# Properties

 Label 1728.2.c.e.1727.3 Level $1728$ Weight $2$ Character 1728.1727 Analytic conductor $13.798$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 432) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1727.3 Root $$0.866025 + 0.500000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1727 Dual form 1728.2.c.e.1727.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+3.00000i q^{5} -1.73205i q^{7} +O(q^{10})$$ $$q+3.00000i q^{5} -1.73205i q^{7} +5.19615 q^{11} +2.00000 q^{13} -6.00000i q^{17} -6.92820i q^{19} -4.00000 q^{25} +6.00000i q^{29} -5.19615i q^{31} +5.19615 q^{35} -8.00000 q^{37} +10.3923i q^{43} +10.3923 q^{47} +4.00000 q^{49} -9.00000i q^{53} +15.5885i q^{55} +10.3923 q^{59} +4.00000 q^{61} +6.00000i q^{65} +3.46410i q^{67} +10.3923 q^{71} +1.00000 q^{73} -9.00000i q^{77} -3.46410i q^{79} -5.19615 q^{83} +18.0000 q^{85} +6.00000i q^{89} -3.46410i q^{91} +20.7846 q^{95} -5.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 8q^{13} - 16q^{25} - 32q^{37} + 16q^{49} + 16q^{61} + 4q^{73} + 72q^{85} - 20q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 3.00000i 1.34164i 0.741620 + 0.670820i $$0.234058\pi$$
−0.741620 + 0.670820i $$0.765942\pi$$
$$6$$ 0 0
$$7$$ − 1.73205i − 0.654654i −0.944911 0.327327i $$-0.893852\pi$$
0.944911 0.327327i $$-0.106148\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 5.19615 1.56670 0.783349 0.621582i $$-0.213510\pi$$
0.783349 + 0.621582i $$0.213510\pi$$
$$12$$ 0 0
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 6.00000i − 1.45521i −0.685994 0.727607i $$-0.740633\pi$$
0.685994 0.727607i $$-0.259367\pi$$
$$18$$ 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
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −4.00000 −0.800000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6.00000i 1.11417i 0.830455 + 0.557086i $$0.188081\pi$$
−0.830455 + 0.557086i $$0.811919\pi$$
$$30$$ 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
$$35$$ 5.19615 0.878310
$$36$$ 0 0
$$37$$ −8.00000 −1.31519 −0.657596 0.753371i $$-0.728427\pi$$
−0.657596 + 0.753371i $$0.728427\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 10.3923i 1.58481i 0.609994 + 0.792406i $$0.291172\pi$$
−0.609994 + 0.792406i $$0.708828\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.3923 1.51587 0.757937 0.652328i $$-0.226208\pi$$
0.757937 + 0.652328i $$0.226208\pi$$
$$48$$ 0 0
$$49$$ 4.00000 0.571429
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 9.00000i − 1.23625i −0.786082 0.618123i $$-0.787894\pi$$
0.786082 0.618123i $$-0.212106\pi$$
$$54$$ 0 0
$$55$$ 15.5885i 2.10195i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.3923 1.35296 0.676481 0.736460i $$-0.263504\pi$$
0.676481 + 0.736460i $$0.263504\pi$$
$$60$$ 0 0
$$61$$ 4.00000 0.512148 0.256074 0.966657i $$-0.417571\pi$$
0.256074 + 0.966657i $$0.417571\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 6.00000i 0.744208i
$$66$$ 0 0
$$67$$ 3.46410i 0.423207i 0.977356 + 0.211604i $$0.0678686\pi$$
−0.977356 + 0.211604i $$0.932131\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 10.3923 1.23334 0.616670 0.787222i $$-0.288481\pi$$
0.616670 + 0.787222i $$0.288481\pi$$
$$72$$ 0 0
$$73$$ 1.00000 0.117041 0.0585206 0.998286i $$-0.481362\pi$$
0.0585206 + 0.998286i $$0.481362\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 9.00000i − 1.02565i
$$78$$ 0 0
$$79$$ − 3.46410i − 0.389742i −0.980829 0.194871i $$-0.937571\pi$$
0.980829 0.194871i $$-0.0624288\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −5.19615 −0.570352 −0.285176 0.958475i $$-0.592052\pi$$
−0.285176 + 0.958475i $$0.592052\pi$$
$$84$$ 0 0
$$85$$ 18.0000 1.95237
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 6.00000i 0.635999i 0.948091 + 0.317999i $$0.103011\pi$$
−0.948091 + 0.317999i $$0.896989\pi$$
$$90$$ 0 0
