# Properties

 Label 2268.2.j.f.757.1 Level 2268 Weight 2 Character 2268.757 Analytic conductor 18.110 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2268 = 2^{2} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2268.j (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.1100711784$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## Embedding invariants

 Embedding label 757.1 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 2268.757 Dual form 2268.2.j.f.1513.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-0.500000 - 0.866025i) q^{7} +O(q^{10})$$ $$q+(-0.500000 - 0.866025i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(-1.00000 + 1.73205i) q^{13} -4.00000 q^{19} +(-3.00000 + 5.19615i) q^{23} +(2.50000 + 4.33013i) q^{25} +(3.00000 + 5.19615i) q^{29} +(-4.00000 + 6.92820i) q^{31} +2.00000 q^{37} +(6.00000 - 10.3923i) q^{41} +(2.00000 + 3.46410i) q^{43} +(6.00000 + 10.3923i) q^{47} +(-0.500000 + 0.866025i) q^{49} +6.00000 q^{53} +(5.00000 + 8.66025i) q^{61} +(-4.00000 + 6.92820i) q^{67} -6.00000 q^{71} -10.0000 q^{73} +(-3.00000 + 5.19615i) q^{77} +(2.00000 + 3.46410i) q^{79} +(-6.00000 - 10.3923i) q^{83} -12.0000 q^{89} +2.00000 q^{91} +(5.00000 + 8.66025i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{7} + O(q^{10})$$ $$2q - q^{7} - 6q^{11} - 2q^{13} - 8q^{19} - 6q^{23} + 5q^{25} + 6q^{29} - 8q^{31} + 4q^{37} + 12q^{41} + 4q^{43} + 12q^{47} - q^{49} + 12q^{53} + 10q^{61} - 8q^{67} - 12q^{71} - 20q^{73} - 6q^{77} + 4q^{79} - 12q^{83} - 24q^{89} + 4q^{91} + 10q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$1135$$ $$1541$$ $$\chi(n)$$ $$1$$ $$1$$ $$e\left(\frac{1}{3}\right)$$

## 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 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 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$6$$ 0 0
$$7$$ −0.500000 0.866025i −0.188982 0.327327i
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.00000 5.19615i −0.904534 1.56670i −0.821541 0.570149i $$-0.806886\pi$$
−0.0829925 0.996550i $$-0.526448\pi$$
$$12$$ 0 0
$$13$$ −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 0.240192i $$-0.922790\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.00000 + 5.19615i −0.625543 + 1.08347i 0.362892 + 0.931831i $$0.381789\pi$$
−0.988436 + 0.151642i $$0.951544\pi$$
$$24$$ 0 0
$$25$$ 2.50000 + 4.33013i 0.500000 + 0.866025i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i $$0.0214140\pi$$
−0.440652 + 0.897678i $$0.645253\pi$$
$$30$$ 0 0
$$31$$ −4.00000 + 6.92820i −0.718421 + 1.24434i 0.243204 + 0.969975i $$0.421802\pi$$
−0.961625 + 0.274367i $$0.911532\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 2.00000 0.328798 0.164399 0.986394i $$-0.447432\pi$$
0.164399 + 0.986394i $$0.447432\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.00000 10.3923i 0.937043 1.62301i 0.166092 0.986110i $$-0.446885\pi$$
0.770950 0.636895i $$-0.219782\pi$$
$$42$$ 0 0
$$43$$ 2.00000 + 3.46410i 0.304997 + 0.528271i 0.977261 0.212041i $$-0.0680112\pi$$
−0.672264 + 0.740312i $$0.734678\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.00000 + 10.3923i 0.875190 + 1.51587i 0.856560 + 0.516047i $$0.172597\pi$$
0.0186297 + 0.999826i $$0.494070\pi$$
$$48$$ 0 0
$$49$$ −0.500000 + 0.866025i −0.0714286 + 0.123718i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$60$$ 0 0
$$61$$ 5.00000 + 8.66025i 0.640184 + 1.10883i 0.985391 + 0.170305i $$0.0544754\pi$$
