(-0.275608\pi\)
−0.983601 + 0.180357i \(0.942275\pi\)
\(312\) 0 0
\(313\) −12.8383 7.41217i −0.725661 0.418961i 0.0911716 0.995835i \(-0.470939\pi\)
−0.816833 + 0.576875i \(0.804272\pi\)
\(314\) 0 0
\(315\) −9.05475 6.32904i −0.510177 0.356601i
\(316\) 0 0
\(317\) −5.87478 3.39181i −0.329961 0.190503i 0.325863 0.945417i \(-0.394345\pi\)
−0.655824 + 0.754914i \(0.727678\pi\)
\(318\) 0 0
\(319\) −4.94732 + 2.85634i −0.276997 + 0.159924i
\(320\) 0 0
\(321\) 14.4083i 0.804191i
\(322\) 0 0
\(323\) −16.4287 −0.914116
\(324\) 0 0
\(325\) 9.57885 + 16.5911i 0.531339 + 0.920306i
\(326\) 0 0
\(327\) 1.68806 2.92380i 0.0933499 0.161687i
\(328\) 0 0
\(329\) 1.34784 15.5542i 0.0743090 0.857528i
\(330\) 0 0
\(331\) −2.37285 + 4.10989i −0.130424 + 0.225900i −0.923840 0.382779i \(-0.874967\pi\)
0.793416 + 0.608679i \(0.208300\pi\)
\(332\) 0 0
\(333\) 7.74648 4.47243i 0.424504 0.245088i
\(334\) 0 0
\(335\) −31.2044 −1.70488
\(336\) 0 0
\(337\) 16.5173 0.899754 0.449877 0.893090i \(-0.351468\pi\)
0.449877 + 0.893090i \(0.351468\pi\)
\(338\) 0 0
\(339\) 9.27429 5.35451i 0.503710 0.290817i
\(340\) 0 0
\(341\) 2.70549 4.68605i 0.146511 0.253764i
\(342\) 0 0
\(343\) 4.74897 17.9010i 0.256420 0.966565i
\(344\) 0 0
\(345\) −0.978920 + 1.69554i −0.0527033 + 0.0912847i
\(346\) 0 0
\(347\) −9.54986 16.5408i −0.512663 0.887959i −0.999892 0.0146846i \(-0.995326\pi\)
0.487229 0.873274i \(-0.338008\pi\)
\(348\) 0 0
\(349\) −2.49767 −0.133697 −0.0668485 0.997763i \(-0.521294\pi\)
−0.0668485 + 0.997763i \(0.521294\pi\)
\(350\) 0 0
\(351\) 1.54062i 0.0822323i
\(352\) 0 0
\(353\) −22.3071 + 12.8790i −1.18729 + 0.685481i −0.957689 0.287805i \(-0.907075\pi\)
−0.229599 + 0.973285i \(0.573741\pi\)
\(354\) 0 0
\(355\) 12.6297 + 7.29179i 0.670317 + 0.387008i
\(356\) 0 0
\(357\) −0.533015 + 6.15101i −0.0282102 + 0.325546i
\(358\) 0 0
\(359\) 28.7697 + 16.6102i 1.51840 + 0.876651i 0.999765 + 0.0216582i \(0.00689457\pi\)
0.518639 + 0.854993i \(0.326439\pi\)
\(360\) 0 0
\(361\) 15.2817 + 26.4687i 0.804300 + 1.39309i
\(362\) 0 0
\(363\) 8.07258i 0.423701i
\(364\) 0 0
\(365\) 60.6597i 3.17507i
\(366\) 0 0
\(367\) 2.17584 + 3.76866i 0.113578 + 0.196722i 0.917210 0.398403i \(-0.130436\pi\)
−0.803633 + 0.595126i \(0.797102\pi\)
\(368\) 0 0
\(369\) 4.60216 + 2.65706i 0.239579 + 0.138321i
\(370\) 0 0
\(371\) −2.37047 + 3.39135i −0.123068 + 0.176070i
\(372\) 0 0
\(373\) −11.0943 6.40533i −0.574444 0.331655i 0.184479 0.982837i \(-0.440940\pi\)
−0.758922 + 0.651181i \(0.774274\pi\)
\(374\) 0 0
\(375\) 26.8859 15.5226i 1.38838 0.801584i
\(376\) 0 0
\(377\) 5.14391i 0.264925i
\(378\) 0 0
\(379\) 24.0807 1.23694 0.618472 0.785807i \(-0.287752\pi\)
0.618472 + 0.785807i \(0.287752\pi\)
\(380\) 0 0
\(381\) −4.74869 8.22498i −0.243283 0.421378i
\(382\) 0 0
\(383\) 8.78233 15.2114i 0.448756 0.777268i −0.549549 0.835461i \(-0.685201\pi\)
0.998305 + 0.0581930i \(0.0185339\pi\)
\(384\) 0 0
\(385\) 17.1242 8.00242i 0.872731 0.407841i
\(386\) 0 0
\(387\) 1.71386 2.96849i 0.0871203 0.150897i
\(388\) 0 0
\(389\) 23.9022 13.7999i 1.21189 0.699685i 0.248719 0.968576i \(-0.419990\pi\)
0.963171 + 0.268890i \(0.0866570\pi\)
\(390\) 0 0
\(391\) 1.09418 0.0553349
\(392\) 0 0
\(393\) −13.6675 −0.689435
\(394\) 0 0
\(395\) 6.10079 3.52229i 0.306964 0.177226i
\(396\) 0 0
\(397\) 8.65850 14.9970i 0.434558 0.752676i −0.562702 0.826660i \(-0.690238\pi\)
0.997259 + 0.0739841i \(0.0235714\pi\)
\(398\) 0 0
\(399\) 16.8748 7.88583i 0.844794 0.394786i
\(400\) 0 0
