# Properties

 Label 3024.2.k.f.1889.3 Level 3024 Weight 2 Character 3024.1889 Analytic conductor 24.147 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: no (minimal twist has level 378) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1889.3 Root $$-0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 3024.1889 Dual form 3024.2.k.f.1889.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.46410 q^{5} +(-2.00000 - 1.73205i) q^{7} +O(q^{10})$$ $$q+3.46410 q^{5} +(-2.00000 - 1.73205i) q^{7} +6.00000i q^{11} -1.73205i q^{13} -1.73205 q^{17} -6.92820i q^{19} -3.00000i q^{23} +7.00000 q^{25} -3.00000i q^{29} -5.19615i q^{31} +(-6.92820 - 6.00000i) q^{35} -2.00000 q^{37} +6.92820 q^{41} +11.0000 q^{43} +6.92820 q^{47} +(1.00000 + 6.92820i) q^{49} -3.00000i q^{53} +20.7846i q^{55} +8.66025 q^{59} -13.8564i q^{61} -6.00000i q^{65} +7.00000 q^{67} +3.00000i q^{71} +6.92820i q^{73} +(10.3923 - 12.0000i) q^{77} -8.00000 q^{79} +3.46410 q^{83} -6.00000 q^{85} +5.19615 q^{89} +(-3.00000 + 3.46410i) q^{91} -24.0000i q^{95} -6.92820i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{7} + O(q^{10})$$ $$4q - 8q^{7} + 28q^{25} - 8q^{37} + 44q^{43} + 4q^{49} + 28q^{67} - 32q^{79} - 24q^{85} - 12q^{91} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3024\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1135$$ $$2593$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 3.46410 1.54919 0.774597 0.632456i $$-0.217953\pi$$
0.774597 + 0.632456i $$0.217953\pi$$
$$6$$ 0 0
$$7$$ −2.00000 1.73205i −0.755929 0.654654i
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 6.00000i 1.80907i 0.426401 + 0.904534i $$0.359781\pi$$
−0.426401 + 0.904534i $$0.640219\pi$$
$$12$$ 0 0
$$13$$ 1.73205i 0.480384i −0.970725 0.240192i $$-0.922790\pi$$
0.970725 0.240192i $$-0.0772105\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −1.73205 −0.420084 −0.210042 0.977692i $$-0.567360\pi$$
−0.210042 + 0.977692i $$0.567360\pi$$
$$18$$ 0 0
$$19$$ 6.92820i 1.58944i −0.606977 0.794719i $$-0.707618\pi$$
0.606977 0.794719i $$-0.292382\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 3.00000i 0.625543i −0.949828 0.312772i $$-0.898743\pi$$
0.949828 0.312772i $$-0.101257\pi$$
$$24$$ 0 0
$$25$$ 7.00000 1.40000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 3.00000i 0.557086i −0.960424 0.278543i $$-0.910149\pi$$
0.960424 0.278543i $$-0.0898515\pi$$
$$30$$ 0 0
$$31$$ 5.19615i 0.933257i −0.884454 0.466628i $$-0.845469\pi$$
0.884454 0.466628i $$-0.154531\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −6.92820 6.00000i −1.17108 1.01419i
$$36$$ 0 0
$$37$$ −2.00000 −0.328798 −0.164399 0.986394i $$-0.552568\pi$$
−0.164399 + 0.986394i $$0.552568\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 6.92820 1.08200 0.541002 0.841021i $$-0.318045\pi$$
0.541002 + 0.841021i $$0.318045\pi$$
$$42$$ 0 0
$$43$$ 11.0000 1.67748 0.838742 0.544529i $$-0.183292\pi$$
0.838742 + 0.544529i $$0.183292\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 6.92820 1.01058 0.505291 0.862949i $$-0.331385\pi$$
0.505291 + 0.862949i $$0.331385\pi$$
$$48$$ 0 0
$$49$$ 1.00000 + 6.92820i 0.142857 + 0.989743i
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 3.00000i 0.412082i −0.978543 0.206041i $$-0.933942\pi$$
0.978543 0.206041i $$-0.0660580\pi$$
$$54$$ 0 0
$$55$$ 20.7846i 2.80260i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 8.66025 1.12747 0.563735 0.825956i $$-0.309364\pi$$
0.563735 + 0.825956i $$0.309364\pi$$
$$60$$ 0 0
$$61$$ 13.8564i 1.77413i −0.461644 0.887066i $$-0.652740\pi$$
0.461644 0.887066i $$-0.347260\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.00000i 0.744208i
$$66$$ 0 0
