# Properties

 Label 1728.3.b.h.1567.7 Level $1728$ Weight $3$ Character 1728.1567 Analytic conductor $47.085$ Analytic rank $0$ Dimension $8$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1728.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$47.0845896815$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}\cdot 3^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 1567.7 Root $$-0.965926 - 0.258819i$$ of defining polynomial Character $$\chi$$ $$=$$ 1728.1567 Dual form 1728.3.b.h.1567.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+6.63103i q^{5} -13.4853i q^{7} +O(q^{10})$$ $$q+6.63103i q^{5} -13.4853i q^{7} +18.4582 q^{11} -18.9706 q^{25} +29.3939i q^{29} -61.4264i q^{31} +89.4213 q^{35} -132.853 q^{49} -105.901i q^{53} +122.397i q^{55} -117.576 q^{59} +143.794 q^{73} -248.914i q^{77} -58.0000i q^{79} +165.037 q^{83} +128.941 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 16 q^{25} - 384 q^{49} + 200 q^{73} + 760 q^{97}+O(q^{100})$$ 8 * q - 16 * q^25 - 384 * q^49 + 200 * q^73 + 760 * q^97

## 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
$$5$$ 6.63103i 1.32621i 0.748528 + 0.663103i $$0.230761\pi$$
−0.748528 + 0.663103i $$0.769239\pi$$
$$6$$ 0 0
$$7$$ − 13.4853i − 1.92647i −0.268662 0.963234i $$-0.586582\pi$$
0.268662 0.963234i $$-0.413418\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 18.4582 1.67802 0.839010 0.544116i $$-0.183135\pi$$
0.839010 + 0.544116i $$0.183135\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ −18.9706 −0.758823
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 29.3939i 1.01358i 0.862069 + 0.506791i $$0.169168\pi$$
−0.862069 + 0.506791i $$0.830832\pi$$
$$30$$ 0 0
$$31$$ − 61.4264i − 1.98150i −0.135711 0.990748i $$-0.543332\pi$$
0.135711 0.990748i $$-0.456668\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 89.4213 2.55489
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −132.853 −2.71128
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ − 105.901i − 1.99814i −0.0431471 0.999069i $$-0.513738\pi$$
0.0431471 0.999069i $$-0.486262\pi$$
$$54$$ 0 0
$$55$$ 122.397i 2.22540i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −117.576 −1.99281 −0.996403 0.0847458i $$-0.972992\pi$$
−0.996403 + 0.0847458i $$0.972992\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 143.794 1.96978 0.984890 0.173181i $$-0.0554046\pi$$
0.984890 + 0.173181i $$0.0554046\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 248.914i − 3.23265i
$$78$$ 0 0
$$79$$ − 58.0000i − 0.734177i −0.930186 0.367089i $$-0.880355\pi$$
0.930186 0.367089i $$-0.119645\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 165.037 1.98840 0.994200 0.107547i $$-0.0342996\pi$$
0.994200 + 0.107547i $$0.0342996\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 128.941 1.32929 0.664645 0.747159i $$-0.268583\pi$$
0.664645 + 0.747159i $$0.268583\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 130.252i − 1.28962i −0.764341 0.644812i $$-0.776936\pi$$
0.764341 0.644812i $$-0.223064\pi$$
$$102$$ 0 0
$$103$$ − 10.0000i − 0.0970874i −0.998821 0.0485437i $$-0.984542\pi$$
0.998821 0.0485437i $$-0.0154580\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −23.5014 −0.219639 −0.109820 0.993952i $$-0.535027\pi$$
−0.109820 + 0.993952i $$0.535027\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 219.706 1.81575
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 39.9814i 0.319851i
$$126$$ 0 0
$$127$$ 208.338i 1.64046i 0.572036 + 0.820229i $$0.306154\pi$$
−0.572036 + 0.820229i $$0.693846\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 255.698 1.95189 0.975947 0.218007i $$-0.0699555\pi$$
0.975947 + 0.218007i $$0.0699555\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −194.912 −1.34422
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ − 285.440i − 1.91571i −0.287258 0.957853i $$-0.592744\pi$$
0.287258 0.957853i $$-0.407256\pi$$
$$150$$ 0 0
$$151$$ − 106.574i − 0.705785i −0.935664 0.352893i $$-0.885198\pi$$
0.935664 0.352893i $$-0.114802\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 407.320 2.62787
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$168$$ 0 0
$$169$$ 169.000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 211.301i 1.22140i 0.791864 + 0.610698i $$0.209111\pi$$
−0.791864 + 0.610698i $$0.790889\pi$$
$$174$$ 0 0
$$175$$ 255.823i 1.46185i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 336.357 1.87909 0.939545 0.342424i $$-0.111248\pi$$
0.939545 + 0.342424i $$0.111248\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ −365.617 −1.89439 −0.947195 0.320658i $$-0.896096\pi$$
−0.947195 + 0.320658i $$0.896096\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 3.45535i 0.0175399i 0.999962 + 0.00876993i $$0.00279159\pi$$
−0.999962 + 0.00876993i $$0.997208\pi$$
$$198$$ 0 0
$$199$$ − 195.985i − 0.984848i −0.870355 0.492424i $$-0.836111\pi$$
0.870355 0.492424i $$-0.163889\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 396.385 1.95263
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −828.352 −3.81729
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ − 230.000i − 1.03139i −0.856772 0.515695i $$-0.827534\pi$$
0.856772 0.515695i $$-0.172466\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −293.939 −1.29488 −0.647442 0.762115i $$-0.724161\pi$$
−0.647442 + 0.762115i $$0.724161\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 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
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ 382.000 1.58506 0.792531 0.609831i $$-0.208763\pi$$
0.792531 + 0.609831i $$0.208763\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 880.951i − 3.59572i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −176.363 −0.702642 −0.351321 0.936255i $$-0.614267\pi$$
