# Properties

 Label 512.2.e.d.129.1 Level $512$ Weight $2$ Character 512.129 Analytic conductor $4.088$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$512 = 2^{9}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 512.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.08834058349$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## Embedding invariants

 Embedding label 129.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 512.129 Dual form 512.2.e.d.385.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(-1.00000 + 1.00000i) q^{5} -3.00000i q^{9} +O(q^{10})$$ $$q+(-1.00000 + 1.00000i) q^{5} -3.00000i q^{9} +(5.00000 + 5.00000i) q^{13} +8.00000 q^{17} +3.00000i q^{25} +(3.00000 + 3.00000i) q^{29} +(-7.00000 + 7.00000i) q^{37} -8.00000i q^{41} +(3.00000 + 3.00000i) q^{45} +7.00000 q^{49} +(9.00000 - 9.00000i) q^{53} +(-11.0000 - 11.0000i) q^{61} -10.0000 q^{65} +6.00000i q^{73} -9.00000 q^{81} +(-8.00000 + 8.00000i) q^{85} +10.0000i q^{89} -8.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + O(q^{10})$$ $$2 q - 2 q^{5} + 10 q^{13} + 16 q^{17} + 6 q^{29} - 14 q^{37} + 6 q^{45} + 14 q^{49} + 18 q^{53} - 22 q^{61} - 20 q^{65} - 18 q^{81} - 16 q^{85} - 16 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/512\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$511$$ $$\chi(n)$$ $$e\left(\frac{3}{4}\right)$$ $$1$$

## 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
$$3$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$4$$ 0 0
$$5$$ −1.00000 + 1.00000i −0.447214 + 0.447214i −0.894427 0.447214i $$-0.852416\pi$$
0.447214 + 0.894427i $$0.352416\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 3.00000i 1.00000i
$$10$$ 0 0
$$11$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$12$$ 0 0
$$13$$ 5.00000 + 5.00000i 1.38675 + 1.38675i 0.832050 + 0.554700i $$0.187167\pi$$
0.554700 + 0.832050i $$0.312833\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 8.00000 1.94029 0.970143 0.242536i $$-0.0779791\pi$$
0.970143 + 0.242536i $$0.0779791\pi$$
$$18$$ 0 0
$$19$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 3.00000i 0.600000i
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.00000 + 3.00000i 0.557086 + 0.557086i 0.928477 0.371391i $$-0.121119\pi$$
−0.371391 + 0.928477i $$0.621119\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.00000 + 7.00000i −1.15079 + 1.15079i −0.164399 + 0.986394i $$0.552568\pi$$
−0.986394 + 0.164399i $$0.947432\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 8.00000i 1.24939i −0.780869 0.624695i $$-0.785223\pi$$
0.780869 0.624695i $$-0.214777\pi$$
$$42$$ 0 0
$$43$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$44$$ 0 0
$$45$$ 3.00000 + 3.00000i 0.447214 + 0.447214i
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 9.00000 9.00000i 1.23625 1.23625i 0.274721 0.961524i $$-0.411414\pi$$
0.961524 0.274721i $$-0.0885855\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$60$$ 0 0
$$61$$ −11.0000 11.0000i −1.40841 1.40841i −0.768221 0.640184i $$-0.778858\pi$$
−0.640184 0.768221i $$-0.721142\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −10.0000 −1.24035
$$66$$ 0 0
$$67$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 6.00000i 0.702247i 0.936329 + 0.351123i $$0.114200\pi$$
−0.936329 + 0.351123i $$0.885800\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ −9.00000 −1.00000
$$82$$ 0 0
$$83$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$84$$ 0 0
$$85$$ −8.00000 + 8.00000i −0.867722 + 0.867722i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 10.0000i 1.06000i 0.847998 + 0.529999i $$0.177808\pi$$
−0.847998 + 0.529999i $$0.822192\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −8.00000 −0.812277 −0.406138 0.913812i $$-0.633125\pi$$
−0.406138 + 0.913812i $$0.633125\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 9.00000 9.00000i 0.895533 0.895533i −0.0995037 0.995037i $$-0.531726\pi$$
0.995037 + 0.0995037i $$0.0317255\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$108$$ 0 0
$$109$$ −13.0000 13.0000i −1.24517 1.24517i −0.957826 0.287348i $$-0.907226\pi$$
−0.287348 0.957826i $$-0.592774\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 15.0000 15.0000i 1.38675 1.38675i
