# Properties

 Label 1728.2.a.bd.1.1 Level $1728$ Weight $2$ Character 1728.1 Self dual yes Analytic conductor $13.798$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1728.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.7981494693$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 864) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.60555 q^{5} -3.60555 q^{7} +O(q^{10})$$ $$q-3.60555 q^{5} -3.60555 q^{7} +1.00000 q^{11} -4.00000 q^{13} -7.21110 q^{19} +6.00000 q^{23} +8.00000 q^{25} +7.21110 q^{29} +3.60555 q^{31} +13.0000 q^{35} -10.0000 q^{37} -7.21110 q^{41} +7.21110 q^{43} +10.0000 q^{47} +6.00000 q^{49} +3.60555 q^{53} -3.60555 q^{55} -4.00000 q^{59} +14.4222 q^{65} +7.21110 q^{67} -8.00000 q^{71} -3.00000 q^{73} -3.60555 q^{77} +14.4222 q^{79} +9.00000 q^{83} +7.21110 q^{89} +14.4222 q^{91} +26.0000 q^{95} +7.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 2q^{11} - 8q^{13} + 12q^{23} + 16q^{25} + 26q^{35} - 20q^{37} + 20q^{47} + 12q^{49} - 8q^{59} - 16q^{71} - 6q^{73} + 18q^{83} + 52q^{95} + 14q^{97} + O(q^{100})$$

## 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
$$5$$ −3.60555 −1.61245 −0.806226 0.591608i $$-0.798493\pi$$
−0.806226 + 0.591608i $$0.798493\pi$$
$$6$$ 0 0
$$7$$ −3.60555 −1.36277 −0.681385 0.731925i $$-0.738622\pi$$
−0.681385 + 0.731925i $$0.738622\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511 0.150756 0.988571i $$-0.451829\pi$$
0.150756 + 0.988571i $$0.451829\pi$$
$$12$$ 0 0
$$13$$ −4.00000 −1.10940 −0.554700 0.832050i $$-0.687167\pi$$
−0.554700 + 0.832050i $$0.687167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ −7.21110 −1.65434 −0.827170 0.561951i $$-0.810051\pi$$
−0.827170 + 0.561951i $$0.810051\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 8.00000 1.60000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 7.21110 1.33907 0.669534 0.742781i $$-0.266494\pi$$
0.669534 + 0.742781i $$0.266494\pi$$
$$30$$ 0 0
$$31$$ 3.60555 0.647576 0.323788 0.946130i $$-0.395044\pi$$
0.323788 + 0.946130i $$0.395044\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 13.0000 2.19740
$$36$$ 0 0
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −7.21110 −1.12619 −0.563093 0.826394i $$-0.690389\pi$$
−0.563093 + 0.826394i $$0.690389\pi$$
$$42$$ 0 0
$$43$$ 7.21110 1.09968 0.549841 0.835269i $$-0.314688\pi$$
0.549841 + 0.835269i $$0.314688\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 10.0000 1.45865 0.729325 0.684167i $$-0.239834\pi$$
0.729325 + 0.684167i $$0.239834\pi$$
$$48$$ 0 0
$$49$$ 6.00000 0.857143
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 3.60555 0.495261 0.247630 0.968855i $$-0.420348\pi$$
0.247630 + 0.968855i $$0.420348\pi$$
$$54$$ 0 0
$$55$$ −3.60555 −0.486172
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 14.4222 1.78885
$$66$$ 0 0
$$67$$ 7.21110 0.880976 0.440488 0.897758i $$-0.354805\pi$$
0.440488 + 0.897758i $$0.354805\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −8.00000 −0.949425 −0.474713 0.880141i $$-0.657448\pi$$
−0.474713 + 0.880141i $$0.657448\pi$$
$$72$$ 0 0
$$73$$ −3.00000 −0.351123 −0.175562 0.984468i $$-0.556174\pi$$
−0.175562 + 0.984468i $$0.556174\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −3.60555 −0.410891
$$78$$ 0 0
$$79$$ 14.4222 1.62262 0.811312 0.584613i $$-0.198754\pi$$
0.811312 + 0.584613i $$0.198754\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 9.00000 0.987878 0.493939 0.869496i $$-0.335557\pi$$
0.493939 + 0.869496i $$0.335557\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 7.21110 0.764375 0.382188 0.924085i $$-0.375171\pi$$
0.382188 + 0.924085i $$0.375171\pi$$