\(401\) −3.87616 + 6.71371i −0.193566 + 0.335266i −0.946430 0.322910i \(-0.895339\pi\)
0.752863 + 0.658177i \(0.228672\pi\)
\(402\) 0 0
\(403\) 2.43613 + 4.21950i 0.121352 + 0.210188i
\(404\) 0 0
\(405\) −4.17553 −0.207484
\(406\) 0 0
\(407\) 15.3044i 0.758611i
\(408\) 0 0
\(409\) −4.01694 + 2.31918i −0.198625 + 0.114676i −0.596014 0.802974i \(-0.703250\pi\)
0.397389 + 0.917650i \(0.369916\pi\)
\(410\) 0 0
\(411\) −10.3829 5.99460i −0.512153 0.295692i
\(412\) 0 0
\(413\) 9.21921 13.1896i 0.453648 0.649018i
\(414\) 0 0
\(415\) 9.85758 + 5.69128i 0.483889 + 0.279374i
\(416\) 0 0
\(417\) 3.32454 + 5.75828i 0.162804 + 0.281984i
\(418\) 0 0
\(419\) 14.2419i 0.695760i −0.937539 0.347880i \(-0.886902\pi\)
0.937539 0.347880i \(-0.113098\pi\)
\(420\) 0 0
\(421\) 20.0126i 0.975356i −0.873024 0.487678i \(-0.837844\pi\)
0.873024 0.487678i \(-0.162156\pi\)
\(422\) 0 0
\(423\) −2.95047 5.11037i −0.143457 0.248475i
\(424\) 0 0
\(425\) −25.1304 14.5091i −1.21900 0.703793i
\(426\) 0 0
\(427\) 2.08292 24.0370i 0.100800 1.16323i
\(428\) 0 0
\(429\) 2.28281 + 1.31798i 0.110215 + 0.0636327i
\(430\) 0 0
\(431\) 12.1099 6.99165i 0.583313 0.336776i −0.179136 0.983824i \(-0.557330\pi\)
0.762449 + 0.647049i \(0.223997\pi\)
\(432\) 0 0
\(433\) 0.984888i 0.0473307i 0.999720 + 0.0236653i \(0.00753362\pi\)
−0.999720 + 0.0236653i \(0.992466\pi\)
\(434\) 0 0
\(435\) −13.9415 −0.668442
\(436\) 0 0
\(437\) −1.65050 2.85875i −0.0789542 0.136753i
\(438\) 0 0
\(439\) 3.43693 5.95294i 0.164036 0.284118i −0.772277 0.635286i \(-0.780882\pi\)
0.936312 + 0.351168i \(0.114215\pi\)
\(440\) 0 0
\(441\) −2.40502 6.57388i −0.114525 0.313042i
\(442\) 0 0
\(443\) 1.70483 2.95285i 0.0809989 0.140294i −0.822680 0.568504i \(-0.807522\pi\)
0.903679 + 0.428210i \(0.140856\pi\)
\(444\) 0 0
\(445\) 7.67582 4.43164i 0.363869 0.210080i
\(446\) 0 0
\(447\) −8.11897 −0.384014
\(448\) 0 0
\(449\) 32.9924 1.55701 0.778503 0.627641i \(-0.215979\pi\)
0.778503 + 0.627641i \(0.215979\pi\)
\(450\) 0 0
\(451\) −7.87416 + 4.54615i −0.370780 + 0.214070i
\(452\) 0 0
\(453\) −0.618020 + 1.07044i −0.0290371 + 0.0502937i
\(454\) 0 0
\(455\) −1.46935 + 16.9563i −0.0688842 + 0.794926i
\(456\) 0 0
\(457\) 11.4224 19.7842i 0.534319 0.925467i −0.464877 0.885375i \(-0.653902\pi\)
0.999196 0.0400919i \(-0.0127651\pi\)
\(458\) 0 0
\(459\) 1.16679 + 2.02094i 0.0544610 + 0.0943292i
\(460\) 0 0
\(461\) −15.8743 −0.739338 −0.369669 0.929164i \(-0.620529\pi\)
−0.369669 + 0.929164i \(0.620529\pi\)
\(462\) 0 0
\(463\) 2.72059i 0.126436i −0.998000 0.0632182i \(-0.979864\pi\)
0.998000 0.0632182i \(-0.0201364\pi\)
\(464\) 0 0
\(465\) 11.4361 6.60261i 0.530334 0.306189i
\(466\) 0 0
\(467\) −20.7726 11.9931i −0.961240 0.554972i −0.0646858 0.997906i \(-0.520605\pi\)
−0.896555 + 0.442933i \(0.853938\pi\)
\(468\) 0 0
\(469\) −16.2058 11.3274i −0.748313 0.523052i
\(470\) 0 0
\(471\) 8.31880 + 4.80286i 0.383310 + 0.221304i
\(472\) 0 0
\(473\) 2.93236 + 5.07900i 0.134830 + 0.233533i
\(474\) 0 0
\(475\) 87.5443i 4.01681i
\(476\) 0 0
\(477\) 1.56389i 0.0716058i
\(478\) 0 0
\(479\) −9.87511 17.1042i −0.451205 0.781511i 0.547256 0.836965i \(-0.315672\pi\)
−0.998461 + 0.0554547i \(0.982339\pi\)
\(480\) 0 0
\(481\) −11.9344 6.89033i −0.544162 0.314172i
\(482\) 0 0
\(483\) −1.12389 + 0.525210i −0.0511386 + 0.0238979i
\(484\) 0 0
\(485\) 7.05874 + 4.07536i 0.320521 + 0.185053i
\(486\) 0 0
\(487\) −32.1435 + 18.5581i −1.45656 + 0.840946i −0.998840 0.0481495i \(-0.984668\pi\)
−0.457721 + 0.889096i \(0.651334\pi\)
\(488\) 0 0
\(489\) 19.4092i 0.877715i