−0.345207 + 0.938527i $$0.612191\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 + 6.92820i −0.488678 + 0.846415i −0.999915 0.0130248i $$-0.995854\pi$$
0.511237 + 0.859440i $$0.329187\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −6.00000 −0.712069 −0.356034 0.934473i $$-0.615871\pi$$
−0.356034 + 0.934473i $$0.615871\pi$$
$$72$$ 0 0
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.00000 + 5.19615i −0.341882 + 0.592157i
$$78$$ 0 0
$$79$$ 2.00000 + 3.46410i 0.225018 + 0.389742i 0.956325 0.292306i $$-0.0944227\pi$$
−0.731307 + 0.682048i $$0.761089\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −6.00000 10.3923i −0.658586 1.14070i −0.980982 0.194099i $$-0.937822\pi$$
0.322396 0.946605i $$-0.395512\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −12.0000 −1.27200 −0.635999 0.771690i $$-0.719412\pi$$
−0.635999 + 0.771690i $$0.719412\pi$$
$$90$$ 0 0
$$91$$ 2.00000 0.209657
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 5.00000 + 8.66025i 0.507673 + 0.879316i 0.999961 + 0.00888289i $$0.00282755\pi$$
−0.492287 + 0.870433i $$0.663839\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −6.00000 10.3923i −0.597022 1.03407i −0.993258 0.115924i $$-0.963017\pi$$
0.396236 0.918149i $$-0.370316\pi$$
$$102$$ 0 0
$$103$$ −4.00000 + 6.92820i −0.394132 + 0.682656i −0.992990 0.118199i $$-0.962288\pi$$
0.598858 + 0.800855i $$0.295621\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 6.00000 0.580042 0.290021 0.957020i $$-0.406338\pi$$
0.290021 + 0.957020i $$0.406338\pi$$
$$108$$ 0 0
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −3.00000 + 5.19615i −0.282216 + 0.488813i −0.971930 0.235269i $$-0.924403\pi$$
0.689714 + 0.724082i $$0.257736\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −12.5000 + 21.6506i −1.13636 + 1.96824i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −4.00000 −0.354943 −0.177471 0.984126i $$-0.556792\pi$$
−0.177471 + 0.984126i $$0.556792\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 6.00000 10.3923i 0.524222 0.907980i −0.475380 0.879781i $$-0.657689\pi$$
0.999602 0.0281993i $$-0.00897729\pi$$
$$132$$ 0 0
$$133$$ 2.00000 + 3.46410i 0.173422 + 0.300376i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000 + 5.19615i 0.256307 + 0.443937i 0.965250 0.261329i $$-0.0841608\pi$$
−0.708942 + 0.705266i $$0.750827\pi$$
$$138$$ 0 0
$$139$$ 2.00000 3.46410i 0.169638 0.293821i −0.768655 0.639664i $$-0.779074\pi$$
0.938293 + 0.345843i $$0.112407\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 12.0000 1.00349
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −3.00000 + 5.19615i −0.245770 + 0.425685i −0.962348 0.271821i $$-0.912374\pi$$
0.716578 + 0.697507i $$0.245707\pi$$
$$150$$ 0 0
$$151$$ −4.00000 6.92820i −0.325515 0.563809i 0.656101 0.754673i $$-0.272204\pi$$
−0.981617 + 0.190864i $$0.938871\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −7.00000 + 12.1244i −0.558661 + 0.967629i 0.438948 + 0.898513i $$0.355351\pi$$
−0.997609 + 0.0691164i $$0.977982\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 6.00000 0.472866
$$162$$ 0 0
$$163$$ −16.0000 −1.25322 −0.626608 0.779334i $$-0.715557\pi$$
−0.626608 + 0.779334i $$0.715557\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −6.00000 + 10.3923i −0.464294 + 0.804181i −0.999169 0.0407502i $$-0.987025\pi$$
0.534875 + 0.844931i $$0.320359\pi$$
$$168$$ 0 0
$$169$$ 4.50000 + 7.79423i 0.346154 + 0.599556i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −6.00000 10.3923i −0.456172 0.790112i 0.542583 0.840002i $$-0.317446\pi$$
−0.998755 + 0.0498898i $$0.984113\pi$$