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 11.0000i 1.00000i
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −8.00000 8.00000i −0.715542 0.715542i
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 8.00000i 0.683486i −0.939793 0.341743i $$-0.888983\pi$$
0.939793 0.341743i $$-0.111017\pi$$
$$138$$ 0 0
$$139$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −17.0000 + 17.0000i −1.39269 + 1.39269i −0.573462 + 0.819232i $$0.694400\pi$$
−0.819232 + 0.573462i $$0.805600\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 24.0000i 1.94029i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 5.00000 + 5.00000i 0.399043 + 0.399043i 0.877896 0.478852i $$-0.158947\pi$$
−0.478852 + 0.877896i $$0.658947\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 37.0000i 2.84615i
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −11.0000 11.0000i −0.836315 0.836315i 0.152057 0.988372i $$-0.451410\pi$$
−0.988372 + 0.152057i $$0.951410\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$180$$ 0 0
$$181$$ −1.00000 + 1.00000i −0.0743294 + 0.0743294i −0.743294 0.668965i $$-0.766738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 14.0000i 1.02930i
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 24.0000 1.72756 0.863779 0.503871i $$-0.168091\pi$$
0.863779 + 0.503871i $$0.168091\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 15.0000 15.0000i 1.06871 1.06871i 0.0712470 0.997459i $$-0.477302\pi$$
0.997459 0.0712470i $$-0.0226979\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 8.00000 + 8.00000i 0.558744 + 0.558744i
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 40.0000 + 40.0000i 2.69069 + 2.69069i
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 9.00000 0.600000
$$226$$ 0 0
$$227$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$228$$ 0 0
$$229$$ −17.0000 + 17.0000i −1.12339 + 1.12339i −0.132164 + 0.991228i $$0.542192\pi$$
−0.991228 + 0.132164i $$0.957808\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 26.0000i 1.70332i −0.524097 0.851658i $$-0.675597\pi$$
0.524097 0.851658i $$-0.324403\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 8.00000 0.515325 0.257663 0.966235i $$-0.417048\pi$$
0.257663 + 0.966235i $$0.417048\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −7.00000 + 7.00000i −0.447214 + 0.447214i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2.00000 0.124757 0.0623783 0.998053i $$-0.480131\pi$$
0.0623783 + 0.998053i $$0.480131\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 9.00000 9.00000i 0.557086 0.557086i
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 18.0000i 1.10573i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 3.00000 + 3.00000i 0.182913 + 0.182913i 0.792624 0.609711i $$-0.208714\pi$$
−0.609711 + 0.792624i $$0.708714\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −23.0000 + 23.0000i −1.38194 + 1.38194i −0.540758 + 0.841178i $$0.681862\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 10.0000i 0.596550i −0.954480 0.298275i $$-0.903589\pi$$
0.954480 0.298275i $$-0.0964112\pi$$
$$282$$ 0 0
$$283$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 47.0000 2.76471
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 15.0000 15.0000i 0.876309 0.876309i −0.116841 0.993151i $$-0.537277\pi$$
0.993151 + 0.116841i $$0.0372769\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 22.0000 1.25972
$$306$$ 0 0
$$307$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ 24.0000i 1.35656i −0.734803 0.678280i $$-0.762726\pi$$
0.734803 0.678280i $$-0.237274\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3.00000 + 3.00000i 0.168497 + 0.168497i 0.786318 0.617822i $$-0.211985\pi$$
−0.617822 + 0.786318i $$0.711985\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −15.0000 + 15.0000i −0.832050 + 0.832050i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$332$$ 0 0
$$333$$ 21.0000 + 21.0000i 1.15079 + 1.15079i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$348$$ 0 0
$$349$$ −13.0000 13.0000i −0.695874 0.695874i 0.267644 0.963518i $$-0.413755\pi$$
−0.963518 + 0.267644i $$0.913755\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ 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
$$361$$ 19.0000i 1.00000i
