# Properties

 Label 117.2.q.b.10.1 Level $117$ Weight $2$ Character 117.10 Analytic conductor $0.934$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 117.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.934249703649$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## Embedding invariants

 Embedding label 10.1 Root $$0.500000 - 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 117.10 Dual form 117.2.q.b.82.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.00000 + 1.73205i) q^{4} +(4.50000 + 2.59808i) q^{7} +O(q^{10})$$ $$q+(-1.00000 + 1.73205i) q^{4} +(4.50000 + 2.59808i) q^{7} +(-2.50000 - 2.59808i) q^{13} +(-2.00000 - 3.46410i) q^{16} +(-3.00000 - 1.73205i) q^{19} +5.00000 q^{25} +(-9.00000 + 5.19615i) q^{28} -8.66025i q^{31} +(-6.00000 + 3.46410i) q^{37} +(6.50000 - 11.2583i) q^{43} +(10.0000 + 17.3205i) q^{49} +(7.00000 - 1.73205i) q^{52} +(-6.50000 + 11.2583i) q^{61} +8.00000 q^{64} +(-10.5000 + 6.06218i) q^{67} -1.73205i q^{73} +(6.00000 - 3.46410i) q^{76} +13.0000 q^{79} +(-4.50000 - 18.1865i) q^{91} +(-16.5000 - 9.52628i) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 9 q^{7}+O(q^{10})$$ 2 * q - 2 * q^4 + 9 * q^7 $$2 q - 2 q^{4} + 9 q^{7} - 5 q^{13} - 4 q^{16} - 6 q^{19} + 10 q^{25} - 18 q^{28} - 12 q^{37} + 13 q^{43} + 20 q^{49} + 14 q^{52} - 13 q^{61} + 16 q^{64} - 21 q^{67} + 12 q^{76} + 26 q^{79} - 9 q^{91} - 33 q^{97}+O(q^{100})$$ 2 * q - 2 * q^4 + 9 * q^7 - 5 * q^13 - 4 * q^16 - 6 * q^19 + 10 * q^25 - 18 * q^28 - 12 * q^37 + 13 * q^43 + 20 * q^49 + 14 * q^52 - 13 * q^61 + 16 * q^64 - 21 * q^67 + 12 * q^76 + 26 * q^79 - 9 * q^91 - 33 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$$.

 $$n$$ $$28$$ $$92$$ $$\chi(n)$$ $$e\left(\frac{5}{6}\right)$$ $$1$$

## 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 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$3$$ 0 0
$$4$$ −1.00000 + 1.73205i −0.500000 + 0.866025i
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ 4.50000 + 2.59808i 1.70084 + 0.981981i 0.944911 + 0.327327i $$0.106148\pi$$
0.755929 + 0.654654i $$0.227186\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$12$$ 0 0
$$13$$ −2.50000 2.59808i −0.693375 0.720577i
$$14$$ 0 0
$$15$$ 0 0
$$16$$ −2.00000 3.46410i −0.500000 0.866025i
$$17$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$18$$ 0 0
$$19$$ −3.00000 1.73205i −0.688247 0.397360i 0.114708 0.993399i $$-0.463407\pi$$
−0.802955 + 0.596040i $$0.796740\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ −9.00000 + 5.19615i −1.70084 + 0.981981i
$$29$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$30$$ 0 0
$$31$$ 8.66025i 1.55543i −0.628619 0.777714i $$-0.716379\pi$$
0.628619 0.777714i $$-0.283621\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.00000 + 3.46410i −0.986394 + 0.569495i −0.904194 0.427121i $$-0.859528\pi$$
−0.0821995 + 0.996616i $$0.526194\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$42$$ 0 0
$$43$$ 6.50000 11.2583i 0.991241 1.71688i 0.381246 0.924473i $$-0.375495\pi$$
0.609994 0.792406i $$-0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ 10.0000 + 17.3205i 1.42857 + 2.47436i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 7.00000 1.73205i 0.970725 0.240192i
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$60$$ 0 0
$$61$$ −6.50000 + 11.2583i −0.832240 + 1.44148i 0.0640184 + 0.997949i $$0.479608\pi$$
−0.896258 + 0.443533i $$0.853725\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −10.5000 + 6.06218i −1.28278 + 0.740613i −0.977356 0.211604i $$-0.932131\pi$$
−0.305424 + 0.952217i $$0.598798\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$72$$ 0 0
$$73$$ 1.73205i 0.202721i −0.994850 0.101361i $$-0.967680\pi$$
0.994850 0.101361i $$-0.0323196\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 6.00000 3.46410i 0.688247 0.397360i
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 13.0000 1.46261 0.731307 0.682048i $$-0.238911\pi$$
