# Properties

 Label 825.2.f.b.626.4 Level $825$ Weight $2$ Character 825.626 Analytic conductor $6.588$ Analytic rank $0$ Dimension $4$ CM discriminant -11 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 825.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.58765816676$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 165) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 626.4 Root $$-1.65831 - 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.626 Dual form 825.2.f.b.626.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+(1.65831 + 0.500000i) q^{3} -2.00000 q^{4} +(2.50000 + 1.65831i) q^{9} +O(q^{10})$$ $$q+(1.65831 + 0.500000i) q^{3} -2.00000 q^{4} +(2.50000 + 1.65831i) q^{9} +3.31662i q^{11} +(-3.31662 - 1.00000i) q^{12} +4.00000 q^{16} +9.00000i q^{23} +(3.31662 + 4.00000i) q^{27} -5.00000 q^{31} +(-1.65831 + 5.50000i) q^{33} +(-5.00000 - 3.31662i) q^{36} +9.94987 q^{37} -6.63325i q^{44} +12.0000i q^{47} +(6.63325 + 2.00000i) q^{48} +7.00000 q^{49} +6.00000i q^{53} -3.31662i q^{59} -8.00000 q^{64} -9.94987 q^{67} +(-4.50000 + 14.9248i) q^{69} -16.5831i q^{71} +(3.50000 + 8.29156i) q^{81} -16.5831i q^{89} -18.0000i q^{92} +(-8.29156 - 2.50000i) q^{93} -9.94987 q^{97} +(-5.50000 + 8.29156i) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} + 10 q^{9}+O(q^{10})$$ 4 * q - 8 * q^4 + 10 * q^9 $$4 q - 8 q^{4} + 10 q^{9} + 16 q^{16} - 20 q^{31} - 20 q^{36} + 28 q^{49} - 32 q^{64} - 18 q^{69} + 14 q^{81} - 22 q^{99}+O(q^{100})$$ 4 * q - 8 * q^4 + 10 * q^9 + 16 * q^16 - 20 * q^31 - 20 * q^36 + 28 * q^49 - 32 * q^64 - 18 * q^69 + 14 * q^81 - 22 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$3$$ 1.65831 + 0.500000i 0.957427 + 0.288675i
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 2.50000 + 1.65831i 0.833333 + 0.552771i
$$10$$ 0 0
$$11$$ 3.31662i 1.00000i
$$12$$ −3.31662 1.00000i −0.957427 0.288675i
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 4.00000 1.00000
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 9.00000i 1.87663i 0.345782 + 0.938315i $$0.387614\pi$$
−0.345782 + 0.938315i $$0.612386\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 3.31662 + 4.00000i 0.638285 + 0.769800i
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ −5.00000 −0.898027 −0.449013 0.893525i $$-0.648224\pi$$
−0.449013 + 0.893525i $$0.648224\pi$$
$$32$$ 0 0
$$33$$ −1.65831 + 5.50000i −0.288675 + 0.957427i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ −5.00000 3.31662i −0.833333 0.552771i
$$37$$ 9.94987 1.63575 0.817875 0.575396i $$-0.195152\pi$$
0.817875 + 0.575396i $$0.195152\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 6.63325i 1.00000i
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 12.0000i 1.75038i 0.483779 + 0.875190i $$0.339264\pi$$
−0.483779 + 0.875190i $$0.660736\pi$$
$$48$$ 6.63325 + 2.00000i 0.957427 + 0.288675i
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 6.00000i 0.824163i 0.911147 + 0.412082i $$0.135198\pi$$
−0.911147 + 0.412082i $$0.864802\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 3.31662i 0.431788i −0.976417 0.215894i $$-0.930733\pi$$
0.976417 0.215894i $$-0.0692665\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ −8.00000 −1.00000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.94987 −1.21557 −0.607785 0.794101i $$-0.707942\pi$$
−0.607785 + 0.794101i $$0.707942\pi$$
$$68$$ 0 0
$$69$$ −4.50000 + 14.9248i −0.541736 + 1.79674i
$$70$$ 0 0
$$71$$ 16.5831i 1.96805i −0.178017 0.984027i $$-0.556968\pi$$
0.178017 0.984027i $$-0.443032\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 3.50000 + 8.29156i 0.388889 + 0.921285i
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 16.5831i 1.75781i −0.476999 0.878904i $$-0.658275\pi$$