$$67$$ 7.00000 0.855186 0.427593 0.903971i $$-0.359362\pi$$
0.427593 + 0.903971i $$0.359362\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 3.00000i 0.356034i 0.984027 + 0.178017i $$0.0569683\pi$$
−0.984027 + 0.178017i $$0.943032\pi$$
$$72$$ 0 0
$$73$$ 6.92820i 0.810885i 0.914121 + 0.405442i $$0.132883\pi$$
−0.914121 + 0.405442i $$0.867117\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 10.3923 12.0000i 1.18431 1.36753i
$$78$$ 0 0
$$79$$ −8.00000 −0.900070 −0.450035 0.893011i $$-0.648589\pi$$
−0.450035 + 0.893011i $$0.648589\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 3.46410 0.380235 0.190117 0.981761i $$-0.439113\pi$$
0.190117 + 0.981761i $$0.439113\pi$$
$$84$$ 0 0
$$85$$ −6.00000 −0.650791
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 5.19615 0.550791 0.275396 0.961331i $$-0.411191\pi$$
0.275396 + 0.961331i $$0.411191\pi$$
$$90$$ 0 0
$$91$$ −3.00000 + 3.46410i −0.314485 + 0.363137i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 24.0000i 2.46235i
$$96$$ 0 0
$$97$$ 6.92820i 0.703452i −0.936103 0.351726i $$-0.885595\pi$$
0.936103 0.351726i $$-0.114405\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 8.66025i 0.853320i 0.904412 + 0.426660i $$0.140310\pi$$
−0.904412 + 0.426660i $$0.859690\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 0 0
$$109$$ 4.00000 0.383131 0.191565 0.981480i $$-0.438644\pi$$
0.191565 + 0.981480i $$0.438644\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 6.00000i 0.564433i 0.959351 + 0.282216i $$0.0910696\pi$$
−0.959351 + 0.282216i $$0.908930\pi$$
$$114$$ 0 0
$$115$$ 10.3923i 0.969087i
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 3.46410 + 3.00000i 0.317554 + 0.275010i
$$120$$ 0 0
$$121$$ −25.0000 −2.27273
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 6.92820 0.619677
$$126$$ 0 0
$$127$$ −14.0000 −1.24230 −0.621150 0.783692i $$-0.713334\pi$$
−0.621150 + 0.783692i $$0.713334\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 5.19615 0.453990 0.226995 0.973896i $$-0.427110\pi$$
0.226995 + 0.973896i $$0.427110\pi$$
$$132$$ 0 0
$$133$$ −12.0000 + 13.8564i −1.04053 + 1.20150i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 3.46410i 0.293821i 0.989150 + 0.146911i $$0.0469330\pi$$
−0.989150 + 0.146911i $$0.953067\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 10.3923 0.869048
$$144$$ 0 0
$$145$$ 10.3923i 0.863034i
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 15.0000i 1.22885i −0.788976 0.614424i $$-0.789388\pi$$
0.788976 0.614424i $$-0.210612\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 18.0000i 1.44579i
$$156$$ 0 0
$$157$$ 15.5885i 1.24409i 0.782980 + 0.622047i $$0.213699\pi$$
−0.782980 + 0.622047i $$0.786301\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −5.19615 + 6.00000i −0.409514 + 0.472866i
$$162$$ 0 0
$$163$$ −11.0000 −0.861586 −0.430793 0.902451i $$-0.641766\pi$$
−0.430793 + 0.902451i $$0.641766\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 17.3205 1.34030 0.670151 0.742225i $$-0.266230\pi$$
0.670151 + 0.742225i $$0.266230\pi$$
$$168$$ 0 0
$$169$$ 10.0000 0.769231
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −20.7846 −1.58022 −0.790112 0.612962i $$-0.789978\pi$$
−0.790112 + 0.612962i $$0.789978\pi$$
$$174$$ 0 0
$$175$$ −14.0000 12.1244i −1.05830 0.916515i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 6.00000i 0.448461i 0.974536 + 0.224231i $$0.0719869\pi$$
−0.974536 + 0.224231i $$0.928013\pi$$
$$180$$ 0 0
$$181$$ 1.73205i 0.128742i −0.997926 0.0643712i $$-0.979496\pi$$
0.997926 0.0643712i $$-0.0205042\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −6.92820 −0.509372
$$186$$ 0 0
$$187$$ 10.3923i 0.759961i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 24.0000i 1.73658i −0.496058 0.868290i $$-0.665220\pi$$