$$91$$ − 3.46410i − 0.363137i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 20.7846 2.13246
$$96$$ 0 0
$$97$$ −5.00000 −0.507673 −0.253837 0.967247i $$-0.581693\pi$$
−0.253837 + 0.967247i $$0.581693\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 9.00000i − 0.895533i −0.894150 0.447767i $$-0.852219\pi$$
0.894150 0.447767i $$-0.147781\pi$$
$$102$$ 0 0
$$103$$ 3.46410i 0.341328i 0.985329 + 0.170664i $$0.0545913\pi$$
−0.985329 + 0.170664i $$0.945409\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 5.19615 0.502331 0.251166 0.967944i $$-0.419186\pi$$
0.251166 + 0.967944i $$0.419186\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 12.0000i 1.12887i 0.825479 + 0.564433i $$0.190905\pi$$
−0.825479 + 0.564433i $$0.809095\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −10.3923 −0.952661
$$120$$ 0 0
$$121$$ 16.0000 1.45455
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 3.00000i 0.268328i
$$126$$ 0 0
$$127$$ − 5.19615i − 0.461084i −0.973062 0.230542i $$-0.925950\pi$$
0.973062 0.230542i $$-0.0740499\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −5.19615 −0.453990 −0.226995 0.973896i $$-0.572890\pi$$
−0.226995 + 0.973896i $$0.572890\pi$$
$$132$$ 0 0
$$133$$ −12.0000 −1.04053
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.00000i 0.512615i 0.966595 + 0.256307i $$0.0825059\pi$$
−0.966595 + 0.256307i $$0.917494\pi$$
$$138$$ 0 0
$$139$$ 6.92820i 0.587643i 0.955860 + 0.293821i $$0.0949270\pi$$
−0.955860 + 0.293821i $$0.905073\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 10.3923 0.869048
$$144$$ 0 0
$$145$$ −18.0000 −1.49482
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 15.0000i 1.22885i 0.788976 + 0.614424i $$0.210612\pi$$
−0.788976 + 0.614424i $$0.789388\pi$$
$$150$$ 0 0
$$151$$ − 8.66025i − 0.704761i −0.935857 0.352381i $$-0.885372\pi$$
0.935857 0.352381i $$-0.114628\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 15.5885 1.25210
$$156$$ 0 0
$$157$$ 4.00000 0.319235 0.159617 0.987179i $$-0.448974\pi$$
0.159617 + 0.987179i $$0.448974\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ − 3.46410i − 0.271329i −0.990755 0.135665i $$-0.956683\pi$$
0.990755 0.135665i $$-0.0433170\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −10.3923 −0.804181 −0.402090 0.915600i $$-0.631716\pi$$
−0.402090 + 0.915600i $$0.631716\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ − 3.00000i − 0.228086i −0.993476 0.114043i $$-0.963620\pi$$
0.993476 0.114043i $$-0.0363801\pi$$
$$174$$ 0 0
$$175$$ 6.92820i 0.523723i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −5.19615 −0.388379 −0.194189 0.980964i $$-0.562208\pi$$
−0.194189 + 0.980964i $$0.562208\pi$$
$$180$$ 0 0
$$181$$ 8.00000 0.594635 0.297318 0.954779i $$-0.403908\pi$$
0.297318 + 0.954779i $$0.403908\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 24.0000i − 1.76452i
$$186$$ 0 0
$$187$$ − 31.1769i − 2.27988i
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 10.3923 0.751961 0.375980 0.926628i $$-0.377306\pi$$
0.375980 + 0.926628i $$0.377306\pi$$
$$192$$ 0 0
$$193$$ 11.0000 0.791797 0.395899 0.918294i $$-0.370433\pi$$
0.395899 + 0.918294i $$0.370433\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 15.0000i 1.06871i 0.845262 + 0.534353i $$0.179445\pi$$
−0.845262 + 0.534353i $$0.820555\pi$$
$$198$$ 0 0
$$199$$ − 22.5167i − 1.59616i −0.602549 0.798082i $$-0.705848\pi$$
0.602549 0.798082i $$-0.294152\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 10.3923 0.729397
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 36.0000i − 2.49017i
$$210$$ 0 0
$$211$$ − 13.8564i − 0.953914i −0.878927 0.476957i $$-0.841740\pi$$
0.878927 0.476957i $$-0.158260\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −31.1769 −2.12625
$$216$$ 0 0
$$217$$ −9.00000 −0.610960
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ − 12.0000i − 0.807207i