\(490\) 0 0
\(491\) 20.5746 0.928517 0.464259 0.885700i \(-0.346321\pi\)
0.464259 + 0.885700i \(0.346321\pi\)
\(492\) 0 0
\(493\) 3.89573 + 6.74761i 0.175455 + 0.303897i
\(494\) 0 0
\(495\) 3.57210 6.18706i 0.160554 0.278088i
\(496\) 0 0
\(497\) 3.91219 + 8.37162i 0.175486 + 0.375518i
\(498\) 0 0
\(499\) 0.517579 0.896473i 0.0231700 0.0401316i −0.854208 0.519932i \(-0.825957\pi\)
0.877378 + 0.479800i \(0.159291\pi\)
\(500\) 0 0
\(501\) −15.3704 + 8.87410i −0.686699 + 0.396466i
\(502\) 0 0
\(503\) −19.7898 −0.882382 −0.441191 0.897413i \(-0.645444\pi\)
−0.441191 + 0.897413i \(0.645444\pi\)
\(504\) 0 0
\(505\) 57.5425 2.56061
\(506\) 0 0
\(507\) 9.20280 5.31324i 0.408711 0.235969i
\(508\) 0 0
\(509\) −2.11849 + 3.66933i −0.0939004 + 0.162640i −0.909149 0.416471i \(-0.863267\pi\)
0.815249 + 0.579111i \(0.196600\pi\)
\(510\) 0 0
\(511\) −22.0199 + 31.5031i −0.974104 + 1.39362i
\(512\) 0 0
\(513\) 3.52006 6.09693i 0.155415 0.269186i
\(514\) 0 0
\(515\) −16.0394 27.7811i −0.706781 1.22418i
\(516\) 0 0
\(517\) 10.0963 0.444037
\(518\) 0 0
\(519\) 7.14150i 0.313477i
\(520\) 0 0
\(521\) −33.6570 + 19.4319i −1.47454 + 0.851326i −0.999588 0.0286855i \(-0.990868\pi\)
−0.474952 + 0.880012i \(0.657535\pi\)
\(522\) 0 0
\(523\) −14.8181 8.55526i −0.647952 0.374095i 0.139719 0.990191i \(-0.455380\pi\)
−0.787671 + 0.616096i \(0.788713\pi\)
\(524\) 0 0
\(525\) 32.7772 + 2.84030i 1.43051 + 0.123961i
\(526\) 0 0
\(527\) −6.39126 3.69000i −0.278408 0.160739i
\(528\) 0 0
\(529\) −11.3901 19.7282i −0.495221 0.857747i
\(530\) 0 0
\(531\) 6.08229i 0.263949i
\(532\) 0 0
\(533\) 8.18704i 0.354620i
\(534\) 0 0
\(535\) 30.0811 + 52.1019i 1.30052 + 2.25256i
\(536\) 0 0
\(537\) −20.7121 11.9581i −0.893791 0.516031i
\(538\) 0 0
\(539\) 11.7983 + 2.06023i 0.508187 + 0.0887402i
\(540\) 0 0
\(541\) −37.6652 21.7460i −1.61935 0.934935i −0.987087 0.160185i \(-0.948791\pi\)
−0.632268 0.774750i \(-0.717876\pi\)
\(542\) 0 0
\(543\) −17.8030 + 10.2786i −0.764002 + 0.441097i
\(544\) 0 0
\(545\) 14.0971i 0.603852i
\(546\) 0 0
\(547\) −10.8290 −0.463016 −0.231508 0.972833i \(-0.574366\pi\)
−0.231508 + 0.972833i \(0.574366\pi\)
\(548\) 0 0
\(549\) −4.55959 7.89744i −0.194598 0.337054i
\(550\) 0 0
\(551\) 11.7530 20.3567i 0.500693 0.867226i
\(552\) 0 0
\(553\) 4.44702 + 0.385356i 0.189106 + 0.0163870i
\(554\) 0 0
\(555\) −18.6748 + 32.3456i −0.792699 + 1.37300i
\(556\) 0 0
\(557\) 7.04197 4.06568i 0.298378 0.172269i −0.343336 0.939213i \(-0.611557\pi\)
0.641714 + 0.766944i \(0.278224\pi\)
\(558\) 0 0
\(559\) −5.28082 −0.223355
\(560\) 0 0
\(561\) −3.99268 −0.168571
\(562\) 0 0
\(563\) −19.6081 + 11.3207i −0.826381 + 0.477111i −0.852612 0.522545i \(-0.824983\pi\)
0.0262311 + 0.999656i \(0.491649\pi\)
\(564\) 0 0
\(565\) −22.3579 + 38.7251i −0.940605 + 1.62918i
\(566\) 0 0
\(567\) −2.16853 1.51575i −0.0910696 0.0636554i
\(568\) 0 0
\(569\) 10.1485 17.5778i 0.425449 0.736900i −0.571013 0.820941i \(-0.693449\pi\)
0.996462 + 0.0840413i \(0.0267828\pi\)
\(570\) 0 0
\(571\) 18.3819 + 31.8383i 0.769257 + 1.33239i 0.937967 + 0.346726i \(0.112707\pi\)
−0.168710 + 0.985666i \(0.553960\pi\)
\(572\) 0 0
\(573\) −2.01521 −0.0841868
\(574\) 0 0
\(575\) 5.83059i 0.243153i
\(576\) 0 0
\(577\) 1.39915 0.807801i 0.0582475 0.0336292i −0.470593 0.882350i \(-0.655960\pi\)
0.528841 + 0.848721i \(0.322627\pi\)
\(578\) 0 0
\(579\) −3.09232 1.78535i −0.128512 0.0741967i
\(580\) 0 0
\(581\) 3.05348 + 6.53409i 0.126680 + 0.271080i
\(582\) 0 0
\(583\) −2.31729 1.33789i −0.0959723 0.0554097i