$$90$$ 0 0
$$91$$ 14.4222 1.51186
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 26.0000 2.66754
$$96$$ 0 0
$$97$$ 7.00000 0.710742 0.355371 0.934725i $$-0.384354\pi$$
0.355371 + 0.934725i $$0.384354\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −10.8167 −1.07630 −0.538149 0.842850i $$-0.680876\pi$$
−0.538149 + 0.842850i $$0.680876\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 17.0000 1.64345 0.821726 0.569883i $$-0.193011\pi$$
0.821726 + 0.569883i $$0.193011\pi$$
$$108$$ 0 0
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −7.21110 −0.678363 −0.339182 0.940721i $$-0.610150\pi$$
−0.339182 + 0.940721i $$0.610150\pi$$
$$114$$ 0 0
$$115$$ −21.6333 −2.01732
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −10.0000 −0.909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −10.8167 −0.967471
$$126$$ 0 0
$$127$$ 3.60555 0.319941 0.159970 0.987122i $$-0.448860\pi$$
0.159970 + 0.987122i $$0.448860\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1.00000 0.0873704 0.0436852 0.999045i $$-0.486090\pi$$
0.0436852 + 0.999045i $$0.486090\pi$$
$$132$$ 0 0
$$133$$ 26.0000 2.25449
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −21.6333 −1.84826 −0.924129 0.382080i $$-0.875208\pi$$
−0.924129 + 0.382080i $$0.875208\pi$$
$$138$$ 0 0
$$139$$ −14.4222 −1.22328 −0.611638 0.791138i $$-0.709489\pi$$
−0.611638 + 0.791138i $$0.709489\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −4.00000 −0.334497
$$144$$ 0 0
$$145$$ −26.0000 −2.15918
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 10.8167 0.886135 0.443067 0.896488i $$-0.353890\pi$$
0.443067 + 0.896488i $$0.353890\pi$$
$$150$$ 0 0
$$151$$ −10.8167 −0.880247 −0.440123 0.897937i $$-0.645065\pi$$
−0.440123 + 0.897937i $$0.645065\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −13.0000 −1.04419
$$156$$ 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
$$161$$ −21.6333 −1.70494
$$162$$ 0 0
$$163$$ 14.4222 1.12963 0.564817 0.825216i $$-0.308947\pi$$
0.564817 + 0.825216i $$0.308947\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ 3.00000 0.230769
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 10.8167 0.822375 0.411187 0.911551i $$-0.365114\pi$$
0.411187 + 0.911551i $$0.365114\pi$$
$$174$$ 0 0
$$175$$ −28.8444 −2.18043
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 15.0000 1.12115 0.560576 0.828103i $$-0.310580\pi$$
0.560576 + 0.828103i $$0.310580\pi$$
$$180$$ 0 0
$$181$$ −8.00000 −0.594635 −0.297318 0.954779i $$-0.596092\pi$$
−0.297318 + 0.954779i $$0.596092\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 36.0555 2.65085
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 12.0000 0.868290 0.434145 0.900843i $$-0.357051\pi$$
0.434145 + 0.900843i $$0.357051\pi$$
$$192$$ 0 0
$$193$$ −15.0000 −1.07972 −0.539862 0.841754i $$-0.681524\pi$$
−0.539862 + 0.841754i $$0.681524\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.0278 1.28442 0.642212 0.766527i $$-0.278017\pi$$
0.642212 + 0.766527i $$0.278017\pi$$
$$198$$ 0 0
$$199$$ 3.60555 0.255591 0.127795 0.991801i $$-0.459210\pi$$
0.127795 + 0.991801i $$0.459210\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −26.0000 −1.82484
$$204$$ 0 0
$$205$$ 26.0000 1.81592
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −7.21110 −0.498802
$$210$$ 0 0
$$211$$ −7.21110 −0.496433 −0.248216 0.968705i $$-0.579844\pi$$
−0.248216 + 0.968705i $$0.579844\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −26.0000 −1.77319
$$216$$ 0 0
$$217$$ −13.0000 −0.882498
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −14.4222 −0.965782 −0.482891 0.875680i $$-0.660413\pi$$