$$222$$ 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
$$227$$ −10.3923 −0.689761 −0.344881 0.938647i $$-0.612081\pi$$
−0.344881 + 0.938647i $$0.612081\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 6.00000i 0.393073i 0.980497 + 0.196537i $$0.0629694\pi$$
−0.980497 + 0.196537i $$0.937031\pi$$
$$234$$ 0 0
$$235$$ 31.1769i 2.03376i
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −20.7846 −1.34444 −0.672222 0.740349i $$-0.734660\pi$$
−0.672222 + 0.740349i $$0.734660\pi$$
$$240$$ 0 0
$$241$$ 22.0000 1.41714 0.708572 0.705638i $$-0.249340\pi$$
0.708572 + 0.705638i $$0.249340\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 12.0000i 0.766652i
$$246$$ 0 0
$$247$$ − 13.8564i − 0.881662i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 10.3923 0.655956 0.327978 0.944685i $$-0.393633\pi$$
0.327978 + 0.944685i $$0.393633\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 6.00000i 0.374270i 0.982334 + 0.187135i $$0.0599201\pi$$
−0.982334 + 0.187135i $$0.940080\pi$$
$$258$$ 0 0
$$259$$ 13.8564i 0.860995i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 27.0000 1.65860
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 30.0000i 1.82913i 0.404436 + 0.914566i $$0.367468\pi$$
−0.404436 + 0.914566i $$0.632532\pi$$
$$270$$ 0 0
$$271$$ 1.73205i 0.105215i 0.998615 + 0.0526073i $$0.0167532\pi$$
−0.998615 + 0.0526073i $$0.983247\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −20.7846 −1.25336
$$276$$ 0 0
$$277$$ −28.0000 −1.68236 −0.841178 0.540758i $$-0.818138\pi$$
−0.841178 + 0.540758i $$0.818138\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ − 6.00000i − 0.357930i −0.983855 0.178965i $$-0.942725\pi$$
0.983855 0.178965i $$-0.0572749\pi$$
$$282$$ 0 0
$$283$$ − 20.7846i − 1.23552i −0.786368 0.617758i $$-0.788041\pi$$
0.786368 0.617758i $$-0.211959\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −19.0000 −1.11765
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ − 6.00000i − 0.350524i −0.984522 0.175262i $$-0.943923\pi$$
0.984522 0.175262i $$-0.0560772\pi$$
$$294$$ 0 0
$$295$$ 31.1769i 1.81519i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 18.0000 1.03750
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 12.0000i 0.687118i
$$306$$ 0 0
$$307$$ 31.1769i 1.77936i 0.456584 + 0.889680i $$0.349073\pi$$
−0.456584 + 0.889680i $$0.650927\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −20.7846 −1.17859 −0.589294 0.807919i $$-0.700594\pi$$
−0.589294 + 0.807919i $$0.700594\pi$$
$$312$$ 0 0
$$313$$ −31.0000 −1.75222 −0.876112 0.482108i $$-0.839871\pi$$
−0.876112 + 0.482108i $$0.839871\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 15.0000i − 0.842484i −0.906948 0.421242i $$-0.861594\pi$$
0.906948 0.421242i $$-0.138406\pi$$
$$318$$ 0 0
$$319$$ 31.1769i 1.74557i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −41.5692 −2.31297
$$324$$ 0 0
$$325$$ −8.00000 −0.443760
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ − 18.0000i − 0.992372i
$$330$$ 0 0
$$331$$ − 3.46410i − 0.190404i −0.995458 0.0952021i $$-0.969650\pi$$
0.995458 0.0952021i $$-0.0303497\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −10.3923 −0.567792
$$336$$ 0 0
$$337$$ −2.00000 −0.108947 −0.0544735 0.998515i $$-0.517348\pi$$
−0.0544735 + 0.998515i $$0.517348\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 27.0000i − 1.46213i
$$342$$ 0 0
$$343$$ − 19.0526i − 1.02874i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −15.5885 −0.836832 −0.418416 0.908255i $$-0.637415\pi$$
−0.418416 + 0.908255i $$0.637415\pi$$
$$348$$ 0 0
$$349$$ 4.00000 0.214115 0.107058 0.994253i $$-0.465857\pi$$
0.107058 + 0.994253i $$0.465857\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 36.0000i 1.91609i 0.286623 + 0.958043i $$0.407467\pi$$
−0.286623 + 0.958043i $$0.592533\pi$$
$$354$$ 0 0
$$355$$ 31.1769i 1.65470i