\(584\) 0 0
\(585\) 3.21646 + 5.57107i 0.132984 + 0.230335i
\(586\) 0 0
\(587\) 3.68747i 0.152198i 0.997100 + 0.0760991i \(0.0242465\pi\)
−0.997100 + 0.0760991i \(0.975753\pi\)
\(588\) 0 0
\(589\) 22.2646i 0.917396i
\(590\) 0 0
\(591\) 8.76965 + 15.1895i 0.360735 + 0.624812i
\(592\) 0 0
\(593\) 32.3781 + 18.6935i 1.32961 + 0.767650i 0.985239 0.171184i \(-0.0547591\pi\)
0.344370 + 0.938834i \(0.388092\pi\)
\(594\) 0 0
\(595\) −10.9144 23.3556i −0.447447 0.957485i
\(596\) 0 0
\(597\) −15.3345 8.85336i −0.627598 0.362344i
\(598\) 0 0
\(599\) 8.61435 4.97350i 0.351973 0.203212i −0.313581 0.949561i \(-0.601529\pi\)
0.665554 + 0.746350i \(0.268195\pi\)
\(600\) 0 0
\(601\) 35.9296i 1.46560i 0.680445 + 0.732799i \(0.261786\pi\)
−0.680445 + 0.732799i \(0.738214\pi\)
\(602\) 0 0
\(603\) −7.47317 −0.304331
\(604\) 0 0
\(605\) −16.8536 29.1914i −0.685198 1.18680i
\(606\) 0 0
\(607\) −14.9355 + 25.8690i −0.606212 + 1.04999i 0.385646 + 0.922647i \(0.373978\pi\)
−0.991859 + 0.127344i \(0.959355\pi\)
\(608\) 0 0
\(609\) −7.24039 5.06085i −0.293395 0.205076i
\(610\) 0 0
\(611\) −4.54557 + 7.87315i −0.183894 + 0.318514i
\(612\) 0 0
\(613\) −32.8160 + 18.9463i −1.32542 + 0.765234i −0.984588 0.174889i \(-0.944043\pi\)
−0.340836 + 0.940123i \(0.610710\pi\)
\(614\) 0 0
\(615\) −22.1892 −0.894756
\(616\) 0 0
\(617\) 39.0332 1.57142 0.785709 0.618597i \(-0.212298\pi\)
0.785709 + 0.618597i \(0.212298\pi\)
\(618\) 0 0
\(619\) 21.5338 12.4325i 0.865517 0.499706i −0.000339137 1.00000i \(-0.500108\pi\)
0.865856 + 0.500294i \(0.166775\pi\)
\(620\) 0 0
\(621\) −0.234442 + 0.406066i −0.00940784 + 0.0162949i
\(622\) 0 0
\(623\) 5.59509 + 0.484842i 0.224163 + 0.0194248i
\(624\) 0 0
\(625\) −33.7275 + 58.4177i −1.34910 + 2.33671i
\(626\) 0 0
\(627\) 6.02273 + 10.4317i 0.240525 + 0.416601i
\(628\) 0 0
\(629\) 20.8735 0.832281
\(630\) 0 0
\(631\) 43.3823i 1.72702i −0.504330 0.863511i \(-0.668260\pi\)
0.504330 0.863511i \(-0.331740\pi\)
\(632\) 0 0
\(633\) −3.66372 + 2.11525i −0.145620 + 0.0840736i
\(634\) 0 0
\(635\) 34.3436 + 19.8283i 1.36289 + 0.786862i
\(636\) 0 0
\(637\) −6.91837 + 8.27276i −0.274116 + 0.327779i
\(638\) 0 0
\(639\) 3.02471 + 1.74631i 0.119656 + 0.0690831i
\(640\) 0 0
\(641\) −3.32559 5.76010i −0.131353 0.227510i 0.792845 0.609423i \(-0.208599\pi\)
−0.924198 + 0.381913i \(0.875265\pi\)
\(642\) 0 0
\(643\) 18.1066i 0.714055i 0.934094 + 0.357027i \(0.116210\pi\)
−0.934094 + 0.357027i \(0.883790\pi\)
\(644\) 0 0
\(645\) 14.3125i 0.563555i
\(646\) 0 0
\(647\) −24.6421 42.6815i −0.968783 1.67798i −0.699089 0.715035i \(-0.746411\pi\)
−0.269694 0.962946i \(-0.586923\pi\)
\(648\) 0 0
\(649\) 9.01239 + 5.20331i 0.353767 + 0.204248i
\(650\) 0 0
\(651\) 8.33602 + 0.722357i 0.326714 + 0.0283114i
\(652\) 0 0
\(653\) −22.1087 12.7645i −0.865181 0.499512i 0.000563051 1.00000i \(-0.499821\pi\)
−0.865744 + 0.500488i \(0.833154\pi\)
\(654\) 0 0
\(655\) 49.4233 28.5345i 1.93113 1.11494i
\(656\) 0 0
\(657\) 14.5274i 0.566769i
\(658\) 0 0
\(659\) 12.0942 0.471125 0.235562 0.971859i \(-0.424307\pi\)
0.235562 + 0.971859i \(0.424307\pi\)
\(660\) 0 0
\(661\) −5.42541 9.39708i −0.211024 0.365504i 0.741011 0.671493i \(-0.234346\pi\)
−0.952035 + 0.305988i \(0.901013\pi\)
\(662\) 0 0
\(663\) 1.79758 3.11350i 0.0698122 0.120918i
\(664\) 0 0
\(665\) −44.5573 + 63.7466i −1.72786 + 2.47199i
\(666\) 0 0
\(667\) −0.782767 + 1.35579i −0.0303089 + 0.0524965i
\(668\) 0 0
\(669\) −1.24302 + 0.717661i −0.0480581 + 0.0277464i
\(670\) 0 0
\(671\) 15.6026 0.602333