0.496058 0.868290i $$-0.334780\pi$$
$$192$$ 0 0
$$193$$ 13.0000 0.935760 0.467880 0.883792i $$-0.345018\pi$$
0.467880 + 0.883792i $$0.345018\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.00000i 0.427482i −0.976890 0.213741i $$-0.931435\pi$$
0.976890 0.213741i $$-0.0685649\pi$$
$$198$$ 0 0
$$199$$ 8.66025i 0.613909i −0.951724 0.306955i $$-0.900690\pi$$
0.951724 0.306955i $$-0.0993100\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −5.19615 + 6.00000i −0.364698 + 0.421117i
$$204$$ 0 0
$$205$$ 24.0000 1.67623
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 41.5692 2.87540
$$210$$ 0 0
$$211$$ −7.00000 −0.481900 −0.240950 0.970538i $$-0.577459\pi$$
−0.240950 + 0.970538i $$0.577459\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 38.1051 2.59875
$$216$$ 0 0
$$217$$ −9.00000 + 10.3923i −0.610960 + 0.705476i
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3.00000i 0.201802i
$$222$$ 0 0
$$223$$ 10.3923i 0.695920i 0.937509 + 0.347960i $$0.113126\pi$$
−0.937509 + 0.347960i $$0.886874\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −25.9808 −1.72440 −0.862202 0.506565i $$-0.830915\pi$$
−0.862202 + 0.506565i $$0.830915\pi$$
$$228$$ 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
$$233$$ 24.0000i 1.57229i 0.618041 + 0.786146i $$0.287927\pi$$
−0.618041 + 0.786146i $$0.712073\pi$$
$$234$$ 0 0
$$235$$ 24.0000 1.56559
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 13.8564i 0.892570i −0.894891 0.446285i $$-0.852747\pi$$
0.894891 0.446285i $$-0.147253\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 3.46410 + 24.0000i 0.221313 + 1.53330i
$$246$$ 0 0
$$247$$ −12.0000 −0.763542
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −3.46410 −0.218652 −0.109326 0.994006i $$-0.534869\pi$$
−0.109326 + 0.994006i $$0.534869\pi$$
$$252$$ 0 0
$$253$$ 18.0000 1.13165
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −20.7846 −1.29651 −0.648254 0.761424i $$-0.724501\pi$$
−0.648254 + 0.761424i $$0.724501\pi$$
$$258$$ 0 0
$$259$$ 4.00000 + 3.46410i 0.248548 + 0.215249i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 9.00000i 0.554964i 0.960731 + 0.277482i $$0.0894999\pi$$
−0.960731 + 0.277482i $$0.910500\pi$$
$$264$$ 0 0
$$265$$ 10.3923i 0.638394i
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −3.46410 −0.211210 −0.105605 0.994408i $$-0.533678\pi$$
−0.105605 + 0.994408i $$0.533678\pi$$
$$270$$ 0 0
$$271$$ 12.1244i 0.736502i 0.929726 + 0.368251i $$0.120043\pi$$
−0.929726 + 0.368251i $$0.879957\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 42.0000i 2.53270i
$$276$$ 0 0
$$277$$ 20.0000 1.20168 0.600842 0.799368i $$-0.294832\pi$$
0.600842 + 0.799368i $$0.294832\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 30.0000i 1.78965i −0.446417 0.894825i $$-0.647300\pi$$
0.446417 0.894825i $$-0.352700\pi$$
$$282$$ 0 0
$$283$$ 20.7846i 1.23552i 0.786368 + 0.617758i $$0.211959\pi$$
−0.786368 + 0.617758i $$0.788041\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −13.8564 12.0000i −0.817918 0.708338i
$$288$$ 0 0
$$289$$ −14.0000 −0.823529
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 17.3205 1.01187 0.505937 0.862570i $$-0.331147\pi$$
0.505937 + 0.862570i $$0.331147\pi$$
$$294$$ 0 0
$$295$$ 30.0000 1.74667
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −5.19615 −0.300501
$$300$$ 0 0
$$301$$ −22.0000 19.0526i −1.26806 1.09817i
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 48.0000i 2.74847i
$$306$$ 0 0
$$307$$ 31.1769i 1.77936i −0.456584 0.889680i $$-0.650927\pi$$
0.456584 0.889680i $$-0.349073\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −13.8564 −0.785725 −0.392862 0.919597i $$-0.628515\pi$$