0.731307 + 0.682048i $$0.238911\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$90$$ 0 0
$$91$$ −4.50000 18.1865i −0.471728 1.90647i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −16.5000 9.52628i −1.67532 0.967247i −0.964579 0.263795i $$-0.915026\pi$$
−0.710742 0.703452i $$-0.751641\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −5.00000 + 8.66025i −0.500000 + 0.866025i
$$101$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$102$$ 0 0
$$103$$ −13.0000 −1.28093 −0.640464 0.767988i $$-0.721258\pi$$
−0.640464 + 0.767988i $$0.721258\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$108$$ 0 0
$$109$$ 8.66025i 0.829502i 0.909935 + 0.414751i $$0.136131\pi$$
−0.909935 + 0.414751i $$0.863869\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 20.7846i 1.96396i
$$113$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −5.50000 + 9.52628i −0.500000 + 0.866025i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 15.0000 + 8.66025i 1.34704 + 0.777714i
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −0.500000 0.866025i −0.0443678 0.0768473i 0.842989 0.537931i $$-0.180794\pi$$
−0.887357 + 0.461084i $$0.847461\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −9.00000 15.5885i −0.780399 1.35169i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$138$$ 0 0
$$139$$ 3.50000 6.06218i 0.296866 0.514187i −0.678551 0.734553i $$-0.737392\pi$$
0.975417 + 0.220366i $$0.0707252\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 13.8564i 1.13899i
$$149$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$150$$ 0 0
$$151$$ 24.2487i 1.97333i 0.162758 + 0.986666i $$0.447961\pi$$
−0.162758 + 0.986666i $$0.552039\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 11.0000 0.877896 0.438948 0.898513i $$-0.355351\pi$$
0.438948 + 0.898513i $$0.355351\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −4.50000 2.59808i −0.352467 0.203497i 0.313304 0.949653i $$-0.398564\pi$$
−0.665771 + 0.746156i $$0.731897\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$168$$ 0 0
$$169$$ −0.500000 + 12.9904i −0.0384615 + 0.999260i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 13.0000 + 22.5167i 0.991241 + 1.71688i
$$173$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$174$$ 0 0
$$175$$ 22.5000 + 12.9904i 1.70084 + 0.981981i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$180$$ 0 0
$$181$$ 26.0000 1.93256 0.966282 0.257485i $$-0.0828937\pi$$
0.966282 + 0.257485i $$0.0828937\pi$$
$$182$$ 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
$$191$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$192$$ 0 0
$$193$$ −13.5000 + 7.79423i −0.971751 + 0.561041i −0.899770 0.436365i $$-0.856266\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −40.0000 −2.85714
$$197$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$198$$ 0 0
$$199$$ 8.50000 14.7224i 0.602549 1.04365i −0.389885 0.920864i $$-0.627485\pi$$
0.992434 0.122782i $$-0.0391815\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ −4.00000 + 13.8564i −0.277350 + 0.960769i
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −6.50000 11.2583i −0.447478 0.775055i 0.550743 0.834675i $$-0.314345\pi$$
−0.998221 + 0.0596196i $$0.981011\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 22.5000 38.9711i 1.52740 2.64553i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 9.00000 5.19615i 0.602685 0.347960i −0.167412 0.985887i $$-0.553541\pi$$
0.770097 + 0.637927i $$0.220208\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$228$$ 0 0
$$229$$ 20.7846i 1.37349i 0.726900 + 0.686743i $$0.240960\pi$$
−0.726900 + 0.686743i $$0.759040\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −24.0000 13.8564i −1.54598 0.892570i −0.998443 0.0557856i $$-0.982234\pi$$
−0.547533 0.836784i $$-0.684433\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ −13.0000 22.5167i −0.832240 1.44148i