$$174$$ 0 0
$$175$$ 2.50000 4.33013i 0.188982 0.327327i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 3.00000 + 5.19615i 0.217072 + 0.375980i 0.953912 0.300088i $$-0.0970159\pi$$
−0.736839 + 0.676068i $$0.763683\pi$$
$$192$$ 0 0
$$193$$ 5.00000 8.66025i 0.359908 0.623379i −0.628037 0.778183i $$-0.716141\pi$$
0.987945 + 0.154805i $$0.0494748\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000 0.427482 0.213741 0.976890i $$-0.431435\pi$$
0.213741 + 0.976890i $$0.431435\pi$$
$$198$$ 0 0
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 3.00000 5.19615i 0.210559 0.364698i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 12.0000 + 20.7846i 0.830057 + 1.43770i
$$210$$ 0 0
$$211$$ 2.00000 3.46410i 0.137686 0.238479i −0.788935 0.614477i $$-0.789367\pi$$
0.926620 + 0.375999i $$0.122700\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 8.00000 0.543075
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −4.00000 6.92820i −0.267860 0.463947i 0.700449 0.713702i $$-0.252983\pi$$
−0.968309 + 0.249756i $$0.919650\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000 + 20.7846i 0.796468 + 1.37952i 0.921903 + 0.387421i $$0.126634\pi$$
−0.125435 + 0.992102i $$0.540033\pi$$
$$228$$ 0 0
$$229$$ −1.00000 + 1.73205i −0.0660819 + 0.114457i −0.897173 0.441679i $$-0.854383\pi$$
0.831092 + 0.556136i $$0.187717\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −6.00000 −0.393073 −0.196537 0.980497i $$-0.562969\pi$$
−0.196537 + 0.980497i $$0.562969\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 9.00000 15.5885i 0.582162 1.00833i −0.413061 0.910703i $$-0.635540\pi$$
0.995223 0.0976302i $$-0.0311262\pi$$
$$240$$ 0 0
$$241$$ 5.00000 + 8.66025i 0.322078 + 0.557856i 0.980917 0.194429i $$-0.0622852\pi$$
−0.658838 + 0.752285i $$0.728952\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4.00000 6.92820i 0.254514 0.440831i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 36.0000 2.26330
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6.00000 + 10.3923i −0.374270 + 0.648254i −0.990217 0.139533i $$-0.955440\pi$$
0.615948 + 0.787787i $$0.288773\pi$$
$$258$$ 0 0
$$259$$ −1.00000 1.73205i −0.0621370 0.107624i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −3.00000 5.19615i −0.184988 0.320408i 0.758585 0.651575i $$-0.225891\pi$$
−0.943572 + 0.331166i $$0.892558\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −24.0000 −1.46331 −0.731653 0.681677i $$-0.761251\pi$$
−0.731653 + 0.681677i $$0.761251\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 15.0000 25.9808i 0.904534 1.56670i
$$276$$ 0 0
$$277$$ 11.0000 + 19.0526i 0.660926 + 1.14476i 0.980373 + 0.197153i $$0.0631696\pi$$
−0.319447 + 0.947604i $$0.603497\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −9.00000 15.5885i −0.536895 0.929929i −0.999069 0.0431402i $$-0.986264\pi$$
0.462174 0.886789i $$-0.347070\pi$$
$$282$$ 0 0
$$283$$ −10.0000 + 17.3205i −0.594438 + 1.02960i 0.399188 + 0.916869i $$0.369292\pi$$
−0.993626 + 0.112728i $$0.964041\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −12.0000 −0.708338
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −6.00000 + 10.3923i −0.350524 + 0.607125i −0.986341 0.164714i $$-0.947330\pi$$
0.635818 + 0.771839i $$0.280663\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −6.00000 10.3923i −0.346989 0.601003i
$$300$$ 0 0
$$301$$ 2.00000 3.46410i 0.115278 0.199667i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 20.0000 1.14146 0.570730 0.821138i $$-0.306660\pi$$