−0.392862 + 0.919597i $$0.628515\pi$$
$$312$$ 0 0
$$313$$ 13.8564i 0.783210i 0.920133 + 0.391605i $$0.128080\pi$$
−0.920133 + 0.391605i $$0.871920\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.0000i 1.01098i 0.862832 + 0.505490i $$0.168688\pi$$
−0.862832 + 0.505490i $$0.831312\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 12.0000i 0.667698i
$$324$$ 0 0
$$325$$ 12.1244i 0.672538i
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −13.8564 12.0000i −0.763928 0.661581i
$$330$$ 0 0
$$331$$ 17.0000 0.934405 0.467202 0.884150i $$-0.345262\pi$$
0.467202 + 0.884150i $$0.345262\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 24.2487 1.32485
$$336$$ 0 0
$$337$$ 13.0000 0.708155 0.354078 0.935216i $$-0.384795\pi$$
0.354078 + 0.935216i $$0.384795\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 31.1769 1.68832
$$342$$ 0 0
$$343$$ 10.0000 15.5885i 0.539949 0.841698i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 18.0000i 0.966291i −0.875540 0.483145i $$-0.839494\pi$$
0.875540 0.483145i $$-0.160506\pi$$
$$348$$ 0 0
$$349$$ 19.0526i 1.01986i −0.860216 0.509930i $$-0.829671\pi$$
0.860216 0.509930i $$-0.170329\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 8.66025 0.460939 0.230469 0.973080i $$-0.425974\pi$$
0.230469 + 0.973080i $$0.425974\pi$$
$$354$$ 0 0
$$355$$ 10.3923i 0.551566i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 27.0000i 1.42501i −0.701669 0.712503i $$-0.747562\pi$$
0.701669 0.712503i $$-0.252438\pi$$
$$360$$ 0 0
$$361$$ −29.0000 −1.52632
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 24.0000i 1.25622i
$$366$$ 0 0
$$367$$ 19.0526i 0.994535i −0.867597 0.497268i $$-0.834337\pi$$
0.867597 0.497268i $$-0.165663\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −5.19615 + 6.00000i −0.269771 + 0.311504i
$$372$$ 0 0
$$373$$ −4.00000 −0.207112 −0.103556 0.994624i $$-0.533022\pi$$
−0.103556 + 0.994624i $$0.533022\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −5.19615 −0.267615
$$378$$ 0 0
$$379$$ −16.0000 −0.821865 −0.410932 0.911666i $$-0.634797\pi$$
−0.410932 + 0.911666i $$0.634797\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 3.46410 0.177007 0.0885037 0.996076i $$-0.471792\pi$$
0.0885037 + 0.996076i $$0.471792\pi$$
$$384$$ 0 0
$$385$$ 36.0000 41.5692i 1.83473 2.11856i
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 6.00000i 0.304212i 0.988364 + 0.152106i $$0.0486055\pi$$
−0.988364 + 0.152106i $$0.951394\pi$$
$$390$$ 0 0
$$391$$ 5.19615i 0.262781i
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −27.7128 −1.39438
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 18.0000i 0.898877i −0.893311 0.449439i $$-0.851624\pi$$
0.893311 0.449439i $$-0.148376\pi$$
$$402$$ 0 0
$$403$$ −9.00000 −0.448322
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 12.0000i 0.594818i
$$408$$ 0 0
$$409$$ 3.46410i 0.171289i −0.996326 0.0856444i $$-0.972705\pi$$
0.996326 0.0856444i $$-0.0272949\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −17.3205 15.0000i −0.852286 0.738102i
$$414$$ 0 0
$$415$$ 12.0000 0.589057
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −29.4449 −1.43848 −0.719238 0.694764i $$-0.755509\pi$$
−0.719238 + 0.694764i $$0.755509\pi$$
$$420$$ 0 0
$$421$$ −40.0000 −1.94948 −0.974740 0.223341i $$-0.928304\pi$$
−0.974740 + 0.223341i $$0.928304\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −12.1244 −0.588118
$$426$$ 0 0
$$427$$ −24.0000 + 27.7128i −1.16144 + 1.34112i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 12.0000i 0.578020i 0.957326 + 0.289010i $$0.0933260\pi$$
−0.957326 + 0.289010i $$0.906674\pi$$
$$432$$ 0 0
$$433$$ 20.7846i 0.998845i 0.866359 + 0.499422i $$0.166454\pi$$
