# Properties

 Label 441.2.e.d.361.1 Level $441$ Weight $2$ Character 441.361 Analytic conductor $3.521$ Analytic rank $0$ Dimension $2$ CM discriminant -7 Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.e (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.52140272914$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 49) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## Embedding invariants

 Embedding label 361.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 441.361 Dual form 441.2.e.d.226.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +3.00000 q^{8} +O(q^{10})$$ $$q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +3.00000 q^{8} +(2.00000 - 3.46410i) q^{11} +(0.500000 + 0.866025i) q^{16} +4.00000 q^{22} +(4.00000 + 6.92820i) q^{23} +(2.50000 - 4.33013i) q^{25} -2.00000 q^{29} +(2.50000 - 4.33013i) q^{32} +(3.00000 + 5.19615i) q^{37} -12.0000 q^{43} +(-2.00000 - 3.46410i) q^{44} +(-4.00000 + 6.92820i) q^{46} +5.00000 q^{50} +(-5.00000 + 8.66025i) q^{53} +(-1.00000 - 1.73205i) q^{58} +7.00000 q^{64} +(-2.00000 + 3.46410i) q^{67} -16.0000 q^{71} +(-3.00000 + 5.19615i) q^{74} +(-4.00000 - 6.92820i) q^{79} +(-6.00000 - 10.3923i) q^{86} +(6.00000 - 10.3923i) q^{88} +8.00000 q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + q^{4} + 6q^{8} + O(q^{10})$$ $$2q + q^{2} + q^{4} + 6q^{8} + 4q^{11} + q^{16} + 8q^{22} + 8q^{23} + 5q^{25} - 4q^{29} + 5q^{32} + 6q^{37} - 24q^{43} - 4q^{44} - 8q^{46} + 10q^{50} - 10q^{53} - 2q^{58} + 14q^{64} - 4q^{67} - 32q^{71} - 6q^{74} - 8q^{79} - 12q^{86} + 12q^{88} + 16q^{92} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.500000 + 0.866025i 0.353553 + 0.612372i 0.986869 0.161521i $$-0.0516399\pi$$
−0.633316 + 0.773893i $$0.718307\pi$$
$$3$$ 0 0
$$4$$ 0.500000 0.866025i 0.250000 0.433013i
$$5$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 3.00000 1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.00000 3.46410i 0.603023 1.04447i −0.389338 0.921095i $$-0.627296\pi$$
0.992361 0.123371i $$-0.0393705\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0.500000 + 0.866025i 0.125000 + 0.216506i
$$17$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$18$$ 0 0
$$19$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 4.00000 + 6.92820i 0.834058 + 1.44463i 0.894795 + 0.446476i $$0.147321\pi$$
−0.0607377 + 0.998154i $$0.519345\pi$$
$$24$$ 0 0
$$25$$ 2.50000 4.33013i 0.500000 0.866025i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$32$$ 2.50000 4.33013i 0.441942 0.765466i
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 3.00000 + 5.19615i 0.493197 + 0.854242i 0.999969 0.00783774i $$-0.00249486\pi$$
−0.506772 + 0.862080i $$0.669162\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ −12.0000 −1.82998 −0.914991 0.403473i $$-0.867803\pi$$
−0.914991 + 0.403473i $$0.867803\pi$$
$$44$$ −2.00000 3.46410i −0.301511 0.522233i
$$45$$ 0 0
$$46$$ −4.00000 + 6.92820i −0.589768 + 1.02151i
$$47$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 5.00000 0.707107
$$51$$ 0 0
$$52$$ 0 0
$$53$$ −5.00000 + 8.66025i −0.686803 + 1.18958i 0.286064 + 0.958211i $$0.407653\pi$$
−0.972867 + 0.231367i $$0.925680\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −1.00000 1.73205i −0.131306 0.227429i
$$59$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$60$$ 0 0
$$61$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 7.00000 0.875000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −2.00000 + 3.46410i −0.244339 + 0.423207i −0.961946 0.273241i $$-0.911904\pi$$
0.717607 + 0.696449i $$0.245238\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −16.0000 −1.89885 −0.949425 0.313993i $$-0.898333\pi$$
−0.949425 + 0.313993i $$0.898333\pi$$
$$72$$ 0 0
$$73$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$74$$ −3.00000 + 5.19615i −0.348743 + 0.604040i
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −4.00000 6.92820i −0.450035 0.779484i 0.548352 0.836247i $$-0.315255\pi$$