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −10.3923 −0.548485 −0.274242 0.961661i $$-0.588427\pi$$
−0.274242 + 0.961661i $$0.588427\pi$$
$$360$$ 0 0
$$361$$ −29.0000 −1.52632
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 3.00000i 0.157027i
$$366$$ 0 0
$$367$$ 8.66025i 0.452062i 0.974120 + 0.226031i $$0.0725750\pi$$
−0.974120 + 0.226031i $$0.927425\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −15.5885 −0.809312
$$372$$ 0 0
$$373$$ −2.00000 −0.103556 −0.0517780 0.998659i $$-0.516489\pi$$
−0.0517780 + 0.998659i $$0.516489\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000i 0.618031i
$$378$$ 0 0
$$379$$ − 27.7128i − 1.42351i −0.702427 0.711756i $$-0.747900\pi$$
0.702427 0.711756i $$-0.252100\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 31.1769 1.59307 0.796533 0.604595i $$-0.206665\pi$$
0.796533 + 0.604595i $$0.206665\pi$$
$$384$$ 0 0
$$385$$ 27.0000 1.37605
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ − 9.00000i − 0.456318i −0.973624 0.228159i $$-0.926729\pi$$
0.973624 0.228159i $$-0.0732706\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 10.3923 0.522894
$$396$$ 0 0
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 24.0000i 1.19850i 0.800561 + 0.599251i $$0.204535\pi$$
−0.800561 + 0.599251i $$0.795465\pi$$
$$402$$ 0 0
$$403$$ − 10.3923i − 0.517678i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −41.5692 −2.06051
$$408$$ 0 0
$$409$$ 25.0000 1.23617 0.618085 0.786111i $$-0.287909\pi$$
0.618085 + 0.786111i $$0.287909\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ − 18.0000i − 0.885722i
$$414$$ 0 0
$$415$$ − 15.5885i − 0.765207i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −10.3923 −0.507697 −0.253849 0.967244i $$-0.581697\pi$$
−0.253849 + 0.967244i $$0.581697\pi$$
$$420$$ 0 0
$$421$$ −4.00000 −0.194948 −0.0974740 0.995238i $$-0.531076\pi$$
−0.0974740 + 0.995238i $$0.531076\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 24.0000i 1.16417i
$$426$$ 0 0
$$427$$ − 6.92820i − 0.335279i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 31.1769 1.50174 0.750870 0.660451i $$-0.229635\pi$$
0.750870 + 0.660451i $$0.229635\pi$$
$$432$$ 0 0
$$433$$ 11.0000 0.528626 0.264313 0.964437i $$-0.414855\pi$$
0.264313 + 0.964437i $$0.414855\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ − 15.5885i − 0.743996i −0.928233 0.371998i $$-0.878673\pi$$
0.928233 0.371998i $$-0.121327\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −31.1769 −1.48126 −0.740630 0.671913i $$-0.765473\pi$$
−0.740630 + 0.671913i $$0.765473\pi$$
$$444$$ 0 0
$$445$$ −18.0000 −0.853282
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 36.0000i − 1.69895i −0.527633 0.849473i $$-0.676920\pi$$
0.527633 0.849473i $$-0.323080\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 10.3923 0.487199
$$456$$ 0 0
$$457$$ 1.00000 0.0467780 0.0233890 0.999726i $$-0.492554\pi$$
0.0233890 + 0.999726i $$0.492554\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9.00000i 0.419172i 0.977790 + 0.209586i $$0.0672116\pi$$
−0.977790 + 0.209586i $$0.932788\pi$$
$$462$$ 0 0
$$463$$ − 32.9090i − 1.52941i −0.644381 0.764705i $$-0.722885\pi$$
0.644381 0.764705i $$-0.277115\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −25.9808 −1.20225 −0.601123 0.799156i $$-0.705280\pi$$
−0.601123 + 0.799156i $$0.705280\pi$$
$$468$$ 0 0
$$469$$ 6.00000 0.277054
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 54.0000i 2.48292i
$$474$$ 0 0
$$475$$ 27.7128i 1.27155i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ −10.3923 −0.474837 −0.237418 0.971408i $$-0.576301\pi$$
−0.237418 + 0.971408i $$0.576301\pi$$
$$480$$ 0 0
$$481$$ −16.0000 −0.729537
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ − 15.0000i − 0.681115i
$$486$$ 0 0
$$487$$ 24.2487i 1.09881i 0.835555 + 0.549407i $$0.185146\pi$$