−0.866359 + 0.499422i $$0.833546\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −20.7846 −0.994263
$$438$$ 0 0
$$439$$ 5.19615i 0.247999i −0.992282 0.123999i $$-0.960428\pi$$
0.992282 0.123999i $$-0.0395721\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 18.0000i 0.855206i 0.903967 + 0.427603i $$0.140642\pi$$
−0.903967 + 0.427603i $$0.859358\pi$$
$$444$$ 0 0
$$445$$ 18.0000 0.853282
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6.00000i 0.283158i 0.989927 + 0.141579i $$0.0452178\pi$$
−0.989927 + 0.141579i $$0.954782\pi$$
$$450$$ 0 0
$$451$$ 41.5692i 1.95742i
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −10.3923 + 12.0000i −0.487199 + 0.562569i
$$456$$ 0 0
$$457$$ −17.0000 −0.795226 −0.397613 0.917553i $$-0.630161\pi$$
−0.397613 + 0.917553i $$0.630161\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 6.92820 0.322679 0.161339 0.986899i $$-0.448419\pi$$
0.161339 + 0.986899i $$0.448419\pi$$
$$462$$ 0 0
$$463$$ 4.00000 0.185896 0.0929479 0.995671i $$-0.470371\pi$$
0.0929479 + 0.995671i $$0.470371\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 31.1769 1.44270 0.721348 0.692573i $$-0.243523\pi$$
0.721348 + 0.692573i $$0.243523\pi$$
$$468$$ 0 0
$$469$$ −14.0000 12.1244i −0.646460 0.559851i
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 66.0000i 3.03468i
$$474$$ 0 0
$$475$$ 48.4974i 2.22521i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 24.2487 1.10795 0.553976 0.832533i $$-0.313110\pi$$
0.553976 + 0.832533i $$0.313110\pi$$
$$480$$ 0 0
$$481$$ 3.46410i 0.157949i
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 24.0000i 1.08978i
$$486$$ 0 0
$$487$$ −28.0000 −1.26880 −0.634401 0.773004i $$-0.718753\pi$$
−0.634401 + 0.773004i $$0.718753\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 18.0000i 0.812329i 0.913800 + 0.406164i $$0.133134\pi$$
−0.913800 + 0.406164i $$0.866866\pi$$
$$492$$ 0 0
$$493$$ 5.19615i 0.234023i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 5.19615 6.00000i 0.233079 0.269137i
$$498$$ 0 0
$$499$$ −28.0000 −1.25345 −0.626726 0.779240i $$-0.715605\pi$$
−0.626726 + 0.779240i $$0.715605\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −17.3205 −0.772283 −0.386142 0.922440i $$-0.626192\pi$$
−0.386142 + 0.922440i $$0.626192\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −3.46410 −0.153544 −0.0767718 0.997049i $$-0.524461\pi$$
−0.0767718 + 0.997049i $$0.524461\pi$$
$$510$$ 0 0
$$511$$ 12.0000 13.8564i 0.530849 0.612971i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 30.0000i 1.32196i
$$516$$ 0 0
$$517$$ 41.5692i 1.82821i
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 1.73205 0.0758825 0.0379413 0.999280i $$-0.487920\pi$$
0.0379413 + 0.999280i $$0.487920\pi$$
$$522$$ 0 0
$$523$$ 17.3205i 0.757373i 0.925525 + 0.378686i $$0.123624\pi$$
−0.925525 + 0.378686i $$0.876376\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 9.00000i 0.392046i
$$528$$ 0 0
$$529$$ 14.0000 0.608696
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 12.0000i 0.519778i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −41.5692 + 6.00000i −1.79051 + 0.258438i
$$540$$ 0 0
$$541$$ 38.0000 1.63375 0.816874 0.576816i $$-0.195705\pi$$
0.816874 + 0.576816i $$0.195705\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 13.8564 0.593543
$$546$$ 0 0
$$547$$ −8.00000 −0.342055 −0.171028 0.985266i $$-0.554709\pi$$
−0.171028 + 0.985266i $$0.554709\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −20.7846 −0.885454
$$552$$ 0 0
$$553$$ 16.0000 + 13.8564i 0.680389 + 0.589234i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 39.0000i 1.65248i 0.563316 + 0.826242i $$0.309525\pi$$
−0.563316 + 0.826242i $$0.690475\pi$$