−0.998388 + 0.0567635i $$0.981922\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −6.00000 10.3923i −0.646997 1.12063i
$$87$$ 0 0
$$88$$ 6.00000 10.3923i 0.639602 1.10782i
$$89$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −2.50000 4.33013i −0.250000 0.433013i
$$101$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$102$$ 0 0
$$103$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ −10.0000 17.3205i −0.966736 1.67444i −0.704875 0.709331i $$-0.748997\pi$$
−0.261861 0.965106i $$-0.584336\pi$$
$$108$$ 0 0
$$109$$ −9.00000 + 15.5885i −0.862044 + 1.49310i 0.00790932 + 0.999969i $$0.497482\pi$$
−0.869953 + 0.493135i $$0.835851\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −2.00000 −0.188144 −0.0940721 0.995565i $$-0.529988\pi$$
−0.0940721 + 0.995565i $$0.529988\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.00000 + 1.73205i −0.0928477 + 0.160817i
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.50000 4.33013i −0.227273 0.393648i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 16.0000 1.41977 0.709885 0.704317i $$-0.248747\pi$$
0.709885 + 0.704317i $$0.248747\pi$$
$$128$$ −1.50000 2.59808i −0.132583 0.229640i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −5.00000 + 8.66025i −0.427179 + 0.739895i −0.996621 0.0821359i $$-0.973826\pi$$
0.569442 + 0.822031i $$0.307159\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −8.00000 13.8564i −0.671345 1.16280i
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ 11.0000 + 19.0526i 0.901155 + 1.56085i 0.825997 + 0.563675i $$0.190613\pi$$
0.0751583 + 0.997172i $$0.476054\pi$$
$$150$$ 0 0
$$151$$ 12.0000 20.7846i 0.976546 1.69143i 0.301811 0.953368i $$-0.402409\pi$$
0.674735 0.738060i $$-0.264258\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$158$$ 4.00000 6.92820i 0.318223 0.551178i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 10.0000 + 17.3205i 0.783260 + 1.35665i 0.930033 + 0.367477i $$0.119778\pi$$
−0.146772 + 0.989170i $$0.546888\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −13.0000 −1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −6.00000 + 10.3923i −0.457496 + 0.792406i
$$173$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 2.00000 3.46410i 0.149487 0.258919i −0.781551 0.623841i $$-0.785571\pi$$
0.931038 + 0.364922i $$0.118904\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 12.0000 + 20.7846i 0.884652 + 1.53226i
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 4.00000 + 6.92820i 0.289430 + 0.501307i 0.973674 0.227946i $$-0.0732010\pi$$
−0.684244 + 0.729253i $$0.739868\pi$$
$$192$$ 0 0
$$193$$ −9.00000 + 15.5885i −0.647834 + 1.12208i 0.335805 + 0.941932i $$0.390992\pi$$
−0.983639 + 0.180150i $$0.942342\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 26.0000 1.85242 0.926212 0.377004i $$-0.123046\pi$$
0.926212 + 0.377004i $$0.123046\pi$$
$$198$$ 0 0
$$199$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$200$$ 7.50000 12.9904i 0.530330 0.918559i
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −12.0000 −0.826114 −0.413057 0.910705i $$-0.635539\pi$$
−0.413057 + 0.910705i $$0.635539\pi$$
$$212$$ 5.00000 + 8.66025i 0.343401 + 0.594789i
$$213$$ 0 0
$$214$$ 10.0000 17.3205i 0.683586 1.18401i
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −18.0000 −1.21911
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −1.00000 1.73205i −0.0665190 0.115214i
$$227$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$228$$ 0 0
$$229$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 11.0000 + 19.0526i 0.720634 + 1.24817i 0.960746 + 0.277429i $$0.0894825\pi$$
−0.240112 + 0.970745i $$0.577184\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −16.0000 −1.03495 −0.517477 0.855697i $$-0.673129\pi$$
−0.517477 + 0.855697i $$0.673129\pi$$
$$240$$ 0 0
$$241$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$242$$ 2.50000 4.33013i 0.160706 0.278351i
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 32.0000 2.01182
$$254$$ 8.00000 + 13.8564i 0.501965 + 0.869428i