−0.835555 + 0.549407i $$0.814854\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 5.19615 0.234499 0.117250 0.993102i $$-0.462592\pi$$
0.117250 + 0.993102i $$0.462592\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 18.0000i − 0.807410i
$$498$$ 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
$$503$$ −41.5692 −1.85348 −0.926740 0.375703i $$-0.877401\pi$$
−0.926740 + 0.375703i $$0.877401\pi$$
$$504$$ 0 0
$$505$$ 27.0000 1.20148
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 9.00000i 0.398918i 0.979906 + 0.199459i $$0.0639185\pi$$
−0.979906 + 0.199459i $$0.936082\pi$$
$$510$$ 0 0
$$511$$ − 1.73205i − 0.0766214i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −10.3923 −0.457940
$$516$$ 0 0
$$517$$ 54.0000 2.37492
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 18.0000i − 0.788594i −0.918983 0.394297i $$-0.870988\pi$$
0.918983 0.394297i $$-0.129012\pi$$
$$522$$ 0 0
$$523$$ − 17.3205i − 0.757373i −0.925525 0.378686i $$-0.876376\pi$$
0.925525 0.378686i $$-0.123624\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −31.1769 −1.35809
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 15.5885i 0.673948i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 20.7846 0.895257
$$540$$ 0 0
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ − 48.0000i − 2.05609i
$$546$$ 0 0
$$547$$ − 24.2487i − 1.03680i −0.855138 0.518400i $$-0.826528\pi$$
0.855138 0.518400i $$-0.173472\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 41.5692 1.77091
$$552$$ 0 0
$$553$$ −6.00000 −0.255146
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 39.0000i − 1.65248i −0.563316 0.826242i $$-0.690475\pi$$
0.563316 0.826242i $$-0.309525\pi$$
$$558$$ 0 0
$$559$$ 20.7846i 0.879095i
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 15.5885 0.656975 0.328488 0.944508i $$-0.393461\pi$$
0.328488 + 0.944508i $$0.393461\pi$$
$$564$$ 0 0
$$565$$ −36.0000 −1.51453
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ − 24.0000i − 1.00613i −0.864248 0.503066i $$-0.832205\pi$$
0.864248 0.503066i $$-0.167795\pi$$
$$570$$ 0 0
$$571$$ 24.2487i 1.01478i 0.861717 + 0.507388i $$0.169389\pi$$
−0.861717 + 0.507388i $$0.830611\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −10.0000 −0.416305 −0.208153 0.978096i $$-0.566745\pi$$
−0.208153 + 0.978096i $$0.566745\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 9.00000i 0.373383i
$$582$$ 0 0
$$583$$ − 46.7654i − 1.93682i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 5.19615 0.214468 0.107234 0.994234i $$-0.465801\pi$$
0.107234 + 0.994234i $$0.465801\pi$$
$$588$$ 0 0
$$589$$ −36.0000 −1.48335
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 6.00000i 0.246390i 0.992382 + 0.123195i $$0.0393141\pi$$
−0.992382 + 0.123195i $$0.960686\pi$$
$$594$$ 0 0
$$595$$ − 31.1769i − 1.27813i
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −31.1769 −1.27385 −0.636927 0.770924i $$-0.719795\pi$$
−0.636927 + 0.770924i $$0.719795\pi$$
$$600$$ 0 0
$$601$$ −19.0000 −0.775026 −0.387513 0.921864i $$-0.626666\pi$$
−0.387513 + 0.921864i $$0.626666\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 48.0000i 1.95148i
$$606$$ 0 0
$$607$$ − 38.1051i − 1.54664i −0.634017 0.773320i $$-0.718595\pi$$
0.634017 0.773320i $$-0.281405\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 20.7846 0.840855
$$612$$ 0 0
$$613$$ 10.0000 0.403896 0.201948 0.979396i $$-0.435273\pi$$
0.201948 + 0.979396i $$0.435273\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 12.0000i − 0.483102i −0.970388 0.241551i $$-0.922344\pi$$
0.970388 0.241551i $$-0.0776561\pi$$
$$618$$ 0 0
$$619$$ 24.2487i 0.974638i 0.873224 + 0.487319i $$0.162025\pi$$
−0.873224 + 0.487319i $$0.837975\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 10.3923 0.416359
$$624$$ 0 0
$$625$$ −29.0000 −1.16000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 48.0000i 1.91389i
$$630$$ 0 0