−0.482891 + 0.875680i $$0.660413\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 12.0000 0.796468 0.398234 0.917284i $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 0 0
$$229$$ 10.0000 0.660819 0.330409 0.943838i $$-0.392813\pi$$
0.330409 + 0.943838i $$0.392813\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 21.6333 1.41725 0.708623 0.705588i $$-0.249317\pi$$
0.708623 + 0.705588i $$0.249317\pi$$
$$234$$ 0 0
$$235$$ −36.0555 −2.35200
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −6.00000 −0.388108 −0.194054 0.980991i $$-0.562164\pi$$
−0.194054 + 0.980991i $$0.562164\pi$$
$$240$$ 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
$$245$$ −21.6333 −1.38210
$$246$$ 0 0
$$247$$ 28.8444 1.83533
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −24.0000 −1.51487 −0.757433 0.652913i $$-0.773547\pi$$
−0.757433 + 0.652913i $$0.773547\pi$$
$$252$$ 0 0
$$253$$ 6.00000 0.377217
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 14.4222 0.899632 0.449816 0.893121i $$-0.351489\pi$$
0.449816 + 0.893121i $$0.351489\pi$$
$$258$$ 0 0
$$259$$ 36.0555 2.24038
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −18.0000 −1.10993 −0.554964 0.831875i $$-0.687268\pi$$
−0.554964 + 0.831875i $$0.687268\pi$$
$$264$$ 0 0
$$265$$ −13.0000 −0.798584
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −21.6333 −1.31901 −0.659503 0.751702i $$-0.729233\pi$$
−0.659503 + 0.751702i $$0.729233\pi$$
$$270$$ 0 0
$$271$$ 10.8167 0.657065 0.328532 0.944493i $$-0.393446\pi$$
0.328532 + 0.944493i $$0.393446\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 8.00000 0.482418
$$276$$ 0 0
$$277$$ 8.00000 0.480673 0.240337 0.970690i $$-0.422742\pi$$
0.240337 + 0.970690i $$0.422742\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 7.21110 0.428656 0.214328 0.976762i $$-0.431244\pi$$
0.214328 + 0.976762i $$0.431244\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 26.0000 1.53473
$$288$$ 0 0
$$289$$ −17.0000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 21.6333 1.26383 0.631916 0.775037i $$-0.282269\pi$$
0.631916 + 0.775037i $$0.282269\pi$$
$$294$$ 0 0
$$295$$ 14.4222 0.839693
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −24.0000 −1.38796
$$300$$ 0 0
$$301$$ −26.0000 −1.49862
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −18.0000 −1.02069 −0.510343 0.859971i $$-0.670482\pi$$
−0.510343 + 0.859971i $$0.670482\pi$$
$$312$$ 0 0
$$313$$ 1.00000 0.0565233 0.0282617 0.999601i $$-0.491003\pi$$
0.0282617 + 0.999601i $$0.491003\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −10.8167 −0.607524 −0.303762 0.952748i $$-0.598243\pi$$
−0.303762 + 0.952748i $$0.598243\pi$$
$$318$$ 0 0
$$319$$ 7.21110 0.403744
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −32.0000 −1.77504
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −36.0555 −1.98780
$$330$$ 0 0
$$331$$ 7.21110 0.396358 0.198179 0.980166i $$-0.436497\pi$$
0.198179 + 0.980166i $$0.436497\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −26.0000 −1.42053
$$336$$ 0 0
$$337$$ 26.0000 1.41631 0.708155 0.706057i $$-0.249528\pi$$
0.708155 + 0.706057i $$0.249528\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 3.60555 0.195252
$$342$$ 0 0
$$343$$ 3.60555 0.194681
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1.00000 −0.0536828 −0.0268414 0.999640i $$-0.508545\pi$$
−0.0268414 + 0.999640i $$0.508545\pi$$
$$348$$ 0 0
$$349$$ 2.00000 0.107058 0.0535288 0.998566i $$-0.482953\pi$$
0.0535288 + 0.998566i $$0.482953\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 7.21110 0.383808 0.191904 0.981414i $$-0.438534\pi$$