0.476999 0.878904i $$-0.341725\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 18.0000i 1.87663i
$$93$$ −8.29156 2.50000i −0.859795 0.259238i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −9.94987 −1.01026 −0.505128 0.863044i $$-0.668555\pi$$
−0.505128 + 0.863044i $$0.668555\pi$$
$$98$$ 0 0
$$99$$ −5.50000 + 8.29156i −0.552771 + 0.833333i
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 19.8997 1.96078 0.980390 0.197066i $$-0.0631413\pi$$
0.980390 + 0.197066i $$0.0631413\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ −6.63325 8.00000i −0.638285 0.769800i
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 16.5000 + 4.97494i 1.56611 + 0.472200i
$$112$$ 0 0
$$113$$ 21.0000i 1.97551i −0.156001 0.987757i $$-0.549860\pi$$
0.156001 0.987757i $$-0.450140\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 10.0000 0.898027
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 3.31662 11.0000i 0.288675 0.957427i
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 3.00000i 0.256307i 0.991754 + 0.128154i $$0.0409051\pi$$
−0.991754 + 0.128154i $$0.959095\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ −6.00000 + 19.8997i −0.505291 + 1.67586i
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 10.0000 + 6.63325i 0.833333 + 0.552771i
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 11.6082 + 3.50000i 0.957427 + 0.288675i
$$148$$ −19.8997 −1.63575
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 9.94987 0.794086 0.397043 0.917800i $$-0.370036\pi$$
0.397043 + 0.917800i $$0.370036\pi$$
$$158$$ 0 0
$$159$$ −3.00000 + 9.94987i −0.237915 + 0.789076i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −19.8997 −1.55867 −0.779334 0.626608i $$-0.784443\pi$$
−0.779334 + 0.626608i $$0.784443\pi$$
$$164$$ 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
$$172$$ 0 0
$$173$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 13.2665i 1.00000i
$$177$$ 1.65831 5.50000i 0.124646 0.413405i
$$178$$ 0 0
$$179$$ 16.5831i 1.23948i −0.784807 0.619740i $$-0.787238\pi$$
0.784807 0.619740i $$-0.212762\pi$$
$$180$$ 0 0
$$181$$ 25.0000 1.85824 0.929118 0.369784i $$-0.120568\pi$$
0.929118 + 0.369784i $$0.120568\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 24.0000i 1.75038i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 23.2164i 1.67988i −0.542681 0.839939i $$-0.682591\pi$$
0.542681 0.839939i $$-0.317409\pi$$
$$192$$ −13.2665 4.00000i −0.957427 0.288675i
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ −14.0000 −1.00000
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −20.0000 −1.41776 −0.708881 0.705328i $$-0.750800\pi$$
−0.708881 + 0.705328i $$0.750800\pi$$
$$200$$ 0 0
$$201$$ −16.5000 4.97494i −1.16382 0.350905i
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −14.9248 + 22.5000i −1.03735 + 1.56386i
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 12.0000i 0.824163i
$$213$$ 8.29156 27.5000i 0.568128 1.88427i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 29.8496 1.99888 0.999439 0.0334825i $$-0.0106598\pi$$
0.999439 + 0.0334825i $$0.0106598\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ −5.00000 −0.330409 −0.165205 0.986259i $$-0.552828\pi$$
−0.165205 + 0.986259i $$0.552828\pi$$
$$230$$ 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
$$236$$ 6.63325i 0.431788i
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 1.65831 + 15.5000i 0.106381 + 0.994325i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 16.5831i 1.04672i −0.852112 0.523359i $$-0.824679\pi$$
0.852112 0.523359i $$-0.175321\pi$$
$$252$$ 0 0
$$253$$ −29.8496 −1.87663
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ 18.0000i 1.12281i 0.827541 + 0.561405i $$0.189739\pi$$