$$631$$ 12.1244i 0.482663i 0.970443 + 0.241331i $$0.0775841\pi$$
−0.970443 + 0.241331i $$0.922416\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 15.5885 0.618609
$$636$$ 0 0
$$637$$ 8.00000 0.316972
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 24.0000i 0.947943i 0.880540 + 0.473972i $$0.157180\pi$$
−0.880540 + 0.473972i $$0.842820\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 20.7846 0.817127 0.408564 0.912730i $$-0.366030\pi$$
0.408564 + 0.912730i $$0.366030\pi$$
$$648$$ 0 0
$$649$$ 54.0000 2.11969
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 9.00000i 0.352197i 0.984373 + 0.176099i $$0.0563478\pi$$
−0.984373 + 0.176099i $$0.943652\pi$$
$$654$$ 0 0
$$655$$ − 15.5885i − 0.609091i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 36.3731 1.41689 0.708447 0.705764i $$-0.249396\pi$$
0.708447 + 0.705764i $$0.249396\pi$$
$$660$$ 0 0
$$661$$ −20.0000 −0.777910 −0.388955 0.921257i $$-0.627164\pi$$
−0.388955 + 0.921257i $$0.627164\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 36.0000i − 1.39602i
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 20.7846 0.802381
$$672$$ 0 0
$$673$$ 23.0000 0.886585 0.443292 0.896377i $$-0.353810\pi$$
0.443292 + 0.896377i $$0.353810\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 18.0000i 0.691796i 0.938272 + 0.345898i $$0.112426\pi$$
−0.938272 + 0.345898i $$0.887574\pi$$
$$678$$ 0 0
$$679$$ 8.66025i 0.332350i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −10.3923 −0.397650 −0.198825 0.980035i $$-0.563713\pi$$
−0.198825 + 0.980035i $$0.563713\pi$$
$$684$$ 0 0
$$685$$ −18.0000 −0.687745
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 18.0000i − 0.685745i
$$690$$ 0 0
$$691$$ 20.7846i 0.790684i 0.918534 + 0.395342i $$0.129374\pi$$
−0.918534 + 0.395342i $$0.870626\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −20.7846 −0.788405
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 21.0000i 0.793159i 0.918000 + 0.396580i $$0.129803\pi$$
−0.918000 + 0.396580i $$0.870197\pi$$
$$702$$ 0 0
$$703$$ 55.4256i 2.09042i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −15.5885 −0.586264
$$708$$ 0 0
$$709$$ 32.0000 1.20179 0.600893 0.799330i $$-0.294812\pi$$
0.600893 + 0.799330i $$0.294812\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 31.1769i 1.16595i
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −20.7846 −0.775135 −0.387568 0.921841i $$-0.626685\pi$$
−0.387568 + 0.921841i $$0.626685\pi$$
$$720$$ 0 0
$$721$$ 6.00000 0.223452
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 24.0000i − 0.891338i
$$726$$ 0 0
$$727$$ 46.7654i 1.73443i 0.497932 + 0.867216i $$0.334093\pi$$
−0.497932 + 0.867216i $$0.665907\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 62.3538 2.30624
$$732$$ 0 0
$$733$$ −4.00000 −0.147743 −0.0738717 0.997268i $$-0.523536\pi$$
−0.0738717 + 0.997268i $$0.523536\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 18.0000i 0.663039i
$$738$$ 0 0
$$739$$ − 10.3923i − 0.382287i −0.981562 0.191144i $$-0.938780\pi$$
0.981562 0.191144i $$-0.0612196\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 31.1769 1.14377 0.571885 0.820334i $$-0.306212\pi$$
0.571885 + 0.820334i $$0.306212\pi$$
$$744$$ 0 0
$$745$$ −45.0000 −1.64867
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ − 9.00000i − 0.328853i
$$750$$ 0 0
$$751$$ 22.5167i 0.821645i 0.911715 + 0.410822i $$0.134758\pi$$
−0.911715 + 0.410822i $$0.865242\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 25.9808 0.945537
$$756$$ 0 0
$$757$$ 26.0000 0.944986 0.472493 0.881334i $$-0.343354\pi$$
0.472493 + 0.881334i $$0.343354\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 24.0000i 0.869999i 0.900431 + 0.435000i $$0.143252\pi$$
−0.900431 + 0.435000i $$0.856748\pi$$
$$762$$ 0 0
$$763$$ 27.7128i 1.00327i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 20.7846 0.750489