0.191904 + 0.981414i $$0.438534\pi$$
$$354$$ 0 0
$$355$$ 28.8444 1.53090
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −10.0000 −0.527780 −0.263890 0.964553i $$-0.585006\pi$$
−0.263890 + 0.964553i $$0.585006\pi$$
$$360$$ 0 0
$$361$$ 33.0000 1.73684
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 10.8167 0.566170
$$366$$ 0 0
$$367$$ −10.8167 −0.564625 −0.282312 0.959323i $$-0.591101\pi$$
−0.282312 + 0.959323i $$0.591101\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −13.0000 −0.674926
$$372$$ 0 0
$$373$$ 16.0000 0.828449 0.414224 0.910175i $$-0.364053\pi$$
0.414224 + 0.910175i $$0.364053\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −28.8444 −1.48556
$$378$$ 0 0
$$379$$ 14.4222 0.740819 0.370409 0.928869i $$-0.379217\pi$$
0.370409 + 0.928869i $$0.379217\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −8.00000 −0.408781 −0.204390 0.978889i $$-0.565521\pi$$
−0.204390 + 0.978889i $$0.565521\pi$$
$$384$$ 0 0
$$385$$ 13.0000 0.662541
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 10.8167 0.548426 0.274213 0.961669i $$-0.411583\pi$$
0.274213 + 0.961669i $$0.411583\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −52.0000 −2.61640
$$396$$ 0 0
$$397$$ −4.00000 −0.200754 −0.100377 0.994949i $$-0.532005\pi$$
−0.100377 + 0.994949i $$0.532005\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −14.4222 −0.720211 −0.360105 0.932912i $$-0.617259\pi$$
−0.360105 + 0.932912i $$0.617259\pi$$
$$402$$ 0 0
$$403$$ −14.4222 −0.718421
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −10.0000 −0.495682
$$408$$ 0 0
$$409$$ 7.00000 0.346128 0.173064 0.984911i $$-0.444633\pi$$
0.173064 + 0.984911i $$0.444633\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 14.4222 0.709670
$$414$$ 0 0
$$415$$ −32.4500 −1.59291
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 4.00000 0.195413 0.0977064 0.995215i $$-0.468849\pi$$
0.0977064 + 0.995215i $$0.468849\pi$$
$$420$$ 0 0
$$421$$ −32.0000 −1.55958 −0.779792 0.626038i $$-0.784675\pi$$
−0.779792 + 0.626038i $$0.784675\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 0 0
$$433$$ −35.0000 −1.68199 −0.840996 0.541041i $$-0.818030\pi$$
−0.840996 + 0.541041i $$0.818030\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −43.2666 −2.06972
$$438$$ 0 0
$$439$$ 18.0278 0.860418 0.430209 0.902729i $$-0.358440\pi$$
0.430209 + 0.902729i $$0.358440\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 0 0
$$445$$ −26.0000 −1.23252
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 7.21110 0.340313 0.170156 0.985417i $$-0.445573\pi$$
0.170156 + 0.985417i $$0.445573\pi$$
$$450$$ 0 0
$$451$$ −7.21110 −0.339558
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −52.0000 −2.43780
$$456$$ 0 0
$$457$$ 23.0000 1.07589 0.537947 0.842978i $$-0.319200\pi$$
0.537947 + 0.842978i $$0.319200\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 10.8167 0.503782 0.251891 0.967756i $$-0.418948\pi$$
0.251891 + 0.967756i $$0.418948\pi$$
$$462$$ 0 0
$$463$$ 25.2389 1.17295 0.586475 0.809968i $$-0.300515\pi$$
0.586475 + 0.809968i $$0.300515\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −23.0000 −1.06431 −0.532157 0.846646i $$-0.678618\pi$$
−0.532157 + 0.846646i $$0.678618\pi$$
$$468$$ 0 0
$$469$$ −26.0000 −1.20057
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 7.21110 0.331567
$$474$$ 0 0
$$475$$ −57.6888 −2.64694
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 42.0000 1.91903 0.959514 0.281659i $$-0.0908848\pi$$
0.959514 + 0.281659i $$0.0908848\pi$$
$$480$$ 0 0
$$481$$ 40.0000 1.82384
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −25.2389 −1.14604
$$486$$ 0 0