$$255$$ 0 0
$$256$$ 8.50000 14.7224i 0.531250 0.920152i
$$257$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 16.0000 27.7128i 0.986602 1.70885i 0.352014 0.935995i $$-0.385497\pi$$
0.634588 0.772851i $$-0.281170\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2.00000 + 3.46410i 0.122169 + 0.211604i
$$269$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$270$$ 0 0
$$271$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ −10.0000 −0.604122
$$275$$ −10.0000 17.3205i −0.603023 1.04447i
$$276$$ 0 0
$$277$$ 5.00000 8.66025i 0.300421 0.520344i −0.675810 0.737075i $$-0.736206\pi$$
0.976231 + 0.216731i $$0.0695395\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$284$$ −8.00000 + 13.8564i −0.474713 + 0.822226i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 8.50000 + 14.7224i 0.500000 + 0.866025i
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 9.00000 + 15.5885i 0.523114 + 0.906061i
$$297$$ 0 0
$$298$$ −11.0000 + 19.0526i −0.637213 + 1.10369i
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 24.0000 1.38104
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$312$$ 0 0
$$313$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −8.00000 −0.450035
$$317$$ −17.0000 29.4449i −0.954815 1.65379i −0.734791 0.678294i $$-0.762720\pi$$
−0.220024 0.975494i $$-0.570614\pi$$
$$318$$ 0 0
$$319$$ −4.00000 + 6.92820i −0.223957 + 0.387905i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −10.0000 + 17.3205i −0.553849 + 0.959294i
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −18.0000 31.1769i −0.989369 1.71364i −0.620625 0.784107i $$-0.713121\pi$$
−0.368744 0.929531i $$-0.620212\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 30.0000 1.63420 0.817102 0.576493i $$-0.195579\pi$$
0.817102 + 0.576493i $$0.195579\pi$$
$$338$$ −6.50000 11.2583i −0.353553 0.612372i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −36.0000 −1.94099
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2.00000 3.46410i 0.107366 0.185963i −0.807337 0.590091i $$-0.799092\pi$$
0.914702 + 0.404128i $$0.132425\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −10.0000 17.3205i −0.533002 0.923186i
$$353$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 4.00000 0.211407
$$359$$ 4.00000 + 6.92820i 0.211112 + 0.365657i 0.952063 0.305903i $$-0.0989582\pi$$
−0.740951 + 0.671559i $$0.765625\pi$$
$$360$$ 0 0
$$361$$ 9.50000 16.4545i 0.500000 0.866025i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$368$$ −4.00000 + 6.92820i −0.208514 + 0.361158i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −11.0000 19.0526i −0.569558 0.986504i −0.996610 0.0822766i $$-0.973781\pi$$
0.427051 0.904227i $$-0.359552\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ −12.0000 −0.616399 −0.308199 0.951322i $$-0.599726\pi$$
−0.308199 + 0.951322i $$0.599726\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −4.00000 + 6.92820i −0.204658 + 0.354478i
$$383$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −18.0000 −0.916176
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −19.0000 + 32.9090i −0.963338 + 1.66855i −0.249323 + 0.968420i $$0.580208\pi$$
−0.714015 + 0.700130i $$0.753125\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 13.0000 + 22.5167i 0.654931 + 1.13437i
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 5.00000 0.250000
$$401$$ −17.0000 29.4449i −0.848939 1.47041i −0.882156 0.470958i $$-0.843908\pi$$
0.0332161 0.999448i $$-0.489425\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000 1.18964
$$408$$ 0 0
$$409$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −26.0000 −1.26716 −0.633581 0.773676i $$-0.718416\pi$$
−0.633581 + 0.773676i $$0.718416\pi$$
$$422$$ −6.00000 10.3923i −0.292075 0.505889i
$$423$$ 0 0
$$424$$ −15.0000 + 25.9808i −0.728464 + 1.26174i
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −20.0000 −0.966736
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 16.0000 27.7128i 0.770693 1.33488i −0.166491 0.986043i $$-0.553244\pi$$