$$558$$ 0 0
$$559$$ 19.0526i 0.805837i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −36.3731 −1.53294 −0.766471 0.642279i $$-0.777989\pi$$
−0.766471 + 0.642279i $$0.777989\pi$$
$$564$$ 0 0
$$565$$ 20.7846i 0.874415i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 18.0000i 0.754599i −0.926091 0.377300i $$-0.876853\pi$$
0.926091 0.377300i $$-0.123147\pi$$
$$570$$ 0 0
$$571$$ −41.0000 −1.71580 −0.857898 0.513820i $$-0.828230\pi$$
−0.857898 + 0.513820i $$0.828230\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 21.0000i 0.875761i
$$576$$ 0 0
$$577$$ 38.1051i 1.58634i 0.609002 + 0.793168i $$0.291570\pi$$
−0.609002 + 0.793168i $$0.708430\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −6.92820 6.00000i −0.287430 0.248922i
$$582$$ 0 0
$$583$$ 18.0000 0.745484
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 25.9808 1.07234 0.536170 0.844110i $$-0.319870\pi$$
0.536170 + 0.844110i $$0.319870\pi$$
$$588$$ 0 0
$$589$$ −36.0000 −1.48335
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 6.92820 0.284507 0.142254 0.989830i $$-0.454565\pi$$
0.142254 + 0.989830i $$0.454565\pi$$
$$594$$ 0 0
$$595$$ 12.0000 + 10.3923i 0.491952 + 0.426043i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 9.00000i 0.367730i −0.982952 0.183865i $$-0.941139\pi$$
0.982952 0.183865i $$-0.0588609\pi$$
$$600$$ 0 0
$$601$$ 31.1769i 1.27173i 0.771799 + 0.635866i $$0.219357\pi$$
−0.771799 + 0.635866i $$0.780643\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −86.6025 −3.52089
$$606$$ 0 0
$$607$$ 1.73205i 0.0703018i 0.999382 + 0.0351509i $$0.0111912\pi$$
−0.999382 + 0.0351509i $$0.988809\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000i 0.485468i
$$612$$ 0 0
$$613$$ −28.0000 −1.13091 −0.565455 0.824779i $$-0.691299\pi$$
−0.565455 + 0.824779i $$0.691299\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000i 0.724653i 0.932051 + 0.362326i $$0.118017\pi$$
−0.932051 + 0.362326i $$0.881983\pi$$
$$618$$ 0 0
$$619$$ 38.1051i 1.53157i 0.643094 + 0.765787i $$0.277650\pi$$
−0.643094 + 0.765787i $$0.722350\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −10.3923 9.00000i −0.416359 0.360577i
$$624$$ 0 0
$$625$$ −11.0000 −0.440000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 3.46410 0.138123
$$630$$ 0 0
$$631$$ −2.00000 −0.0796187 −0.0398094 0.999207i $$-0.512675\pi$$
−0.0398094 + 0.999207i $$0.512675\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −48.4974 −1.92456
$$636$$ 0 0
$$637$$ 12.0000 1.73205i 0.475457 0.0686264i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 10.3923i 0.409832i −0.978780 0.204916i $$-0.934308\pi$$
0.978780 0.204916i $$-0.0656922\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 10.3923 0.408564 0.204282 0.978912i $$-0.434514\pi$$
0.204282 + 0.978912i $$0.434514\pi$$
$$648$$ 0 0
$$649$$ 51.9615i 2.03967i
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 27.0000i 1.05659i 0.849060 + 0.528296i $$0.177169\pi$$
−0.849060 + 0.528296i $$0.822831\pi$$
$$654$$ 0 0
$$655$$ 18.0000 0.703318
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 48.0000i 1.86981i 0.354892 + 0.934907i $$0.384518\pi$$
−0.354892 + 0.934907i $$0.615482\pi$$
$$660$$ 0 0
$$661$$ 48.4974i 1.88633i −0.332323 0.943166i $$-0.607832\pi$$
0.332323 0.943166i $$-0.392168\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −41.5692 + 48.0000i −1.61199 + 1.86136i
$$666$$ 0 0
$$667$$ −9.00000 −0.348481
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 83.1384 3.20952
$$672$$ 0 0
$$673$$ 31.0000 1.19496 0.597481 0.801883i $$-0.296168\pi$$
0.597481 + 0.801883i $$0.296168\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 10.3923 0.399409 0.199704 0.979856i $$-0.436002\pi$$