\(672\) 0 0
\(673\) −48.1931 −1.85771 −0.928854 0.370446i \(-0.879205\pi\)
−0.928854 + 0.370446i \(0.879205\pi\)
\(674\) 0 0
\(675\) 10.7691 6.21752i 0.414501 0.239312i
\(676\) 0 0
\(677\) 2.56093 4.43567i 0.0984246 0.170476i −0.812608 0.582810i \(-0.801953\pi\)
0.911033 + 0.412334i \(0.135286\pi\)
\(678\) 0 0
\(679\) 2.18651 + 4.67888i 0.0839107 + 0.179559i
\(680\) 0 0
\(681\) 8.00718 13.8688i 0.306836 0.531455i
\(682\) 0 0
\(683\) 11.7191 + 20.2980i 0.448418 + 0.776682i 0.998283 0.0585709i \(-0.0186544\pi\)
−0.549866 + 0.835253i \(0.685321\pi\)
\(684\) 0 0
\(685\) 50.0612 1.91274
\(686\) 0 0
\(687\) 16.2500i 0.619975i
\(688\) 0 0
\(689\) 2.08658 1.20468i 0.0794922 0.0458948i
\(690\) 0 0
\(691\) −11.9534 6.90129i −0.454728 0.262537i 0.255097 0.966915i \(-0.417893\pi\)
−0.709825 + 0.704378i \(0.751226\pi\)
\(692\) 0 0
\(693\) 4.10109 1.91650i 0.155788 0.0728020i
\(694\) 0 0
\(695\) −24.0439 13.8817i −0.912035 0.526564i
\(696\) 0 0
\(697\) 6.20044 + 10.7395i 0.234859 + 0.406787i
\(698\) 0 0
\(699\) 11.8611i 0.448627i
\(700\) 0 0
\(701\) 11.8718i 0.448393i 0.974544 + 0.224196i \(0.0719757\pi\)
−0.974544 + 0.224196i \(0.928024\pi\)
\(702\) 0 0
\(703\) −31.4865 54.5362i −1.18754 2.05687i
\(704\) 0 0
\(705\) 21.3385 + 12.3198i 0.803654 + 0.463990i
\(706\) 0 0
\(707\) 29.8843 + 20.8883i 1.12391 + 0.785587i
\(708\) 0 0
\(709\) 27.1241 + 15.6601i 1.01867 + 0.588127i 0.913717 0.406350i \(-0.133199\pi\)
0.104949 + 0.994478i \(0.466532\pi\)
\(710\) 0 0
\(711\) 1.46108 0.843557i 0.0547949 0.0316358i
\(712\) 0 0
\(713\) 1.48286i 0.0555335i
\(714\) 0 0
\(715\) −11.0065 −0.411621
\(716\) 0 0
\(717\) −0.423292 0.733164i −0.0158081 0.0273805i
\(718\) 0 0
\(719\) −12.6273 + 21.8712i −0.470920 + 0.815657i −0.999447 0.0332594i \(-0.989411\pi\)
0.528527 + 0.848917i \(0.322745\pi\)
\(720\) 0 0
\(721\) 1.75479 20.2503i 0.0653518 0.754161i
\(722\) 0 0
\(723\) 0.439609 0.761425i 0.0163492 0.0283177i
\(724\) 0 0
\(725\) 35.9563 20.7594i 1.33538 0.770984i
\(726\) 0 0
\(727\) −17.9342 −0.665144 −0.332572 0.943078i \(-0.607916\pi\)
−0.332572 + 0.943078i \(0.607916\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 6.92720 3.99942i 0.256212 0.147924i
\(732\) 0 0
\(733\) −6.16779 + 10.6829i −0.227813 + 0.394583i −0.957160 0.289561i \(-0.906491\pi\)
0.729347 + 0.684144i \(0.239824\pi\)
\(734\) 0 0
\(735\) 22.4215 + 18.7508i 0.827031 + 0.691632i
\(736\) 0 0
\(737\) 6.39319 11.0733i 0.235496 0.407891i
\(738\) 0 0
\(739\) 21.7463 + 37.6656i 0.799949 + 1.38555i 0.919648 + 0.392743i \(0.128474\pi\)
−0.119699 + 0.992810i \(0.538193\pi\)
\(740\) 0 0
\(741\) −10.8462 −0.398444
\(742\) 0 0
\(743\) 32.8397i 1.20477i 0.798205 + 0.602386i \(0.205783\pi\)
−0.798205 + 0.602386i \(0.794217\pi\)
\(744\) 0 0
\(745\) 29.3591 16.9505i 1.07563 0.621018i
\(746\) 0 0
\(747\) 2.36080 + 1.36301i 0.0863771 + 0.0498698i
\(748\) 0 0
\(749\) −3.29101 + 37.9784i −0.120251 + 1.38770i
\(750\) 0 0
\(751\) −7.62670 4.40328i −0.278302 0.160678i 0.354352 0.935112i \(-0.384701\pi\)
−0.632655 + 0.774434i \(0.718035\pi\)
\(752\) 0 0
\(753\) 9.07203 + 15.7132i 0.330603 + 0.572622i
\(754\) 0 0
\(755\) 5.16112i 0.187832i
\(756\) 0 0
\(757\) 16.9328i 0.615433i −0.951478 0.307717i \(-0.900435\pi\)
0.951478 0.307717i \(-0.0995649\pi\)
\(758\) 0 0
\(759\) −0.401124 0.694766i −0.0145599 0.0252184i
\(760\) 0 0
\(761\) 10.9108 + 6.29937i 0.395517 + 0.228352i 0.684548 0.728968i \(-0.260000\pi\)
−0.289031 + 0.957320i \(0.593333\pi\)
\(762\) 0 0
\(763\) −5.11734 + 7.32121i −0.185260 + 0.265045i