$$487$$ −14.4222 −0.653532 −0.326766 0.945105i $$-0.605959\pi$$
−0.326766 + 0.945105i $$0.605959\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 7.00000 0.315906 0.157953 0.987447i $$-0.449511\pi$$
0.157953 + 0.987447i $$0.449511\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 28.8444 1.29385
$$498$$ 0 0
$$499$$ −21.6333 −0.968440 −0.484220 0.874946i $$-0.660897\pi$$
−0.484220 + 0.874946i $$0.660897\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 42.0000 1.87269 0.936344 0.351085i $$-0.114187\pi$$
0.936344 + 0.351085i $$0.114187\pi$$
$$504$$ 0 0
$$505$$ 39.0000 1.73548
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −10.8167 −0.479440 −0.239720 0.970842i $$-0.577056\pi$$
−0.239720 + 0.970842i $$0.577056\pi$$
$$510$$ 0 0
$$511$$ 10.8167 0.478501
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 10.0000 0.439799
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 43.2666 1.89554 0.947772 0.318947i $$-0.103329\pi$$
0.947772 + 0.318947i $$0.103329\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 28.8444 1.24939
$$534$$ 0 0
$$535$$ −61.2944 −2.64999
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −4.00000 −0.171973 −0.0859867 0.996296i $$-0.527404\pi$$
−0.0859867 + 0.996296i $$0.527404\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 7.21110 0.308890
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −52.0000 −2.21527
$$552$$ 0 0
$$553$$ −52.0000 −2.21126
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 39.6611 1.68049 0.840247 0.542204i $$-0.182410\pi$$
0.840247 + 0.542204i $$0.182410\pi$$
$$558$$ 0 0
$$559$$ −28.8444 −1.21999
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 39.0000 1.64365 0.821827 0.569737i $$-0.192955\pi$$
0.821827 + 0.569737i $$0.192955\pi$$
$$564$$ 0 0
$$565$$ 26.0000 1.09383
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 14.4222 0.604610 0.302305 0.953211i $$-0.402244\pi$$
0.302305 + 0.953211i $$0.402244\pi$$
$$570$$ 0 0
$$571$$ 14.4222 0.603550 0.301775 0.953379i $$-0.402421\pi$$
0.301775 + 0.953379i $$0.402421\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 48.0000 2.00174
$$576$$ 0 0
$$577$$ 6.00000 0.249783 0.124892 0.992170i $$-0.460142\pi$$
0.124892 + 0.992170i $$0.460142\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −32.4500 −1.34625
$$582$$ 0 0
$$583$$ 3.60555 0.149327
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −3.00000 −0.123823 −0.0619116 0.998082i $$-0.519720\pi$$
−0.0619116 + 0.998082i $$0.519720\pi$$
$$588$$ 0 0
$$589$$ −26.0000 −1.07131
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 21.6333 0.888373 0.444187 0.895934i $$-0.353493\pi$$
0.444187 + 0.895934i $$0.353493\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −14.0000 −0.572024 −0.286012 0.958226i $$-0.592330\pi$$
−0.286012 + 0.958226i $$0.592330\pi$$
$$600$$ 0 0
$$601$$ −17.0000 −0.693444 −0.346722 0.937968i $$-0.612705\pi$$
−0.346722 + 0.937968i $$0.612705\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 36.0555 1.46587
$$606$$ 0 0
$$607$$ 14.4222 0.585379 0.292690 0.956208i $$-0.405450\pi$$
0.292690 + 0.956208i $$0.405450\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −40.0000 −1.61823
$$612$$ 0 0
$$613$$ 18.0000 0.727013 0.363507 0.931592i $$-0.381579\pi$$
0.363507 + 0.931592i $$0.381579\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −21.6333 −0.870924 −0.435462 0.900207i $$-0.643415\pi$$
−0.435462 + 0.900207i $$0.643415\pi$$
$$618$$ 0 0
$$619$$ 43.2666 1.73903 0.869516 0.493905i $$-0.164431\pi$$
0.869516 + 0.493905i $$0.164431\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −26.0000 −1.04167
$$624$$ 0 0