0.199704 + 0.979856i $$0.436002\pi$$
$$678$$ 0 0
$$679$$ −12.0000 + 13.8564i −0.460518 + 0.531760i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 42.0000i 1.60709i 0.595247 + 0.803543i $$0.297054\pi$$
−0.595247 + 0.803543i $$0.702946\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −5.19615 −0.197958
$$690$$ 0 0
$$691$$ 10.3923i 0.395342i 0.980268 + 0.197671i $$0.0633378\pi$$
−0.980268 + 0.197671i $$0.936662\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 12.0000i 0.455186i
$$696$$ 0 0
$$697$$ −12.0000 −0.454532
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 30.0000i 1.13308i 0.824033 + 0.566542i $$0.191719\pi$$
−0.824033 + 0.566542i $$0.808281\pi$$
$$702$$ 0 0
$$703$$ 13.8564i 0.522604i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −15.5885 −0.583792
$$714$$ 0 0
$$715$$ 36.0000 1.34632
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −6.92820 −0.258378 −0.129189 0.991620i $$-0.541237\pi$$
−0.129189 + 0.991620i $$0.541237\pi$$
$$720$$ 0 0
$$721$$ 15.0000 17.3205i 0.558629 0.645049i
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 21.0000i 0.779920i
$$726$$ 0 0
$$727$$ 25.9808i 0.963573i −0.876289 0.481787i $$-0.839988\pi$$
0.876289 0.481787i $$-0.160012\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −19.0526 −0.704684
$$732$$ 0 0
$$733$$ 36.3731i 1.34347i −0.740792 0.671735i $$-0.765549\pi$$
0.740792 0.671735i $$-0.234451\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 42.0000i 1.54709i
$$738$$ 0 0
$$739$$ −20.0000 −0.735712 −0.367856 0.929883i $$-0.619908\pi$$
−0.367856 + 0.929883i $$0.619908\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 9.00000i 0.330178i −0.986279 0.165089i $$-0.947209\pi$$
0.986279 0.165089i $$-0.0527911\pi$$
$$744$$ 0 0
$$745$$ 51.9615i 1.90372i
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2.00000 0.0729810 0.0364905 0.999334i $$-0.488382\pi$$
0.0364905 + 0.999334i $$0.488382\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 27.7128 1.00857
$$756$$ 0 0
$$757$$ −2.00000 −0.0726912 −0.0363456 0.999339i $$-0.511572\pi$$
−0.0363456 + 0.999339i $$0.511572\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −19.0526 −0.690655 −0.345327 0.938482i $$-0.612232\pi$$
−0.345327 + 0.938482i $$0.612232\pi$$
$$762$$ 0 0
$$763$$ −8.00000 6.92820i −0.289619 0.250818i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 15.0000i 0.541619i
$$768$$ 0 0
$$769$$ 3.46410i 0.124919i −0.998048 0.0624593i $$-0.980106\pi$$
0.998048 0.0624593i $$-0.0198944\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −38.1051 −1.37055 −0.685273 0.728286i $$-0.740317\pi$$
−0.685273 + 0.728286i $$0.740317\pi$$
$$774$$ 0 0
$$775$$ 36.3731i 1.30656i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 48.0000i 1.71978i
$$780$$ 0 0
$$781$$ −18.0000 −0.644091
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 54.0000i 1.92734i
$$786$$ 0 0
$$787$$ 3.46410i 0.123482i −0.998092 0.0617409i $$-0.980335\pi$$
0.998092 0.0617409i $$-0.0196653\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 10.3923 12.0000i 0.369508 0.426671i
$$792$$ 0 0
$$793$$ −24.0000 −0.852265
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 31.1769 1.10434 0.552171 0.833731i $$-0.313799\pi$$
0.552171 + 0.833731i $$0.313799\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −41.5692 −1.46695
$$804$$ 0 0
$$805$$ −18.0000 + 20.7846i −0.634417 + 0.732561i
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 24.0000i 0.843795i −0.906644 0.421898i $$-0.861364\pi$$
0.906644 0.421898i $$-0.138636\pi$$
$$810$$ 0 0
$$811$$ 31.1769i 1.09477i 0.836881 + 0.547385i $$0.184377\pi$$
−0.836881 + 0.547385i $$0.815623\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −38.1051 −1.33476