$$768$$ 0 0
$$769$$ −25.0000 −0.901523 −0.450762 0.892644i $$-0.648848\pi$$
−0.450762 + 0.892644i $$0.648848\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 6.00000i 0.215805i 0.994161 + 0.107903i $$0.0344134\pi$$
−0.994161 + 0.107903i $$0.965587\pi$$
$$774$$ 0 0
$$775$$ 20.7846i 0.746605i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 54.0000 1.93227
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 12.0000i 0.428298i
$$786$$ 0 0
$$787$$ 34.6410i 1.23482i 0.786642 + 0.617409i $$0.211818\pi$$
−0.786642 + 0.617409i $$0.788182\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 20.7846 0.739016
$$792$$ 0 0
$$793$$ 8.00000 0.284088
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 33.0000i 1.16892i 0.811423 + 0.584460i $$0.198694\pi$$
−0.811423 + 0.584460i $$0.801306\pi$$
$$798$$ 0 0
$$799$$ − 62.3538i − 2.20592i
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 5.19615 0.183368
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 6.00000i 0.210949i 0.994422 + 0.105474i $$0.0336361\pi$$
−0.994422 + 0.105474i $$0.966364\pi$$
$$810$$ 0 0
$$811$$ − 20.7846i − 0.729846i −0.931038 0.364923i $$-0.881095\pi$$
0.931038 0.364923i $$-0.118905\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 10.3923 0.364027
$$816$$ 0 0
$$817$$ 72.0000 2.51896
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ − 42.0000i − 1.46581i −0.680331 0.732905i $$-0.738164\pi$$
0.680331 0.732905i $$-0.261836\pi$$
$$822$$ 0 0
$$823$$ 32.9090i 1.14713i 0.819159 + 0.573567i $$0.194441\pi$$
−0.819159 + 0.573567i $$0.805559\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −31.1769 −1.08413 −0.542064 0.840337i $$-0.682357\pi$$
−0.542064 + 0.840337i $$0.682357\pi$$
$$828$$ 0 0
$$829$$ 26.0000 0.903017 0.451509 0.892267i $$-0.350886\pi$$
0.451509 + 0.892267i $$0.350886\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 24.0000i − 0.831551i
$$834$$ 0 0
$$835$$ − 31.1769i − 1.07892i
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 31.1769 1.07635 0.538173 0.842834i $$-0.319115\pi$$
0.538173 + 0.842834i $$0.319115\pi$$
$$840$$ 0 0
$$841$$ −7.00000 −0.241379
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 27.0000i − 0.928828i
$$846$$ 0 0
$$847$$ − 27.7128i − 0.952224i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −50.0000 −1.71197 −0.855984 0.517003i $$-0.827048\pi$$
−0.855984 + 0.517003i $$0.827048\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 30.0000i − 1.02478i −0.858753 0.512390i $$-0.828760\pi$$
0.858753 0.512390i $$-0.171240\pi$$
$$858$$ 0 0
$$859$$ 6.92820i 0.236387i 0.992991 + 0.118194i $$0.0377103\pi$$
−0.992991 + 0.118194i $$0.962290\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 20.7846 0.707516 0.353758 0.935337i $$-0.384904\pi$$
0.353758 + 0.935337i $$0.384904\pi$$
$$864$$ 0 0
$$865$$ 9.00000 0.306009
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 18.0000i − 0.610608i
$$870$$ 0 0
$$871$$ 6.92820i 0.234753i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 5.19615 0.175662
$$876$$ 0 0
$$877$$ −38.0000 −1.28317 −0.641584 0.767052i $$-0.721723\pi$$
−0.641584 + 0.767052i $$0.721723\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 54.0000i 1.81931i 0.415369 + 0.909653i $$0.363653\pi$$
−0.415369 + 0.909653i $$0.636347\pi$$
$$882$$ 0 0
$$883$$ 6.92820i 0.233153i 0.993182 + 0.116576i $$0.0371920\pi$$
−0.993182 + 0.116576i $$0.962808\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 10.3923 0.348939 0.174470 0.984663i $$-0.444179\pi$$
0.174470 + 0.984663i $$0.444179\pi$$
$$888$$ 0 0
$$889$$ −9.00000 −0.301850
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ − 72.0000i − 2.40939i
$$894$$ 0 0
$$895$$ − 15.5885i − 0.521065i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 31.1769 1.03981
$$900$$ 0 0
$$901$$ −54.0000 −1.79900
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 24.0000i 0.797787i