$$625$$ −1.00000 −0.0400000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 25.2389 1.00474 0.502372 0.864652i $$-0.332461\pi$$
0.502372 + 0.864652i $$0.332461\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −13.0000 −0.515889
$$636$$ 0 0
$$637$$ −24.0000 −0.950915
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −36.0555 −1.42411 −0.712054 0.702125i $$-0.752235\pi$$
−0.712054 + 0.702125i $$0.752235\pi$$
$$642$$ 0 0
$$643$$ −43.2666 −1.70627 −0.853134 0.521691i $$-0.825301\pi$$
−0.853134 + 0.521691i $$0.825301\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −4.00000 −0.157014
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −18.0278 −0.705481 −0.352740 0.935721i $$-0.614750\pi$$
−0.352740 + 0.935721i $$0.614750\pi$$
$$654$$ 0 0
$$655$$ −3.60555 −0.140881
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 27.0000 1.05177 0.525885 0.850555i $$-0.323734\pi$$
0.525885 + 0.850555i $$0.323734\pi$$
$$660$$ 0 0
$$661$$ −14.0000 −0.544537 −0.272268 0.962221i $$-0.587774\pi$$
−0.272268 + 0.962221i $$0.587774\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −93.7443 −3.63525
$$666$$ 0 0
$$667$$ 43.2666 1.67529
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −19.0000 −0.732396 −0.366198 0.930537i $$-0.619341\pi$$
−0.366198 + 0.930537i $$0.619341\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −7.21110 −0.277145 −0.138573 0.990352i $$-0.544251\pi$$
−0.138573 + 0.990352i $$0.544251\pi$$
$$678$$ 0 0
$$679$$ −25.2389 −0.968579
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ 78.0000 2.98023
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −14.4222 −0.549442
$$690$$ 0 0
$$691$$ 14.4222 0.548647 0.274323 0.961638i $$-0.411546\pi$$
0.274323 + 0.961638i $$0.411546\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 52.0000 1.97247
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −10.8167 −0.408539 −0.204270 0.978915i $$-0.565482\pi$$
−0.204270 + 0.978915i $$0.565482\pi$$
$$702$$ 0 0
$$703$$ 72.1110 2.71972
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 39.0000 1.46675
$$708$$ 0 0
$$709$$ −4.00000 −0.150223 −0.0751116 0.997175i $$-0.523931\pi$$
−0.0751116 + 0.997175i $$0.523931\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 21.6333 0.810174
$$714$$ 0 0
$$715$$ 14.4222 0.539360
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 4.00000 0.149175 0.0745874 0.997214i $$-0.476236\pi$$
0.0745874 + 0.997214i $$0.476236\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 57.6888 2.14251
$$726$$ 0 0
$$727$$ −32.4500 −1.20350 −0.601751 0.798684i $$-0.705530\pi$$
−0.601751 + 0.798684i $$0.705530\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 46.0000 1.69905 0.849524 0.527549i $$-0.176889\pi$$
0.849524 + 0.527549i $$0.176889\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 7.21110 0.265624
$$738$$ 0 0
$$739$$ −28.8444 −1.06106 −0.530529 0.847667i $$-0.678007\pi$$
−0.530529 + 0.847667i $$0.678007\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 36.0000 1.32071 0.660356 0.750953i $$-0.270405\pi$$
0.660356 + 0.750953i $$0.270405\pi$$
$$744$$ 0 0
$$745$$ −39.0000 −1.42885
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −61.2944 −2.23965
$$750$$ 0 0
$$751$$ 46.8722 1.71039 0.855195 0.518307i $$-0.173437\pi$$
0.855195 + 0.518307i $$0.173437\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 39.0000 1.41936
$$756$$ 0 0
$$757$$ −42.0000 −1.52652 −0.763258 0.646094i $$-0.776401\pi$$
−0.763258 + 0.646094i $$0.776401\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 7.21110 0.261059
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 16.0000 0.577727