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −6.00000 6.00000i −0.314054 0.314054i
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ −24.0000 −1.24939
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 25.0000 25.0000i 1.29445 1.29445i 0.362446 0.932005i $$-0.381942\pi$$
0.932005 0.362446i $$-0.118058\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 30.0000i 1.54508i
$$378$$ 0 0
$$379$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −7.00000 + 7.00000i −0.354914 + 0.354914i −0.861934 0.507020i $$-0.830747\pi$$
0.507020 + 0.861934i $$0.330747\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −13.0000 13.0000i −0.652451 0.652451i 0.301131 0.953583i $$-0.402636\pi$$
−0.953583 + 0.301131i $$0.902636\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 40.0000 1.99750 0.998752 0.0499376i $$-0.0159023\pi$$
0.998752 + 0.0499376i $$0.0159023\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 9.00000 9.00000i 0.447214 0.447214i
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 40.0000i 1.97787i 0.148340 + 0.988936i $$0.452607\pi$$
−0.148340 + 0.988936i $$0.547393\pi$$
$$410$$ 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 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$420$$ 0 0
$$421$$ −1.00000 + 1.00000i −0.0487370 + 0.0487370i −0.731055 0.682318i $$-0.760972\pi$$
0.682318 + 0.731055i $$0.260972\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 24.0000i 1.16417i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −24.0000 −1.15337 −0.576683 0.816968i $$-0.695653\pi$$
−0.576683 + 0.816968i $$0.695653\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ 21.0000i 1.00000i
$$442$$ 0 0
$$443$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$444$$ 0 0
$$445$$ −10.0000 10.0000i −0.474045 0.474045i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −40.0000 −1.88772 −0.943858 0.330350i $$-0.892833\pi$$
−0.943858 + 0.330350i $$0.892833\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 8.00000i 0.374224i −0.982339 0.187112i $$-0.940087\pi$$
0.982339 0.187112i $$-0.0599128\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −29.0000 29.0000i −1.35066 1.35066i −0.884918 0.465746i $$-0.845786\pi$$
−0.465746 0.884918i $$-0.654214\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −27.0000 27.0000i −1.23625 1.23625i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −70.0000 −3.19173
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 8.00000 8.00000i 0.363261 0.363261i
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$492$$ 0 0
$$493$$ 24.0000 + 24.0000i 1.08091 + 1.08091i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 18.0000i 0.800989i
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −27.0000 27.0000i −1.19675 1.19675i −0.975133 0.221621i $$-0.928865\pi$$
−0.221621 0.975133i $$-0.571135\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 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
$$521$$ 40.0000i 1.75243i −0.481919 0.876216i $$-0.660060\pi$$
0.481919 0.876216i $$-0.339940\pi$$
$$522$$ 0 0
$$523$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 23.0000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 40.0000 40.0000i 1.73259 1.73259i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −11.0000 11.0000i −0.472927 0.472927i 0.429934 0.902861i $$-0.358537\pi$$
−0.902861 + 0.429934i $$0.858537\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 26.0000 1.11372
$$546$$ 0 0
$$547$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$548$$ 0 0
$$549$$ −33.0000 + 33.0000i −1.40841 + 1.40841i
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 5.00000 + 5.00000i 0.211857 + 0.211857i 0.805056 0.593199i $$-0.202135\pi$$
−0.593199 + 0.805056i $$0.702135\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$564$$ 0 0
$$565$$ 14.0000 14.0000i 0.588984 0.588984i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 40.0000i 1.67689i 0.544988 + 0.838444i $$0.316534\pi$$
−0.544988 + 0.838444i $$0.683466\pi$$
$$570$$ 0 0
$$571$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −2.00000 −0.0832611 −0.0416305 0.999133i $$-0.513255\pi$$
−0.0416305 + 0.999133i $$0.513255\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 30.0000i 1.24035i
$$586$$ 0 0