0.570730 + 0.821138i $$0.306660\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6.00000 + 10.3923i −0.340229 + 0.589294i −0.984475 0.175525i $$-0.943838\pi$$
0.644246 + 0.764818i $$0.277171\pi$$
$$312$$ 0 0
$$313$$ −13.0000 22.5167i −0.734803 1.27272i −0.954810 0.297218i $$-0.903941\pi$$
0.220006 0.975499i $$-0.429392\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −15.0000 25.9808i −0.842484 1.45922i −0.887788 0.460252i $$-0.847759\pi$$
0.0453045 0.998973i $$-0.485574\pi$$
$$318$$ 0 0
$$319$$ 18.0000 31.1769i 1.00781 1.74557i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −10.0000 −0.554700
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 6.00000 10.3923i 0.330791 0.572946i
$$330$$ 0 0
$$331$$ −10.0000 17.3205i −0.549650 0.952021i −0.998298 0.0583130i $$-0.981428\pi$$
0.448649 0.893708i $$-0.351905\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 11.0000 19.0526i 0.599208 1.03786i −0.393730 0.919226i $$-0.628816\pi$$
0.992938 0.118633i $$-0.0378512\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 48.0000 2.59935
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −9.00000 + 15.5885i −0.483145 + 0.836832i −0.999813 0.0193540i $$-0.993839\pi$$
0.516667 + 0.856186i $$0.327172\pi$$
$$348$$ 0 0
$$349$$ 5.00000 + 8.66025i 0.267644 + 0.463573i 0.968253 0.249973i $$-0.0804216\pi$$
−0.700609 + 0.713545i $$0.747088\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −6.00000 −0.316668 −0.158334 0.987386i $$-0.550612\pi$$
−0.158334 + 0.987386i $$0.550612\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −16.0000 27.7128i −0.835193 1.44660i −0.893873 0.448320i $$-0.852022\pi$$
0.0586798 0.998277i $$-0.481311\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −3.00000 5.19615i −0.155752 0.269771i
$$372$$ 0 0
$$373$$ 5.00000 8.66025i 0.258890 0.448411i −0.707055 0.707159i $$-0.749977\pi$$
0.965945 + 0.258748i $$0.0833099\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −12.0000 −0.618031
$$378$$ 0 0
$$379$$ 20.0000 1.02733 0.513665 0.857991i $$-0.328287\pi$$
0.513665 + 0.857991i $$0.328287\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −12.0000 + 20.7846i −0.613171 + 1.06204i 0.377531 + 0.925997i $$0.376773\pi$$
−0.990702 + 0.136047i $$0.956560\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 15.0000 + 25.9808i 0.760530 + 1.31728i 0.942578 + 0.333987i $$0.108394\pi$$
−0.182047 + 0.983290i $$0.558272\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 14.0000 0.702640 0.351320 0.936255i $$-0.385733\pi$$
0.351320 + 0.936255i $$0.385733\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 3.00000 5.19615i 0.149813 0.259483i −0.781345 0.624099i $$-0.785466\pi$$
0.931158 + 0.364615i $$0.118800\pi$$
$$402$$ 0 0
$$403$$ −8.00000 13.8564i −0.398508 0.690237i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −6.00000 10.3923i −0.297409 0.515127i
$$408$$ 0 0
$$409$$ −7.00000 + 12.1244i −0.346128 + 0.599511i −0.985558 0.169338i $$-0.945837\pi$$
0.639430 + 0.768849i $$0.279170\pi$$
$$410$$ 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
$$419$$ −12.0000 + 20.7846i −0.586238 + 1.01539i 0.408481 + 0.912767i $$0.366058\pi$$
−0.994720 + 0.102628i $$0.967275\pi$$
$$420$$ 0 0
$$421$$ −13.0000 22.5167i −0.633581 1.09739i −0.986814 0.161859i $$-0.948251\pi$$
0.353233 0.935536i $$-0.385082\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 5.00000 8.66025i 0.241967 0.419099i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −6.00000 −0.289010 −0.144505 0.989504i $$-0.546159\pi$$