\(764\) 0 0
\(765\) −8.43848 4.87196i −0.305094 0.176146i
\(766\) 0 0
\(767\) −8.11510 + 4.68526i −0.293019 + 0.169175i
\(768\) 0 0
\(769\) 16.0445i 0.578581i −0.957241 0.289291i \(-0.906581\pi\)
0.957241 0.289291i \(-0.0934194\pi\)
\(770\) 0 0
\(771\) −18.3484 −0.660802
\(772\) 0 0
\(773\) −9.15671 15.8599i −0.329344 0.570440i 0.653038 0.757325i \(-0.273494\pi\)
−0.982382 + 0.186885i \(0.940161\pi\)
\(774\) 0 0
\(775\) −19.6631 + 34.0574i −0.706318 + 1.22338i
\(776\) 0 0
\(777\) −21.4403 + 10.0194i −0.769166 + 0.359443i
\(778\) 0 0
\(779\) 18.7060 32.3998i 0.670213 1.16084i
\(780\) 0 0
\(781\) −5.17518 + 2.98789i −0.185183 + 0.106915i
\(782\) 0 0
\(783\) −3.33885 −0.119321
\(784\) 0 0
\(785\) −40.1090 −1.43155
\(786\) 0 0
\(787\) −3.88976 + 2.24576i −0.138655 + 0.0800526i −0.567723 0.823220i \(-0.692175\pi\)
0.429068 + 0.903272i \(0.358842\pi\)
\(788\) 0 0
\(789\) −1.74385 + 3.02044i −0.0620828 + 0.107531i
\(790\) 0 0
\(791\) −25.6689 + 11.9955i −0.912680 + 0.426510i
\(792\) 0 0
\(793\) −7.02460 + 12.1670i −0.249451 + 0.432062i
\(794\) 0 0
\(795\) −3.26504 5.65522i −0.115799 0.200570i
\(796\) 0 0
\(797\) −31.7698 −1.12534 −0.562672 0.826680i \(-0.690227\pi\)
−0.562672 + 0.826680i \(0.690227\pi\)
\(798\) 0 0
\(799\) 13.7703i 0.487159i
\(800\) 0 0
\(801\) 1.83829 1.06134i 0.0649527 0.0375004i
\(802\) 0 0
\(803\) −21.5259 12.4280i −0.759634 0.438575i
\(804\) 0 0
\(805\) 2.96759 4.24563i 0.104594 0.149639i
\(806\) 0 0
\(807\) −12.7410 7.35605i −0.448506 0.258945i
\(808\) 0 0
\(809\) −12.5784 21.7864i −0.442232 0.765969i 0.555623 0.831435i \(-0.312480\pi\)
−0.997855 + 0.0654659i \(0.979147\pi\)
\(810\) 0 0
\(811\) 48.0042i 1.68565i −0.538184 0.842827i \(-0.680890\pi\)
0.538184 0.842827i \(-0.319110\pi\)
\(812\) 0 0
\(813\) 19.9028i 0.698021i
\(814\) 0 0
\(815\) 40.5219 + 70.1859i 1.41942 + 2.45851i
\(816\) 0 0
\(817\) −20.8985 12.0658i −0.731148 0.422128i
\(818\) 0 0
\(819\) −0.351896 + 4.06089i −0.0122962 + 0.141899i
\(820\) 0 0
\(821\) −14.5854 8.42089i −0.509034 0.293891i 0.223402 0.974726i \(-0.428284\pi\)
−0.732437 + 0.680835i \(0.761617\pi\)
\(822\) 0 0
\(823\) −17.1163 + 9.88209i −0.596636 + 0.344468i −0.767717 0.640789i \(-0.778607\pi\)
0.171081 + 0.985257i \(0.445274\pi\)
\(824\) 0 0
\(825\) 21.2760i 0.740735i
\(826\) 0 0
\(827\) −49.1702 −1.70981 −0.854907 0.518781i \(-0.826386\pi\)
−0.854907 + 0.518781i \(0.826386\pi\)
\(828\) 0 0
\(829\) −7.96007 13.7872i −0.276465 0.478851i 0.694039 0.719937i \(-0.255829\pi\)
−0.970504 + 0.241087i \(0.922496\pi\)
\(830\) 0 0
\(831\) 13.2233 22.9034i 0.458710 0.794510i
\(832\) 0 0
\(833\) 2.80992 16.0915i 0.0973580 0.557539i
\(834\) 0 0
\(835\) 37.0541 64.1795i 1.28231 2.22102i
\(836\) 0 0
\(837\) 2.73883 1.58126i 0.0946678 0.0546565i
\(838\) 0 0
\(839\) 38.3305 1.32332 0.661658 0.749806i \(-0.269853\pi\)
0.661658 + 0.749806i \(0.269853\pi\)
\(840\) 0 0
\(841\) 17.8521 0.615589
\(842\) 0 0
\(843\) 17.5662 10.1419i 0.605013 0.349304i
\(844\) 0 0
\(845\) −22.1856 + 38.4266i −0.763207 + 1.32191i
\(846\) 0 0
\(847\) 1.84387 21.2783i 0.0633561 0.731131i
\(848\) 0 0
\(849\) 3.29336 5.70426i 0.113028 0.195770i
\(850\) 0 0
\(851\) 2.09705 + 3.63220i 0.0718860 + 0.124510i
\(852\) 0 0
\(853\) 22.5158 0.770927 0.385463 0.922723i \(-0.374042\pi\)
0.385463 + 0.922723i \(0.374042\pi\)
\(854\) 0 0
\(855\) 29.3963i 1.00533i
\(856\) 0 0
\(857\) −11.1913 + 6.46129i −0.382287 + 0.220713i −0.678813 0.734311i \(-0.737505\pi\)
0.296526 + 0.955025i \(0.404172\pi\)
\(858\) 0 0