$$906$$ 0 0
$$907$$ − 27.7128i − 0.920189i −0.887870 0.460094i $$-0.847816\pi$$
0.887870 0.460094i $$-0.152184\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 10.3923 0.344312 0.172156 0.985070i $$-0.444927\pi$$
0.172156 + 0.985070i $$0.444927\pi$$
$$912$$ 0 0
$$913$$ −27.0000 −0.893570
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 9.00000i 0.297206i
$$918$$ 0 0
$$919$$ − 15.5885i − 0.514216i −0.966383 0.257108i $$-0.917230\pi$$
0.966383 0.257108i $$-0.0827696\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 20.7846 0.684134
$$924$$ 0 0
$$925$$ 32.0000 1.05215
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 12.0000i 0.393707i 0.980433 + 0.196854i $$0.0630724\pi$$
−0.980433 + 0.196854i $$0.936928\pi$$
$$930$$ 0 0
$$931$$ − 27.7128i − 0.908251i
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 93.5307 3.05878
$$936$$ 0 0
$$937$$ 17.0000 0.555366 0.277683 0.960673i $$-0.410434\pi$$
0.277683 + 0.960673i $$0.410434\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ − 3.00000i − 0.0977972i −0.998804 0.0488986i $$-0.984429\pi$$
0.998804 0.0488986i $$-0.0155711\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −46.7654 −1.51967 −0.759835 0.650116i $$-0.774720\pi$$
−0.759835 + 0.650116i $$0.774720\pi$$
$$948$$ 0 0
$$949$$ 2.00000 0.0649227
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 24.0000i 0.777436i 0.921357 + 0.388718i $$0.127082\pi$$
−0.921357 + 0.388718i $$0.872918\pi$$
$$954$$ 0 0
$$955$$ 31.1769i 1.00886i
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 10.3923 0.335585
$$960$$ 0 0
$$961$$ 4.00000 0.129032
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 33.0000i 1.06231i
$$966$$ 0 0
$$967$$ − 50.2295i − 1.61527i −0.589682 0.807635i $$-0.700747\pi$$
0.589682 0.807635i $$-0.299253\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 25.9808 0.833762 0.416881 0.908961i $$-0.363123\pi$$
0.416881 + 0.908961i $$0.363123\pi$$
$$972$$ 0 0
$$973$$ 12.0000 0.384702
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 54.0000i 1.72761i 0.503824 + 0.863807i $$0.331926\pi$$
−0.503824 + 0.863807i $$0.668074\pi$$
$$978$$ 0 0
$$979$$ 31.1769i 0.996419i
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −10.3923 −0.331463 −0.165732 0.986171i $$-0.552999\pi$$
−0.165732 + 0.986171i $$0.552999\pi$$
$$984$$ 0 0
$$985$$ −45.0000 −1.43382
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ − 32.9090i − 1.04539i −0.852520 0.522694i $$-0.824927\pi$$
0.852520 0.522694i $$-0.175073\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 67.5500 2.14148
$$996$$ 0 0
$$997$$ −26.0000 −0.823428 −0.411714 0.911313i $$-0.635070\pi$$
−0.411714 + 0.911313i $$0.635070\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1728.2.c.e.1727.3 4
3.2 odd 2 inner 1728.2.c.e.1727.1 4
4.3 odd 2 inner 1728.2.c.e.1727.4 4
8.3 odd 2 432.2.c.c.431.2 yes 4
8.5 even 2 432.2.c.c.431.1 4
12.11 even 2 inner 1728.2.c.e.1727.2 4
24.5 odd 2 432.2.c.c.431.3 yes 4
24.11 even 2 432.2.c.c.431.4 yes 4
72.5 odd 6 1296.2.s.i.431.2 4
72.11 even 6 1296.2.s.i.863.1 4
72.13 even 6 1296.2.s.i.431.1 4
72.29 odd 6 1296.2.s.g.863.1 4
72.43 odd 6 1296.2.s.i.863.2 4
72.59 even 6 1296.2.s.g.431.2 4
72.61 even 6 1296.2.s.g.863.2 4
72.67 odd 6 1296.2.s.g.431.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
432.2.c.c.431.1 4 8.5 even 2
432.2.c.c.431.2 yes 4 8.3 odd 2
432.2.c.c.431.3 yes 4 24.5 odd 2
432.2.c.c.431.4 yes 4 24.11 even 2
1296.2.s.g.431.1 4 72.67 odd 6
1296.2.s.g.431.2 4 72.59 even 6
1296.2.s.g.863.1 4 72.29 odd 6
1296.2.s.g.863.2 4 72.61 even 6
1296.2.s.i.431.1 4 72.13 even 6
1296.2.s.i.431.2 4 72.5 odd 6
1296.2.s.i.863.1 4 72.11 even 6
1296.2.s.i.863.2 4 72.43 odd 6
1728.2.c.e.1727.1 4 3.2 odd 2 inner
1728.2.c.e.1727.2 4 12.11 even 2 inner
1728.2.c.e.1727.3 4 1.1 even 1 trivial
1728.2.c.e.1727.4 4 4.3 odd 2 inner