−0.827541 + 0.561405i $$0.810261\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 8.29156 27.5000i 0.507435 1.68297i
$$268$$ 19.8997 1.21557
$$269$$ 13.2665i 0.808873i −0.914566 0.404436i $$-0.867468\pi$$
0.914566 0.404436i $$-0.132532\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 9.00000 29.8496i 0.541736 1.79674i
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ −12.5000 8.29156i −0.748355 0.496403i
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 33.1662i 1.96805i
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ −16.5000 4.97494i −0.967247 0.291636i
$$292$$ 0 0
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −13.2665 + 11.0000i −0.769800 + 0.638285i
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 33.0000 + 9.94987i 1.87730 + 0.566029i
$$310$$ 0 0
$$311$$ 33.1662i 1.88069i 0.340229 + 0.940343i $$0.389495\pi$$
−0.340229 + 0.940343i $$0.610505\pi$$
$$312$$ 0 0
$$313$$ −29.8496 −1.68720 −0.843600 0.536972i $$-0.819568\pi$$
−0.843600 + 0.536972i $$0.819568\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 27.0000i 1.51647i −0.651981 0.758236i $$-0.726062\pi$$
0.651981 0.758236i $$-0.273938\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ −7.00000 16.5831i −0.388889 0.921285i
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −35.0000 −1.92377 −0.961887 0.273447i $$-0.911836\pi$$
−0.961887 + 0.273447i $$0.911836\pi$$
$$332$$ 0 0
$$333$$ 24.8747 + 16.5000i 1.36312 + 0.904194i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 10.5000 34.8246i 0.570282 1.89141i
$$340$$ 0 0
$$341$$ 16.5831i 0.898027i
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 9.00000i 0.479022i −0.970894 0.239511i $$-0.923013\pi$$
0.970894 0.239511i $$-0.0769871\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 33.1662i 1.75781i
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ 19.0000 1.00000
$$362$$ 0 0
$$363$$ −18.2414 5.50000i −0.957427 0.288675i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9.94987 0.519379 0.259690 0.965692i $$-0.416380\pi$$
0.259690 + 0.965692i $$0.416380\pi$$
$$368$$ 36.0000i 1.87663i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 16.5831 + 5.00000i 0.859795 + 0.259238i
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 25.0000 1.28416 0.642082 0.766636i $$-0.278071\pi$$
0.642082 + 0.766636i $$0.278071\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 39.0000i 1.99281i 0.0847358 + 0.996403i $$0.472995\pi$$
−0.0847358 + 0.996403i $$0.527005\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 19.8997 1.01026
$$389$$ 36.4829i 1.84976i 0.380265 + 0.924878i $$0.375833\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 11.0000 16.5831i 0.552771 0.833333i
$$397$$ 39.7995 1.99748 0.998740 0.0501886i $$-0.0159822\pi$$
0.998740 + 0.0501886i $$0.0159822\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 26.5330i 1.32499i 0.749064 + 0.662497i $$0.230503\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 33.0000i 1.63575i
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ −1.50000 + 4.97494i −0.0739895 + 0.245396i
$$412$$ −39.7995 −1.96078
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 33.1662i 1.62028i 0.586238 + 0.810139i $$0.300608\pi$$
−0.586238 + 0.810139i $$0.699392\pi$$
$$420$$ 0 0
$$421$$ 10.0000 0.487370 0.243685 0.969854i $$-0.421644\pi$$
0.243685 + 0.969854i $$0.421644\pi$$
$$422$$ 0 0
$$423$$ −19.8997 + 30.0000i −0.967559 + 1.45865i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 13.2665 + 16.0000i 0.638285 + 0.769800i
$$433$$ 29.8496 1.43448 0.717241 0.696826i $$-0.245405\pi$$
0.717241 + 0.696826i $$0.245405\pi$$
$$434$$ 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
$$441$$ 17.5000 + 11.6082i 0.833333 + 0.552771i
$$442$$ 0 0
$$443$$ 21.0000i 0.997740i 0.866677 + 0.498870i $$0.166252\pi$$