$$587$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ −46.0000 −1.88899 −0.944497 0.328521i $$-0.893450\pi$$
−0.944497 + 0.328521i $$0.893450\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 10.0000i 0.407909i 0.978980 + 0.203954i $$0.0653794\pi$$
−0.978980 + 0.203954i $$0.934621\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −11.0000 11.0000i −0.447214 0.447214i
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −1.00000 + 1.00000i −0.0403896 + 0.0403896i −0.727013 0.686624i $$-0.759092\pi$$
0.686624 + 0.727013i $$0.259092\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 38.0000i 1.52982i −0.644136 0.764911i $$-0.722783\pi$$
0.644136 0.764911i $$-0.277217\pi$$
$$618$$ 0 0
$$619$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −56.0000 + 56.0000i −2.23287 + 2.23287i
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 35.0000 + 35.0000i 1.38675 + 1.38675i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −8.00000 −0.315981 −0.157991 0.987441i $$-0.550502\pi$$
−0.157991 + 0.987441i $$0.550502\pi$$
$$642$$ 0 0
$$643$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 35.0000 + 35.0000i 1.36966 + 1.36966i 0.860927 + 0.508729i $$0.169885\pi$$
0.508729 + 0.860927i $$0.330115\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 18.0000 0.702247
$$658$$ 0 0
$$659$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$660$$ 0 0
$$661$$ 31.0000 31.0000i 1.20576 1.20576i 0.233373 0.972387i $$-0.425024\pi$$
0.972387 0.233373i $$-0.0749763\pi$$
$$662$$ 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
$$673$$ 24.0000 0.925132 0.462566 0.886585i $$-0.346929\pi$$
0.462566 + 0.886585i $$0.346929\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 25.0000 25.0000i 0.960828 0.960828i −0.0384331 0.999261i $$-0.512237\pi$$
0.999261 + 0.0384331i $$0.0122367\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 −0.707107 0.707107i $$-0.750000\pi$$
0.707107 + 0.707107i $$0.250000\pi$$
$$684$$ 0 0
$$685$$ 8.00000 + 8.00000i 0.305664 + 0.305664i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 90.0000 3.42873
$$690$$ 0 0
$$691$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 64.0000i 2.42417i
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 21.0000 + 21.0000i 0.793159 + 0.793159i 0.982006 0.188847i $$-0.0604752\pi$$
−0.188847 + 0.982006i $$0.560475\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −7.00000 + 7.00000i −0.262891 + 0.262891i −0.826227 0.563337i $$-0.809517\pi$$
0.563337 + 0.826227i $$0.309517\pi$$
$$710$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −9.00000 + 9.00000i −0.334252 + 0.334252i
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 27.0000i 1.00000i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −29.0000 29.0000i −1.07114 1.07114i −0.997268 0.0738717i $$-0.976464\pi$$
−0.0738717 0.997268i $$-0.523536\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 34.0000i 1.24566i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −17.0000 + 17.0000i −0.617876 + 0.617876i −0.944986 0.327111i $$-0.893925\pi$$
0.327111 + 0.944986i $$0.393925\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 40.0000i 1.45000i 0.688749 + 0.724999i $$0.258160\pi$$
−0.688749 + 0.724999i $$0.741840\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 24.0000 + 24.0000i 0.867722 + 0.867722i
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 24.0000 0.865462 0.432731 0.901523i $$-0.357550\pi$$
0.432731 + 0.901523i $$0.357550\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −39.0000 + 39.0000i −1.40273 + 1.40273i −0.611448 + 0.791285i $$0.709412\pi$$
−0.791285 + 0.611448i $$0.790588\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −10.0000 −0.356915
$$786$$ 0 0
$$787$$ 0 0 0.707107 0.707107i $$-0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 110.000i 3.90621i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 37.0000 + 37.0000i 1.31061 + 1.31061i 0.920967 + 0.389640i $$0.127401\pi$$
0.389640 + 0.920967i $$0.372599\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 30.0000 1.06000
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 56.0000i 1.96886i 0.175791 + 0.984428i $$0.443752\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 0 0
$$811$$ 0 0