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.00000 + 12.1244i 0.190885 + 0.771454i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ −8.00000 + 13.8564i −0.500000 + 0.866025i
$$257$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$258$$ 0 0
$$259$$ −36.0000 −2.23693
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 24.2487i 1.48123i
$$269$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$270$$ 0 0
$$271$$ 13.5000 7.79423i 0.820067 0.473466i −0.0303728 0.999539i $$-0.509669\pi$$
0.850439 + 0.526073i $$0.176336\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −13.0000 + 22.5167i −0.781094 + 1.35290i 0.150210 + 0.988654i $$0.452005\pi$$
−0.931305 + 0.364241i $$0.881328\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ −12.5000 21.6506i −0.743048 1.28700i −0.951101 0.308879i $$-0.900046\pi$$
0.208053 0.978117i $$-0.433287\pi$$
$$284$$ 0 0
$$285$$ 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
$$292$$ 3.00000 + 1.73205i 0.175562 + 0.101361i
$$293$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 58.5000 33.7750i 3.37188 1.94676i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 13.8564i 0.794719i
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1.73205i 0.0988534i 0.998778 + 0.0494267i $$0.0157394\pi$$
−0.998778 + 0.0494267i $$0.984261\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 13.0000 0.734803 0.367402 0.930062i $$-0.380247\pi$$
0.367402 + 0.930062i $$0.380247\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ −13.0000 + 22.5167i −0.731307 + 1.26666i
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −12.5000 12.9904i −0.693375 0.720577i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 16.5000 + 9.52628i 0.906922 + 0.523612i 0.879440 0.476011i $$-0.157918\pi$$
0.0274825 + 0.999622i $$0.491251\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −29.0000 −1.57973 −0.789865 0.613280i $$-0.789850\pi$$
−0.789865 + 0.613280i $$0.789850\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 67.5500i 3.64736i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$348$$ 0 0
$$349$$ 25.5000 14.7224i 1.36498 0.788074i 0.374701 0.927146i $$-0.377745\pi$$
0.990282 + 0.139072i $$0.0444119\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ −3.50000 6.06218i −0.184211 0.319062i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 36.0000 + 10.3923i 1.88691 + 0.544705i
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 15.5000 + 26.8468i 0.809093 + 1.40139i 0.913493 + 0.406855i $$0.133375\pi$$
−0.104399 + 0.994535i $$0.533292\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6.50000 + 11.2583i −0.336557 + 0.582934i −0.983783 0.179364i $$-0.942596\pi$$
0.647225 + 0.762299i $$0.275929\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 10.5000 6.06218i 0.539349 0.311393i −0.205466 0.978664i $$-0.565871\pi$$
0.744815 + 0.667271i $$0.232538\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 33.0000 19.0526i 1.67532 0.967247i
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −34.5000 19.9186i −1.73151 0.999685i −0.878300 0.478110i $$-0.841322\pi$$
−0.853206 0.521575i $$-0.825345\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ −10.0000 17.3205i −0.500000 0.866025i
$$401$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$402$$ 0 0
$$403$$ −22.5000 + 21.6506i −1.12080 + 1.07849i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −22.5000 12.9904i −1.11255 0.642333i −0.173064 0.984911i $$-0.555367\pi$$
−0.939490 + 0.342578i $$0.888700\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 13.0000 22.5167i 0.640464 1.10932i
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$420$$ 0 0
$$421$$ 36.3731i 1.77271i −0.463002 0.886357i $$-0.653228\pi$$
0.463002 0.886357i $$-0.346772\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −58.5000 + 33.7750i −2.83101 + 1.63449i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$432$$ 0 0
$$433$$ 17.5000 30.3109i 0.840996 1.45665i −0.0480569 0.998845i $$-0.515303\pi$$