−0.866677 + 0.498870i $$0.833748\pi$$
$$444$$ −33.0000 9.94987i −1.56611 0.472200i
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 16.5831i 0.782606i −0.920262 0.391303i $$-0.872024\pi$$
0.920262 0.391303i $$-0.127976\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 42.0000i 1.97551i
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$462$$ 0 0
$$463$$ 29.8496 1.38723 0.693615 0.720346i $$-0.256017\pi$$
0.693615 + 0.720346i $$0.256017\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 3.00000i 0.138823i −0.997588 0.0694117i $$-0.977888\pi$$
0.997588 0.0694117i $$-0.0221122\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 16.5000 + 4.97494i 0.760280 + 0.229233i
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −9.94987 + 15.0000i −0.455573 + 0.686803i
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 22.0000 1.00000
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −9.94987 −0.450872 −0.225436 0.974258i $$-0.572381\pi$$
−0.225436 + 0.974258i $$0.572381\pi$$
$$488$$ 0 0
$$489$$ −33.0000 9.94987i −1.49231 0.449949i
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ −20.0000 −0.898027
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 40.0000 1.79065 0.895323 0.445418i $$-0.146945\pi$$
0.895323 + 0.445418i $$0.146945\pi$$
$$500$$ 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
$$506$$ 0 0
$$507$$ 21.5581 + 6.50000i 0.957427 + 0.288675i
$$508$$ 0 0
$$509$$ 3.31662i 0.147007i −0.997295 0.0735034i $$-0.976582\pi$$
0.997295 0.0735034i $$-0.0234180\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −39.7995 −1.75038
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 43.1161i 1.88895i −0.328581 0.944476i $$-0.606570\pi$$
0.328581 0.944476i $$-0.393430\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ −6.63325 + 22.0000i −0.288675 + 0.957427i
$$529$$ −58.0000 −2.52174
$$530$$ 0 0
$$531$$ 5.50000 8.29156i 0.238680 0.359823i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 8.29156 27.5000i 0.357807 1.18671i
$$538$$ 0 0
$$539$$ 23.2164i 1.00000i
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 41.4578 + 12.5000i 1.77912 + 0.536426i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 12.0000 39.7995i 0.505291 1.67586i
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 11.6082 38.5000i 0.484939 1.60836i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ −20.0000 13.2665i −0.833333 0.552771i
$$577$$ −9.94987 −0.414219 −0.207109 0.978318i $$-0.566406\pi$$
−0.207109 + 0.978318i $$0.566406\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −19.8997 −0.824163
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 48.0000i 1.98117i −0.136892 0.990586i $$-0.543711\pi$$
0.136892 0.990586i $$-0.456289\pi$$
$$588$$ −23.2164 7.00000i −0.957427 0.288675i
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 39.7995 1.63575
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −33.1662 10.0000i −1.35740 0.409273i
$$598$$ 0 0
$$599$$ 33.1662i 1.35514i 0.735460 + 0.677568i $$0.236966\pi$$
−0.735460 + 0.677568i $$0.763034\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ −24.8747 16.5000i −1.01298 0.671932i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 42.0000i 1.69086i −0.534089 0.845428i $$-0.679345\pi$$
0.534089 0.845428i $$-0.320655\pi$$
$$618$$ 0 0
$$619$$ −1.00000 −0.0401934 −0.0200967 0.999798i $$-0.506397\pi$$
−0.0200967 + 0.999798i $$0.506397\pi$$
$$620$$ 0 0
$$621$$ −36.0000 + 29.8496i −1.44463 + 1.19782i
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 0 0
$$628$$ −19.8997 −0.794086
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −7.00000 −0.278666 −0.139333 0.990246i $$-0.544496\pi$$