0.937184 0.348836i $$-0.113423\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 9.00000 + 15.5885i 0.431022 + 0.746552i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −10.0000 17.3205i −0.475114 0.822922i 0.524479 0.851423i $$-0.324260\pi$$
−0.999594 + 0.0285009i $$0.990927\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −2.00000 −0.0943858 −0.0471929 0.998886i $$-0.515028\pi$$
−0.0471929 + 0.998886i $$0.515028\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ −1.00000 + 1.73205i −0.0470360 + 0.0814688i
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3.00000 + 5.19615i 0.140334 + 0.243066i 0.927622 0.373519i $$-0.121849\pi$$
−0.787288 + 0.616585i $$0.788516\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ −40.0000 −1.85896 −0.929479 0.368875i $$-0.879743\pi$$
−0.929479 + 0.368875i $$0.879743\pi$$
$$464$$ −1.00000 1.73205i −0.0464238 0.0804084i
$$465$$ 0 0
$$466$$ −11.0000 + 19.0526i −0.509565 + 0.882593i
$$467$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −24.0000 + 41.5692i −1.10352 + 1.91135i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −8.00000 13.8564i −0.365911 0.633777i
$$479$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −5.00000 −0.227273
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 12.0000 20.7846i 0.543772 0.941841i −0.454911 0.890537i $$-0.650329\pi$$
0.998683 0.0513038i $$-0.0163377\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −44.0000 −1.98569 −0.992846 0.119401i $$-0.961903\pi$$
−0.992846 + 0.119401i $$0.961903\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −18.0000 31.1769i −0.805791 1.39567i −0.915756 0.401735i $$-0.868407\pi$$
0.109965 0.993935i $$-0.464926\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 16.0000 + 27.7128i 0.711287 + 1.23198i
$$507$$ 0 0
$$508$$ 8.00000 13.8564i 0.354943 0.614779i
$$509$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 11.0000 0.486136
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$522$$ 0 0
$$523$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −20.5000 + 35.5070i −0.891304 + 1.54378i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −6.00000 + 10.3923i −0.259161 + 0.448879i
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 17.0000 + 29.4449i 0.730887 + 1.26593i 0.956504 + 0.291718i $$0.0942267\pi$$
−0.225617 + 0.974216i $$0.572440\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 44.0000 1.88130 0.940652 0.339372i $$-0.110215\pi$$
0.940652 + 0.339372i $$0.110215\pi$$
$$548$$ 5.00000 + 8.66025i 0.213589 + 0.369948i
$$549$$ 0 0
$$550$$ 10.0000 17.3205i 0.426401 0.738549i
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 10.0000 0.424859
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 23.0000 39.8372i 0.974541 1.68796i 0.293101 0.956082i $$-0.405313\pi$$
0.681441 0.731873i $$-0.261354\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 13.0000 + 22.5167i 0.548372 + 0.949808i
$$563$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ −48.0000 −2.01404
$$569$$ 11.0000 + 19.0526i 0.461144 + 0.798725i 0.999018 0.0443003i $$-0.0141058\pi$$
−0.537874 + 0.843025i $$0.680772\pi$$
$$570$$ 0 0
$$571$$ −2.00000 + 3.46410i −0.0836974 + 0.144968i −0.904835 0.425762i $$-0.860006\pi$$
0.821138 + 0.570730i $$0.193340\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 40.0000 1.66812
$$576$$ 0 0
$$577$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$578$$ −8.50000 + 14.7224i −0.353553 + 0.612372i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 20.0000 + 34.6410i 0.828315 + 1.43468i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −3.00000 + 5.19615i −0.123299 + 0.213561i
$$593$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 22.0000 0.901155
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 16.0000 27.7128i 0.653742 1.13231i −0.328465 0.944516i $$-0.606531\pi$$