\(859\) −20.1418 11.6289i −0.687229 0.396772i 0.115344 0.993326i \(-0.463203\pi\)
−0.802573 + 0.596554i \(0.796536\pi\)
\(860\) 0 0
\(861\) −11.5238 8.05485i −0.392730 0.274509i
\(862\) 0 0
\(863\) −7.69412 4.44220i −0.261911 0.151214i 0.363295 0.931674i \(-0.381652\pi\)
−0.625206 + 0.780460i \(0.714985\pi\)
\(864\) 0 0
\(865\) −14.9098 25.8245i −0.506947 0.878059i
\(866\) 0 0
\(867\) 11.5544i 0.392409i
\(868\) 0 0
\(869\) 2.88660i 0.0979212i
\(870\) 0 0
\(871\) 5.75667 + 9.97084i 0.195057 + 0.337849i
\(872\) 0 0
\(873\) 1.69050 + 0.976012i 0.0572148 + 0.0330330i
\(874\) 0 0
\(875\) −74.4135 + 34.7746i −2.51564 + 1.17560i
\(876\) 0 0
\(877\) 40.2134 + 23.2172i 1.35791 + 0.783990i 0.989342 0.145610i \(-0.0465145\pi\)
0.368569 + 0.929600i \(0.379848\pi\)
\(878\) 0 0
\(879\) −11.2097 + 6.47192i −0.378094 + 0.218292i
\(880\) 0 0
\(881\) 27.1901i 0.916058i −0.888937 0.458029i \(-0.848556\pi\)
0.888937 0.458029i \(-0.151444\pi\)
\(882\) 0 0
\(883\) 21.7975 0.733543 0.366772 0.930311i \(-0.380463\pi\)
0.366772 + 0.930311i \(0.380463\pi\)
\(884\) 0 0
\(885\) 12.6984 + 21.9942i 0.426852 + 0.739329i
\(886\) 0 0
\(887\) −21.1661 + 36.6607i −0.710688 + 1.23095i 0.253911 + 0.967227i \(0.418283\pi\)
−0.964599 + 0.263720i \(0.915051\pi\)
\(888\) 0 0
\(889\) 10.6383 + 22.7646i 0.356796 + 0.763502i
\(890\) 0 0
\(891\) 0.855485 1.48174i 0.0286598 0.0496403i
\(892\) 0 0
\(893\) −35.9777 + 20.7717i −1.20395 + 0.695099i
\(894\) 0 0
\(895\) 99.8629 3.33805
\(896\) 0 0
\(897\) 0.722374 0.0241194
\(898\) 0 0
\(899\) 9.14454 5.27960i 0.304987 0.176085i
\(900\) 0 0
\(901\) −1.82473 + 3.16053i −0.0607906 + 0.105292i
\(902\) 0 0
\(903\) −5.19555 + 7.43310i −0.172897 + 0.247358i
\(904\) 0 0
\(905\) 42.9186 74.3371i 1.42666 2.47105i
\(906\) 0 0
\(907\) −22.5605 39.0760i −0.749109 1.29750i −0.948250 0.317525i \(-0.897148\pi\)
0.199141 0.979971i \(-0.436185\pi\)
\(908\) 0 0
\(909\) 13.7809 0.457083
\(910\) 0 0
\(911\) 36.2714i 1.20173i −0.799352 0.600863i \(-0.794824\pi\)
0.799352 0.600863i \(-0.205176\pi\)
\(912\) 0 0
\(913\) −4.03926 + 2.33207i −0.133680 + 0.0771801i
\(914\) 0 0
\(915\) 32.9760 + 19.0387i 1.09015 + 0.629400i
\(916\) 0 0
\(917\) 36.0258 + 3.12181i 1.18968 + 0.103091i
\(918\) 0 0
\(919\) −37.7905 21.8183i −1.24659 0.719721i −0.276164 0.961110i \(-0.589063\pi\)
−0.970428 + 0.241390i \(0.922397\pi\)
\(920\) 0 0
\(921\) 5.93863 + 10.2860i 0.195685 + 0.338936i
\(922\) 0 0
\(923\) 5.38082i 0.177112i
\(924\) 0 0
\(925\) 111.230i 3.65721i
\(926\) 0 0
\(927\) −3.84129 6.65331i −0.126165 0.218523i
\(928\) 0 0
\(929\) 10.2075 + 5.89332i 0.334898 + 0.193354i 0.658014 0.753006i \(-0.271397\pi\)
−0.323115 + 0.946360i \(0.604730\pi\)
\(930\) 0 0
\(931\) −46.2809 + 16.9317i −1.51680 + 0.554913i
\(932\) 0 0
\(933\) 10.2511 + 5.91849i 0.335607 + 0.193763i
\(934\) 0 0
\(935\) 14.4380 8.33578i 0.472173 0.272609i
\(936\) 0 0
\(937\) 35.0529i 1.14513i −0.819860 0.572565i \(-0.805949\pi\)
0.819860 0.572565i \(-0.194051\pi\)
\(938\) 0 0
\(939\) 14.8243 0.483774
\(940\) 0 0
\(941\) 14.2985 + 24.7658i 0.466119 + 0.807342i 0.999251 0.0386903i \(-0.0123186\pi\)
−0.533132 + 0.846032i \(0.678985\pi\)
\(942\) 0 0
\(943\) −1.24585 + 2.15788i −0.0405705 + 0.0702702i
\(944\) 0 0
\(945\) 11.0062 + 0.953738i 0.358031 + 0.0310251i
\(946\) 0 0
\(947\) −25.7444 + 44.5906i −0.836580 + 1.44900i 0.0561576 + 0.998422i \(0.482115\pi\)
−0.892738 + 0.450577i \(0.851218\pi\)
\(948\) 0 0
\(949\) 19.3828 11.1906i 0.629191 0.363264i
\(950\) 0 0
\(951\) 6.78361 0.219974
\(952\) 0 0