−0.144505 + 0.989504i $$0.546159\pi$$
$$432$$ 0 0
$$433$$ 26.0000 1.24948 0.624740 0.780833i $$-0.285205\pi$$
0.624740 + 0.780833i $$0.285205\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 12.0000 20.7846i 0.574038 0.994263i
$$438$$ 0 0
$$439$$ −4.00000 6.92820i −0.190910 0.330665i 0.754642 0.656136i $$-0.227810\pi$$
−0.945552 + 0.325471i $$0.894477\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 15.0000 + 25.9808i 0.712672 + 1.23438i 0.963851 + 0.266443i $$0.0858483\pi$$
−0.251179 + 0.967941i $$0.580818\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 30.0000 1.41579 0.707894 0.706319i $$-0.249646\pi$$
0.707894 + 0.706319i $$0.249646\pi$$
$$450$$ 0 0
$$451$$ −72.0000 −3.39035
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 11.0000 + 19.0526i 0.514558 + 0.891241i 0.999857 + 0.0168929i $$0.00537742\pi$$
−0.485299 + 0.874348i $$0.661289\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.00000 + 10.3923i 0.279448 + 0.484018i 0.971248 0.238071i $$-0.0765153\pi$$
−0.691800 + 0.722089i $$0.743182\pi$$
$$462$$ 0 0
$$463$$ 14.0000 24.2487i 0.650635 1.12693i −0.332334 0.943162i $$-0.607836\pi$$
0.982969 0.183771i $$-0.0588306\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 8.00000 0.369406
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 12.0000 20.7846i 0.551761 0.955677i
$$474$$ 0 0
$$475$$ −10.0000 17.3205i −0.458831 0.794719i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −6.00000 10.3923i −0.274147 0.474837i 0.695773 0.718262i $$-0.255062\pi$$
−0.969920 + 0.243426i $$0.921729\pi$$
$$480$$ 0 0
$$481$$ −2.00000 + 3.46410i −0.0911922 + 0.157949i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 32.0000 1.45006 0.725029 0.688718i $$-0.241826\pi$$
0.725029 + 0.688718i $$0.241826\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −9.00000 + 15.5885i −0.406164 + 0.703497i −0.994456 0.105151i $$-0.966467\pi$$
0.588292 + 0.808649i $$0.299801\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3.00000 + 5.19615i 0.134568 + 0.233079i
$$498$$ 0 0
$$499$$ 2.00000 3.46410i 0.0895323 0.155074i −0.817781 0.575529i $$-0.804796\pi$$
0.907314 + 0.420455i $$0.138129\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 12.0000 20.7846i 0.531891 0.921262i −0.467416 0.884037i $$-0.654815\pi$$
0.999307 0.0372243i $$-0.0118516\pi$$
$$510$$ 0 0
$$511$$ 5.00000 + 8.66025i 0.221187 + 0.383107i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 36.0000 62.3538i 1.58328 2.74232i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ −4.00000 −0.174908 −0.0874539 0.996169i $$-0.527873\pi$$
−0.0874539 + 0.996169i $$0.527873\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −6.50000 11.2583i −0.282609 0.489493i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.0000 + 20.7846i 0.519778 + 0.900281i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ 2.00000 0.0859867 0.0429934 0.999075i $$-0.486311\pi$$
0.0429934 + 0.999075i $$0.486311\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −4.00000 6.92820i −0.171028 0.296229i 0.767752 0.640747i $$-0.221375\pi$$
−0.938779 + 0.344519i $$0.888042\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −12.0000 20.7846i −0.511217 0.885454i
$$552$$ 0 0
$$553$$ 2.00000 3.46410i 0.0850487 0.147309i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −42.0000 −1.77960 −0.889799 0.456354i $$-0.849155\pi$$
−0.889799 + 0.456354i $$0.849155\pi$$
$$558$$ 0 0
$$559$$ −8.00000 −0.338364