0.889053 0.457804i $$-0.151364\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −15.0000 8.66025i −0.718370 0.414751i
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 6.50000 + 11.2583i 0.310228 + 0.537331i 0.978412 0.206666i $$-0.0662612\pi$$
−0.668184 + 0.743996i $$0.732928\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 36.0000 + 20.7846i 1.70084 + 0.981981i
$$449$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.5000 6.06218i 0.491169 0.283577i −0.233890 0.972263i $$-0.575146\pi$$
0.725059 + 0.688686i $$0.241812\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$462$$ 0 0
$$463$$ 1.73205i 0.0804952i 0.999190 + 0.0402476i $$0.0128147\pi$$
−0.999190 + 0.0402476i $$0.987185\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ −63.0000 −2.90907
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −15.0000 8.66025i −0.688247 0.397360i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$480$$ 0 0
$$481$$ 24.0000 + 6.92820i 1.09431 + 0.315899i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ −11.0000 19.0526i −0.500000 0.866025i
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 3.00000 + 1.73205i 0.135943 + 0.0784867i 0.566429 0.824110i $$-0.308325\pi$$
−0.430486 + 0.902597i $$0.641658\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −30.0000 + 17.3205i −1.34704 + 0.777714i
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 31.1769i 1.39567i −0.716258 0.697835i $$-0.754147\pi$$
0.716258 0.697835i $$-0.245853\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 2.00000 0.0887357
$$509$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$510$$ 0 0
$$511$$ 4.50000 7.79423i 0.199068 0.344796i
$$512$$ 0 0
$$513$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 0 0
$$523$$ −4.00000 6.92820i −0.174908 0.302949i 0.765222 0.643767i $$-0.222629\pi$$
−0.940129 + 0.340818i $$0.889296\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 11.5000 19.9186i 0.500000 0.866025i
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 36.0000 1.56080
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 43.3013i 1.86167i 0.365444 + 0.930834i $$0.380917\pi$$
−0.365444 + 0.930834i $$0.619083\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 41.0000 1.75303 0.876517 0.481371i $$-0.159861\pi$$
0.876517 + 0.481371i $$0.159861\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 58.5000 + 33.7750i 2.48767 + 1.43626i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 7.00000 + 12.1244i 0.296866 + 0.514187i
$$557$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$558$$ 0 0
$$559$$ −45.5000 + 11.2583i −1.92444 + 0.476177i
$$560$$ 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
$$569$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$570$$ 0 0
$$571$$ 16.0000 0.669579 0.334790 0.942293i $$-0.391335\pi$$
0.334790 + 0.942293i $$0.391335\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 13.8564i 0.576850i 0.957503 + 0.288425i $$0.0931316\pi$$
−0.957503 + 0.288425i $$0.906868\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$588$$ 0 0
$$589$$ −15.0000 + 25.9808i −0.618064 + 1.07052i
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 24.0000 + 13.8564i 0.986394 + 0.569495i
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 13.0000 + 22.5167i 0.530281 + 0.918474i 0.999376 + 0.0353259i $$0.0112469\pi$$
−0.469095 + 0.883148i $$0.655420\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −42.0000 24.2487i −1.70896 0.986666i
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −10.0000 + 17.3205i −0.405887 + 0.703018i −0.994424 0.105453i $$-0.966371\pi$$
0.588537 + 0.808470i $$0.299704\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −28.5000 + 16.4545i −1.15110 + 0.664590i −0.949156 0.314806i $$-0.898061\pi$$
−0.201948 + 0.979396i $$0.564727\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$618$$ 0 0
$$619$$ 46.7654i 1.87966i −0.341644 0.939829i $$-0.610984\pi$$