$$768$$ 0 0
$$769$$ −11.0000 −0.396670 −0.198335 0.980134i $$-0.563553\pi$$
−0.198335 + 0.980134i $$0.563553\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −21.6333 −0.778096 −0.389048 0.921217i $$-0.627196\pi$$
−0.389048 + 0.921217i $$0.627196\pi$$
$$774$$ 0 0
$$775$$ 28.8444 1.03612
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 52.0000 1.86309
$$780$$ 0 0
$$781$$ −8.00000 −0.286263
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −14.4222 −0.514751
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 26.0000 0.924454
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 10.8167 0.383146 0.191573 0.981478i $$-0.438641\pi$$
0.191573 + 0.981478i $$0.438641\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −3.00000 −0.105868
$$804$$ 0 0
$$805$$ 78.0000 2.74914
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 50.4777 1.77470 0.887351 0.461095i $$-0.152543\pi$$
0.887351 + 0.461095i $$0.152543\pi$$
$$810$$ 0 0
$$811$$ −7.21110 −0.253216 −0.126608 0.991953i $$-0.540409\pi$$
−0.126608 + 0.991953i $$0.540409\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −52.0000 −1.82148
$$816$$ 0 0
$$817$$ −52.0000 −1.81925
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 7.21110 0.251669 0.125835 0.992051i $$-0.459839\pi$$
0.125835 + 0.992051i $$0.459839\pi$$
$$822$$ 0 0
$$823$$ 18.0278 0.628408 0.314204 0.949355i $$-0.398262\pi$$
0.314204 + 0.949355i $$0.398262\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 16.0000 0.556375 0.278187 0.960527i $$-0.410266\pi$$
0.278187 + 0.960527i $$0.410266\pi$$
$$828$$ 0 0
$$829$$ 46.0000 1.59765 0.798823 0.601566i $$-0.205456\pi$$
0.798823 + 0.601566i $$0.205456\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 7.21110 0.249550
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ 23.0000 0.793103
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −10.8167 −0.372104
$$846$$ 0 0
$$847$$ 36.0555 1.23888
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −60.0000 −2.05677
$$852$$ 0 0
$$853$$ 42.0000 1.43805 0.719026 0.694983i $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 28.8444 0.985306 0.492653 0.870226i $$-0.336027\pi$$
0.492653 + 0.870226i $$0.336027\pi$$
$$858$$ 0 0
$$859$$ 28.8444 0.984159 0.492079 0.870550i $$-0.336237\pi$$
0.492079 + 0.870550i $$0.336237\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 28.0000 0.953131 0.476566 0.879139i $$-0.341881\pi$$
0.476566 + 0.879139i $$0.341881\pi$$
$$864$$ 0 0
$$865$$ −39.0000 −1.32604
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 14.4222 0.489240
$$870$$ 0 0
$$871$$ −28.8444 −0.977356
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 39.0000 1.31844
$$876$$ 0 0
$$877$$ −34.0000 −1.14810 −0.574049 0.818821i $$-0.694628\pi$$
−0.574049 + 0.818821i $$0.694628\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 43.2666 1.45769 0.728845 0.684679i $$-0.240058\pi$$
0.728845 + 0.684679i $$0.240058\pi$$
$$882$$ 0 0
$$883$$ 50.4777 1.69871 0.849355 0.527822i $$-0.176991\pi$$
0.849355 + 0.527822i $$0.176991\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 36.0000 1.20876 0.604381 0.796696i $$-0.293421\pi$$
0.604381 + 0.796696i $$0.293421\pi$$
$$888$$ 0 0
$$889$$ −13.0000 −0.436006
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −72.1110 −2.41310
$$894$$ 0 0
$$895$$ −54.0833 −1.80780
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 26.0000 0.867149
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 28.8444 0.958821
$$906$$ 0 0
$$907$$ −7.21110 −0.239441 −0.119720 0.992808i $$-0.538200\pi$$
−0.119720 + 0.992808i $$0.538200\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −30.0000 −0.993944 −0.496972 0.867766i $$-0.665555\pi$$