\(953\) −11.8761 −0.384706 −0.192353 0.981326i \(-0.561612\pi\)
−0.192353 + 0.981326i \(0.561612\pi\)
\(954\) 0 0
\(955\) 7.28724 4.20729i 0.235810 0.136145i
\(956\) 0 0
\(957\) 2.85634 4.94732i 0.0923323 0.159924i
\(958\) 0 0
\(959\) 25.9989 + 18.1726i 0.839548 + 0.586823i
\(960\) 0 0
\(961\) 10.4992 18.1852i 0.338684 0.586618i
\(962\) 0 0
\(963\) 7.20414 + 12.4779i 0.232150 + 0.402096i
\(964\) 0 0
\(965\) 14.9096 0.479956
\(966\) 0 0
\(967\) 59.2193i 1.90437i −0.305530 0.952183i \(-0.598833\pi\)
0.305530 0.952183i \(-0.401167\pi\)
\(968\) 0 0
\(969\) 14.2276 8.21434i 0.457058 0.263883i
\(970\) 0 0
\(971\) 4.53638 + 2.61908i 0.145579 + 0.0840503i 0.571020 0.820936i \(-0.306548\pi\)
−0.425441 + 0.904986i \(0.639881\pi\)
\(972\) 0 0
\(973\) −7.44782 15.9375i −0.238766 0.510931i
\(974\) 0 0
\(975\) −16.5911 9.57885i −0.531339 0.306769i
\(976\) 0 0
\(977\) 26.6049 + 46.0810i 0.851165 + 1.47426i 0.880158 + 0.474682i \(0.157437\pi\)
−0.0289924 + 0.999580i \(0.509230\pi\)
\(978\) 0 0
\(979\) 3.63183i 0.116074i
\(980\) 0 0
\(981\) 3.37612i 0.107791i
\(982\) 0 0
\(983\) −13.9737 24.2032i −0.445693 0.771963i 0.552407 0.833574i \(-0.313709\pi\)
−0.998100 + 0.0616113i \(0.980376\pi\)
\(984\) 0 0
\(985\) −63.4241 36.6179i −2.02086 1.16674i
\(986\) 0 0
\(987\) 6.60981 + 14.1442i 0.210393 + 0.450215i
\(988\) 0 0
\(989\) 1.39188 + 0.803601i 0.0442592 + 0.0255530i
\(990\) 0 0
\(991\) 18.7980 10.8530i 0.597139 0.344758i −0.170776 0.985310i \(-0.554628\pi\)
0.767915 + 0.640552i \(0.221294\pi\)
\(992\) 0 0
\(993\) 4.74570i 0.150600i
\(994\) 0 0
\(995\) 73.9349 2.34389
\(996\) 0 0
\(997\) 16.8274 + 29.1458i 0.532928 + 0.923058i 0.999261 + 0.0384485i \(0.0122416\pi\)
−0.466333 + 0.884609i \(0.654425\pi\)
\(998\) 0 0
\(999\) −4.47243 + 7.74648i −0.141501 + 0.245088i
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 672.2.bb.a.271.8 32
3.2 odd 2 2016.2.bs.c.271.1 32
4.3 odd 2 168.2.t.a.19.1 32
7.2 even 3 4704.2.p.a.3919.5 32
7.3 odd 6 inner 672.2.bb.a.367.1 32
7.5 odd 6 4704.2.p.a.3919.24 32
8.3 odd 2 inner 672.2.bb.a.271.1 32
8.5 even 2 168.2.t.a.19.12 yes 32
12.11 even 2 504.2.bk.c.19.16 32
21.17 even 6 2016.2.bs.c.1711.16 32
24.5 odd 2 504.2.bk.c.19.5 32
24.11 even 2 2016.2.bs.c.271.16 32
28.3 even 6 168.2.t.a.115.12 yes 32
28.19 even 6 1176.2.p.a.979.19 32
28.23 odd 6 1176.2.p.a.979.20 32
56.3 even 6 inner 672.2.bb.a.367.8 32
56.5 odd 6 1176.2.p.a.979.18 32
56.19 even 6 4704.2.p.a.3919.6 32
56.37 even 6 1176.2.p.a.979.17 32
56.45 odd 6 168.2.t.a.115.1 yes 32
56.51 odd 6 4704.2.p.a.3919.23 32
84.59 odd 6 504.2.bk.c.451.5 32
168.59 odd 6 2016.2.bs.c.1711.1 32
168.101 even 6 504.2.bk.c.451.16 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.t.a.19.1 32 4.3 odd 2
168.2.t.a.19.12 yes 32 8.5 even 2
168.2.t.a.115.1 yes 32 56.45 odd 6
168.2.t.a.115.12 yes 32 28.3 even 6
504.2.bk.c.19.5 32 24.5 odd 2
504.2.bk.c.19.16 32 12.11 even 2
504.2.bk.c.451.5 32 84.59 odd 6
504.2.bk.c.451.16 32 168.101 even 6
672.2.bb.a.271.1 32 8.3 odd 2 inner
672.2.bb.a.271.8 32 1.1 even 1 trivial
672.2.bb.a.367.1 32 7.3 odd 6 inner
672.2.bb.a.367.8 32 56.3 even 6 inner
1176.2.p.a.979.17 32 56.37 even 6
1176.2.p.a.979.18 32 56.5 odd 6
1176.2.p.a.979.19 32 28.19 even 6
1176.2.p.a.979.20 32 28.23 odd 6
2016.2.bs.c.271.1 32 3.2 odd 2
2016.2.bs.c.271.16 32 24.11 even 2
2016.2.bs.c.1711.1 32 168.59 odd 6
2016.2.bs.c.1711.16 32 21.17 even 6
4704.2.p.a.3919.5 32 7.2 even 3
4704.2.p.a.3919.6 32 56.19 even 6
4704.2.p.a.3919.23 32 56.51 odd 6
4704.2.p.a.3919.24 32 7.5 odd 6