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 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
$$568$$ 0 0
$$569$$ 3.00000 + 5.19615i 0.125767 + 0.217834i 0.922032 0.387113i $$-0.126528\pi$$
−0.796266 + 0.604947i $$0.793194\pi$$
$$570$$ 0 0
$$571$$ 20.0000 34.6410i 0.836974 1.44968i −0.0554391 0.998462i $$-0.517656\pi$$
0.892413 0.451219i $$-0.149011\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −30.0000 −1.25109
$$576$$ 0 0
$$577$$ −22.0000 −0.915872 −0.457936 0.888985i $$-0.651411\pi$$
−0.457936 + 0.888985i $$0.651411\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −6.00000 + 10.3923i −0.248922 + 0.431145i
$$582$$ 0 0
$$583$$ −18.0000 31.1769i −0.745484 1.29122i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −12.0000 20.7846i −0.495293 0.857873i 0.504692 0.863299i $$-0.331606\pi$$
−0.999985 + 0.00542667i $$0.998273\pi$$
$$588$$ 0 0
$$589$$ 16.0000 27.7128i 0.659269 1.14189i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −48.0000 −1.97112 −0.985562 0.169316i $$-0.945844\pi$$
−0.985562 + 0.169316i $$0.945844\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 15.0000 25.9808i 0.612883 1.06155i −0.377869 0.925859i $$-0.623343\pi$$
0.990752 0.135686i $$-0.0433238\pi$$
$$600$$ 0 0
$$601$$ −1.00000 1.73205i −0.0407909 0.0706518i 0.844909 0.534910i $$-0.179654\pi$$
−0.885700 + 0.464258i $$0.846321\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −16.0000 + 27.7128i −0.649420 + 1.12483i 0.333842 + 0.942629i $$0.391655\pi$$
−0.983262 + 0.182199i $$0.941678\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −24.0000 −0.970936
$$612$$ 0 0
$$613$$ 38.0000 1.53481 0.767403 0.641165i $$-0.221549\pi$$
0.767403 + 0.641165i $$0.221549\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −9.00000 + 15.5885i −0.362326 + 0.627568i −0.988343 0.152242i $$-0.951351\pi$$
0.626017 + 0.779809i $$0.284684\pi$$
$$618$$ 0 0
$$619$$ −10.0000 17.3205i −0.401934 0.696170i 0.592025 0.805919i $$-0.298329\pi$$
−0.993959 + 0.109749i $$0.964995\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 6.00000 + 10.3923i 0.240385 + 0.416359i
$$624$$ 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
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.00000 1.73205i −0.0396214 0.0686264i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −9.00000 15.5885i −0.355479 0.615707i 0.631721 0.775196i $$-0.282349\pi$$
−0.987200 + 0.159489i $$0.949015\pi$$
$$642$$ 0 0
$$643$$ 2.00000 3.46410i 0.0788723 0.136611i −0.823891 0.566748i $$-0.808201\pi$$
0.902764 + 0.430137i $$0.141535\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 36.0000 1.41531 0.707653 0.706560i $$-0.249754\pi$$
0.707653 + 0.706560i $$0.249754\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −9.00000 + 15.5885i −0.352197 + 0.610023i −0.986634 0.162951i $$-0.947899\pi$$
0.634437 + 0.772975i $$0.281232\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −3.00000 5.19615i −0.116863 0.202413i 0.801660 0.597781i $$-0.203951\pi$$
−0.918523 + 0.395367i $$0.870617\pi$$
$$660$$ 0 0
$$661$$ −19.0000 + 32.9090i −0.739014 + 1.28001i 0.213925 + 0.976850i $$0.431375\pi$$
−0.952940 + 0.303160i $$0.901958\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −36.0000 −1.39393
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 30.0000 51.9615i 1.15814 2.00595i
$$672$$ 0 0
$$673$$ 17.0000 + 29.4449i 0.655302 + 1.13502i 0.981818 + 0.189824i $$0.0607919\pi$$
−0.326516 + 0.945192i $$0.605875\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 6.00000 + 10.3923i 0.230599 + 0.399409i 0.957984 0.286820i $$-0.0925982\pi$$