−0.139333 + 0.990246i $$0.544496\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 6.00000 19.8997i 0.237915 0.789076i
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 27.5000 41.4578i 1.08788 1.64005i
$$640$$ 0 0
$$641$$ 23.2164i 0.916992i −0.888697 0.458496i $$-0.848388\pi$$
0.888697 0.458496i $$-0.151612\pi$$
$$642$$ 0 0
$$643$$ −29.8496 −1.17715 −0.588577 0.808441i $$-0.700312\pi$$
−0.588577 + 0.808441i $$0.700312\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 27.0000i 1.06148i 0.847535 + 0.530740i $$0.178086\pi$$
−0.847535 + 0.530740i $$0.821914\pi$$
$$648$$ 0 0
$$649$$ 11.0000 0.431788
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 39.7995 1.55867
$$653$$ 51.0000i 1.99578i −0.0648948 0.997892i $$-0.520671\pi$$
0.0648948 0.997892i $$-0.479329\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ −13.0000 −0.505641 −0.252821 0.967513i $$-0.581358\pi$$
−0.252821 + 0.967513i $$0.581358\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 49.5000 + 14.9248i 1.91378 + 0.577027i
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ −26.0000 −1.00000
$$677$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 24.0000i 0.918334i 0.888350 + 0.459167i $$0.151852\pi$$
−0.888350 + 0.459167i $$0.848148\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −8.29156 2.50000i −0.316343 0.0953809i
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ 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
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 26.5330i 1.00000i
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ −3.31662 + 11.0000i −0.124646 + 0.413405i
$$709$$ −19.0000 −0.713560 −0.356780 0.934188i $$-0.616125\pi$$
−0.356780 + 0.934188i $$0.616125\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 45.0000i 1.68526i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 33.1662i 1.23948i
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 16.5831i 0.618446i −0.950990 0.309223i $$-0.899931\pi$$
0.950990 0.309223i $$-0.100069\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ −50.0000 −1.85824
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 9.94987 0.369020 0.184510 0.982831i $$-0.440930\pi$$
0.184510 + 0.982831i $$0.440930\pi$$
$$728$$ 0 0
$$729$$ −5.00000 + 26.5330i −0.185185 + 0.982704i
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 33.0000i 1.21557i
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 23.0000 0.839282 0.419641 0.907690i $$-0.362156\pi$$
0.419641 + 0.907690i $$0.362156\pi$$
$$752$$ 48.0000i 1.75038i
$$753$$ 8.29156 27.5000i 0.302161 1.00216i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −39.7995 −1.44654 −0.723269 0.690567i $$-0.757361\pi$$
−0.723269 + 0.690567i $$0.757361\pi$$
$$758$$ 0 0
$$759$$ −49.5000 14.9248i −1.79674 0.541736i
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 46.4327i 1.67988i
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 26.5330 + 8.00000i 0.957427 + 0.288675i
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ −9.00000 + 29.8496i −0.324127 + 1.07501i
$$772$$ 0 0
$$773$$ 54.0000i 1.94225i −0.238581 0.971123i $$-0.576682\pi$$
0.238581 0.971123i $$-0.423318\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 55.0000 1.96805
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 28.0000 1.00000
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 40.0000 1.41776
$$797$$ 3.00000i 0.106265i −0.998587 0.0531327i $$-0.983079\pi$$
0.998587 0.0531327i $$-0.0169206\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 27.5000 41.4578i 0.971665 1.46484i
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 33.0000 + 9.94987i 1.16382 + 0.350905i
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.63325 22.0000i 0.233501 0.774437i
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$