0.341644 0.939829i $$-0.389016\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −11.0000 + 19.0526i −0.438948 + 0.760280i
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 43.5000 + 25.1147i 1.73171 + 0.999802i 0.875806 + 0.482663i $$0.160330\pi$$
0.855901 + 0.517139i $$0.173003\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 20.0000 69.2820i 0.792429 2.74505i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$642$$ 0 0
$$643$$ 16.5000 + 9.52628i 0.650696 + 0.375680i 0.788723 0.614749i $$-0.210743\pi$$
−0.138027 + 0.990429i $$0.544076\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 9.00000 5.19615i 0.352467 0.203497i
$$653$$ 0 0 0.866025 0.500000i $$-0.166667\pi$$
−0.866025 + 0.500000i $$0.833333\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$660$$ 0 0
$$661$$ 13.5000 7.79423i 0.525089 0.303160i −0.213925 0.976850i $$-0.568625\pi$$
0.739014 + 0.673690i $$0.235292\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 6.50000 + 11.2583i 0.250557 + 0.433977i 0.963679 0.267063i $$-0.0860531\pi$$
−0.713123 + 0.701039i $$0.752720\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −22.0000 13.8564i −0.846154 0.532939i
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ −49.5000 85.7365i −1.89964 3.29027i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −52.0000 −1.98248
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −28.5000 + 16.4545i −1.08419 + 0.625958i −0.932024 0.362397i $$-0.881959\pi$$
−0.152167 + 0.988355i $$0.548625\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ −45.0000 + 25.9808i −1.70084 + 0.981981i
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −4.50000 2.59808i −0.169001 0.0975728i 0.413114 0.910679i $$-0.364441\pi$$
−0.582115 + 0.813107i $$0.697775\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$720$$ 0 0
$$721$$ −58.5000 33.7750i −2.17865 1.25785i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −26.0000 + 45.0333i −0.966282 + 1.67365i
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 49.0000 1.81731 0.908655 0.417548i $$-0.137111\pi$$
0.908655 + 0.417548i $$0.137111\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 53.6936i 1.98322i −0.129275 0.991609i $$-0.541265\pi$$
0.129275 0.991609i $$-0.458735\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 45.0000 25.9808i 1.65535 0.955718i 0.680534 0.732717i $$-0.261748\pi$$
0.974818 0.223001i $$-0.0715853\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 −0.500000 0.866025i $$-0.666667\pi$$
0.500000 + 0.866025i $$0.333333\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 26.0000 + 45.0333i 0.948753 + 1.64329i 0.748056 + 0.663636i $$0.230988\pi$$
0.200698 + 0.979653i $$0.435679\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 13.0000 + 22.5167i 0.472493 + 0.818382i 0.999505 0.0314762i $$-0.0100208\pi$$
−0.527011 + 0.849858i $$0.676688\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$762$$ 0 0
$$763$$ −22.5000 + 38.9711i −0.814555 + 1.41085i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −48.0000 + 27.7128i −1.73092 + 0.999350i −0.847432 + 0.530904i $$0.821852\pi$$
−0.883493 + 0.468445i $$0.844814\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 31.1769i 1.12208i
$$773$$ 0 0 0.500000 0.866025i $$-0.333333\pi$$
−0.500000 + 0.866025i $$0.666667\pi$$
$$774$$ 0 0
$$775$$ 43.3013i 1.55543i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 40.0000 69.2820i 1.42857 2.47436i
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −43.5000 25.1147i −1.55061 0.895244i −0.998092 0.0617409i $$-0.980335\pi$$
−0.552515 0.833503i $$-0.686332\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 45.5000 11.2583i 1.61575 0.399795i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 17.0000 + 29.4449i 0.602549 + 1.04365i
$$797$$ 0 0 −0.866025 0.500000i $$-0.833333\pi$$
0.866025 + 0.500000i $$0.166667\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0