−0.727386 + 0.686229i $$0.759265\pi$$
$$678$$ 0 0
$$679$$ 5.00000 8.66025i 0.191882 0.332350i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 42.0000 1.60709 0.803543 0.595247i $$-0.202946\pi$$
0.803543 + 0.595247i $$0.202946\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −6.00000 + 10.3923i −0.228582 + 0.395915i
$$690$$ 0 0
$$691$$ 2.00000 + 3.46410i 0.0760836 + 0.131781i 0.901557 0.432660i $$-0.142425\pi$$
−0.825473 + 0.564441i $$0.809092\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −6.00000 −0.226617 −0.113308 0.993560i $$-0.536145\pi$$
−0.113308 + 0.993560i $$0.536145\pi$$
$$702$$ 0 0
$$703$$ −8.00000 −0.301726
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −6.00000 + 10.3923i −0.225653 + 0.390843i
$$708$$ 0 0
$$709$$ 5.00000 + 8.66025i 0.187779 + 0.325243i 0.944509 0.328484i $$-0.106538\pi$$
−0.756730 + 0.653727i $$0.773204\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −24.0000 41.5692i −0.898807 1.55678i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 8.00000 0.297936
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −15.0000 + 25.9808i −0.557086 + 0.964901i
$$726$$ 0 0
$$727$$ −4.00000 6.92820i −0.148352 0.256953i 0.782267 0.622944i $$-0.214063\pi$$
−0.930618 + 0.365991i $$0.880730\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 23.0000 39.8372i 0.849524 1.47142i −0.0321090 0.999484i $$-0.510222\pi$$
0.881633 0.471935i $$-0.156444\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 48.0000 1.76810
$$738$$ 0 0
$$739$$ 32.0000 1.17714 0.588570 0.808447i $$-0.299691\pi$$
0.588570 + 0.808447i $$0.299691\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 3.00000 5.19615i 0.110059 0.190628i −0.805735 0.592277i $$-0.798229\pi$$
0.915794 + 0.401648i $$0.131563\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −3.00000 5.19615i −0.109618 0.189863i
$$750$$ 0 0
$$751$$ −16.0000 + 27.7128i −0.583848 + 1.01125i 0.411170 + 0.911559i $$0.365120\pi$$
−0.995018 + 0.0996961i $$0.968213\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −18.0000 + 31.1769i −0.652499 + 1.13016i 0.330015 + 0.943976i $$0.392946\pi$$
−0.982514 + 0.186187i $$0.940387\pi$$
$$762$$ 0 0
$$763$$ −7.00000 12.1244i −0.253417 0.438931i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −13.0000 + 22.5167i −0.468792 + 0.811972i −0.999364 0.0356685i $$-0.988644\pi$$
0.530572 + 0.847640i $$0.321977\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −12.0000 −0.431610 −0.215805 0.976436i $$-0.569238\pi$$
−0.215805 + 0.976436i $$0.569238\pi$$
$$774$$ 0 0
$$775$$ −40.0000 −1.43684
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −24.0000 + 41.5692i −0.859889 + 1.48937i
$$780$$ 0 0
$$781$$ 18.0000 + 31.1769i 0.644091 + 1.11560i
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 2.00000 3.46410i 0.0712923 0.123482i −0.828176 0.560469i $$-0.810621\pi$$
0.899468 + 0.436987i $$0.143954\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ −20.0000 −0.710221
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 18.0000 31.1769i 0.637593 1.10434i −0.348367 0.937358i $$-0.613264\pi$$
0.985959 0.166985i $$-0.0534030\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 30.0000 + 51.9615i 1.05868 + 1.83368i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 54.0000 1.89854 0.949269 0.314464i $$-0.101825\pi$$
0.949269 + 0.314464i $$0.101825\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −8.00000 13.8564i −0.279885 0.484774i
$$818$$ 0 0
$$819$$ 0 0