−0.496972 + 0.867766i $$0.665555\pi$$
$$912$$ 0 0
$$913$$ 9.00000 0.297857
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −3.60555 −0.119066
$$918$$ 0 0
$$919$$ −54.0833 −1.78404 −0.892021 0.451994i $$-0.850713\pi$$
−0.892021 + 0.451994i $$0.850713\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 32.0000 1.05329
$$924$$ 0 0
$$925$$ −80.0000 −2.63038
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −43.2666 −1.41953 −0.709766 0.704438i $$-0.751199\pi$$
−0.709766 + 0.704438i $$0.751199\pi$$
$$930$$ 0 0
$$931$$ −43.2666 −1.41801
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 35.0000 1.14340 0.571700 0.820463i $$-0.306284\pi$$
0.571700 + 0.820463i $$0.306284\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 25.2389 0.822763 0.411382 0.911463i $$-0.365046\pi$$
0.411382 + 0.911463i $$0.365046\pi$$
$$942$$ 0 0
$$943$$ −43.2666 −1.40895
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 27.0000 0.877382 0.438691 0.898638i $$-0.355442\pi$$
0.438691 + 0.898638i $$0.355442\pi$$
$$948$$ 0 0
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −43.2666 −1.40154 −0.700772 0.713386i $$-0.747161\pi$$
−0.700772 + 0.713386i $$0.747161\pi$$
$$954$$ 0 0
$$955$$ −43.2666 −1.40007
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 78.0000 2.51875
$$960$$ 0 0
$$961$$ −18.0000 −0.580645
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 54.0833 1.74100
$$966$$ 0 0
$$967$$ 10.8167 0.347840 0.173920 0.984760i $$-0.444357\pi$$
0.173920 + 0.984760i $$0.444357\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 25.0000 0.802288 0.401144 0.916015i $$-0.368613\pi$$
0.401144 + 0.916015i $$0.368613\pi$$
$$972$$ 0 0
$$973$$ 52.0000 1.66704
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 36.0555 1.15352 0.576759 0.816914i $$-0.304317\pi$$
0.576759 + 0.816914i $$0.304317\pi$$
$$978$$ 0 0
$$979$$ 7.21110 0.230468
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ −6.00000 −0.191370 −0.0956851 0.995412i $$-0.530504\pi$$
−0.0956851 + 0.995412i $$0.530504\pi$$
$$984$$ 0 0
$$985$$ −65.0000 −2.07107
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 43.2666 1.37580
$$990$$ 0 0
$$991$$ −61.2944 −1.94708 −0.973540 0.228517i $$-0.926612\pi$$
−0.973540 + 0.228517i $$0.926612\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −13.0000 −0.412128
$$996$$ 0 0
$$997$$ −36.0000 −1.14013 −0.570066 0.821599i $$-0.693082\pi$$
−0.570066 + 0.821599i $$0.693082\pi$$
$$998$$ 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.a.bd.1.1 2
3.2 odd 2 1728.2.a.bc.1.2 2
4.3 odd 2 1728.2.a.bc.1.1 2
8.3 odd 2 864.2.a.n.1.2 yes 2
8.5 even 2 864.2.a.m.1.2 yes 2
12.11 even 2 inner 1728.2.a.bd.1.2 2
24.5 odd 2 864.2.a.n.1.1 yes 2
24.11 even 2 864.2.a.m.1.1 2
72.5 odd 6 2592.2.i.bb.865.2 4
72.11 even 6 2592.2.i.bc.1729.2 4
72.13 even 6 2592.2.i.bc.865.1 4
72.29 odd 6 2592.2.i.bb.1729.2 4
72.43 odd 6 2592.2.i.bb.1729.1 4
72.59 even 6 2592.2.i.bc.865.2 4
72.61 even 6 2592.2.i.bc.1729.1 4
72.67 odd 6 2592.2.i.bb.865.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
864.2.a.m.1.1 2 24.11 even 2
864.2.a.m.1.2 yes 2 8.5 even 2
864.2.a.n.1.1 yes 2 24.5 odd 2
864.2.a.n.1.2 yes 2 8.3 odd 2
1728.2.a.bc.1.1 2 4.3 odd 2
1728.2.a.bc.1.2 2 3.2 odd 2
1728.2.a.bd.1.1 2 1.1 even 1 trivial
1728.2.a.bd.1.2 2 12.11 even 2 inner
2592.2.i.bb.865.1 4 72.67 odd 6
2592.2.i.bb.865.2 4 72.5 odd 6
2592.2.i.bb.1729.1 4 72.43 odd 6
2592.2.i.bb.1729.2 4 72.29 odd 6
2592.2.i.bc.865.1 4 72.13 even 6
2592.2.i.bc.865.2 4 72.59 even 6
2592.2.i.bc.1729.1 4 72.61 even 6
2592.2.i.bc.1729.2 4 72.11 even 6