0.982208 0.187799i $$-0.0601353\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −12.0000 20.7846i −0.488273 0.845714i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 19.0000 32.9090i 0.767403 1.32918i −0.171564 0.985173i $$-0.554882\pi$$
0.938967 0.344008i $$-0.111785\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 26.0000 1.04672 0.523360 0.852111i $$-0.324678\pi$$
0.523360 + 0.852111i $$0.324678\pi$$
$$618$$ 0 0
$$619$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −12.5000 21.6506i −0.500000 0.866025i
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 16.0000 0.636950 0.318475 0.947931i $$-0.396829\pi$$
0.318475 + 0.947931i $$0.396829\pi$$
$$632$$ −12.0000 20.7846i −0.477334 0.826767i
$$633$$ 0 0
$$634$$ 17.0000 29.4449i 0.675156 1.16940i
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −8.00000 −0.316723
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 23.0000 39.8372i 0.908445 1.57347i 0.0922210 0.995739i $$-0.470603\pi$$
0.816224 0.577735i $$-0.196063\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ 25.0000 + 43.3013i 0.978326 + 1.69451i 0.668493 + 0.743719i $$0.266940\pi$$
0.309833 + 0.950791i $$0.399727\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −44.0000 −1.71400 −0.856998 0.515319i $$-0.827673\pi$$
−0.856998 + 0.515319i $$0.827673\pi$$
$$660$$ 0 0
$$661$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$662$$ 18.0000 31.1769i 0.699590 1.21173i
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −8.00000 13.8564i −0.309761 0.536522i
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 30.0000 1.15642 0.578208 0.815890i $$-0.303752\pi$$
0.578208 + 0.815890i $$0.303752\pi$$
$$674$$ 15.0000 + 25.9808i 0.577778 + 1.00074i
$$675$$ 0 0
$$676$$ −6.50000 + 11.2583i −0.250000 + 0.433013i
$$677$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −26.0000 + 45.0333i −0.994862 + 1.72315i −0.409757 + 0.912194i $$0.634387\pi$$
−0.585105 + 0.810958i $$0.698947\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −6.00000 10.3923i −0.228748 0.396203i
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 14.0000 24.2487i 0.527645 0.913908i
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 3.00000 + 5.19615i 0.112667 + 0.195146i 0.916845 0.399244i $$-0.130727\pi$$
−0.804178 + 0.594389i $$0.797394\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −2.00000 3.46410i −0.0747435 0.129460i
$$717$$ 0 0
$$718$$ −4.00000 + 6.92820i −0.149279 + 0.258558i
$$719$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 19.0000 0.707107
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −5.00000 + 8.66025i −0.185695 + 0.321634i
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 40.0000 1.47442
$$737$$ 8.00000 + 13.8564i 0.294684 + 0.510407i
$$738$$ 0 0
$$739$$ 26.0000 45.0333i 0.956425 1.65658i 0.225354 0.974277i $$-0.427646\pi$$
0.731072 0.682300i $$-0.239020\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 40.0000 1.46746 0.733729 0.679442i $$-0.237778\pi$$
0.733729 + 0.679442i $$0.237778\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 11.0000 19.0526i 0.402739 0.697564i
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 24.0000 + 41.5692i 0.875772 + 1.51688i 0.855938 + 0.517079i $$0.172981\pi$$
0.0198348 + 0.999803i $$0.493686\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −54.0000 −1.96266 −0.981332 0.192323i $$-0.938398\pi$$
−0.981332 + 0.192323i $$0.938398\pi$$
$$758$$ −6.00000 10.3923i −0.217930 0.377466i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 8.00000 0.289430
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 9.00000 + 15.5885i 0.323917 + 0.561041i
$$773$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ −38.0000 −1.36237
$$779$$ 0 0
$$780$$ 0 0
$$781$$ −32.0000 + 55.4256i −1.14505 + 1.98328i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$788$$ 13.0000 22.5167i 0.463106 0.802123i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ −12.5000 21.6506i −0.441942 0.765466i
$$801$$ 0 0
$$802$$ 17.0000 29.4449i 0.600291 1.03973i
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0