−0.351321 + 0.936255i $$0.614267\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 702.235 2.64994
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 323.333i − 1.20198i −0.799257 0.600990i $$-0.794773\pi$$
0.799257 0.600990i $$-0.205227\pi$$
$$270$$ 0 0
$$271$$ − 495.690i − 1.82912i −0.404455 0.914558i $$-0.632539\pi$$
0.404455 0.914558i $$-0.367461\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −350.163 −1.27332
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −289.000 −1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 440.908i 1.50481i 0.658703 + 0.752403i $$0.271105\pi$$
−0.658703 + 0.752403i $$0.728895\pi$$
$$294$$ 0 0
$$295$$ − 779.647i − 2.64287i
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 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
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ −23.4996 −0.0750785 −0.0375392 0.999295i $$-0.511952\pi$$
−0.0375392 + 0.999295i $$0.511952\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 19.8086i 0.0624877i 0.999512 + 0.0312439i $$0.00994685\pi$$
−0.999512 + 0.0312439i $$0.990053\pi$$
$$318$$ 0 0
$$319$$ 542.558i 1.70081i
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −190.000 −0.563798 −0.281899 0.959444i $$-0.590964\pi$$
−0.281899 + 0.959444i $$0.590964\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ − 1133.82i − 3.32499i
$$342$$ 0 0
$$343$$ 1130.78i 3.29673i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 251.130 0.723717 0.361859 0.932233i $$-0.382142\pi$$
0.361859 + 0.932233i $$0.382142\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ −361.000 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 953.502i 2.61233i
$$366$$ 0 0
$$367$$ − 193.780i − 0.528010i −0.964521 0.264005i $$-0.914956\pi$$
0.964521 0.264005i $$-0.0850435\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1428.11 −3.84935
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 1650.56 4.28716
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 541.766i 1.39272i 0.717695 + 0.696358i $$0.245197\pi$$
−0.717695 + 0.696358i $$0.754803\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 384.600 0.973670
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −698.411 −1.70761 −0.853803 0.520595i $$-0.825710\pi$$
−0.853803 + 0.520595i $$0.825710\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 1585.54i 3.83908i
$$414$$ 0 0
$$415$$ 1094.37i 2.63703i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 411.514 0.982134 0.491067 0.871122i $$-0.336607\pi$$
0.491067 + 0.871122i $$0.336607\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 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
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 140.176 0.323732 0.161866 0.986813i $$-0.448249\pi$$
0.161866 + 0.986813i $$0.448249\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 582.249i 1.32631i 0.748483 + 0.663154i $$0.230782\pi$$
−0.748483 + 0.663154i $$0.769218\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 881.816 1.99056 0.995278 0.0970655i $$-0.0309456\pi$$
0.995278 + 0.0970655i $$0.0309456\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −379.881 −0.831250 −0.415625 0.909536i $$-0.636437\pi$$
−0.415625 + 0.909536i $$0.636437\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 81.7720i 0.177380i 0.996059 + 0.0886899i $$0.0282680\pi$$
−0.996059 + 0.0886899i $$0.971732\pi$$
$$462$$ 0 0
$$463$$ − 897.161i − 1.93771i −0.247625 0.968856i $$-0.579650\pi$$
0.247625 0.968856i $$-0.420350\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −206.132 −0.441395 −0.220698 0.975342i $$-0.570833\pi$$
−0.220698 + 0.975342i $$0.570833\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 855.013i 1.76291i
$$486$$ 0 0
$$487$$ 970.000i 1.99179i 0.0905356 + 0.995893i $$0.471142\pi$$
−0.0905356 + 0.995893i $$0.528858\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −838.142 −1.70701 −0.853505 0.521084i $$-0.825528\pi$$
−0.853505 + 0.521084i $$0.825528\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 863.705 1.71031
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 938.372i 1.84356i 0.387713 + 0.921780i $$0.373265\pi$$
−0.387713 + 0.921780i $$0.626735\pi$$
$$510$$ 0 0
$$511$$ − 1939.10i − 3.79472i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 66.3103 0.128758
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$522$$ 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
$$529$$ 529.000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 155.839i − 0.291287i
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −2452.23 −4.54958
$$540$$ 0 0
$$541$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ −782.146 −1.41437
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 473.114i 0.849396i 0.905335 + 0.424698i $$0.139620\pi$$
−0.905335 + 0.424698i $$0.860380\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 256.563 0.455708 0.227854 0.973695i $$-0.426829\pi$$
0.227854 + 0.973695i $$0.426829\pi$$
$$564$$ 0 0
$$565$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −290.000 −0.502600 −0.251300 0.967909i $$-0.580858\pi$$
−0.251300 + 0.967909i $$0.580858\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ − 2225.57i − 3.83059i
$$582$$ 0 0
$$583$$ − 1954.75i − 3.35291i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −1148.82 −1.95711 −0.978556 0.205982i $$-0.933961\pi$$
−0.978556 + 0.205982i $$0.933961\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 683.853 1.13786 0.568929 0.822387i $$-0.307358\pi$$
0.568929 + 0.822387i $$0.307358\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 1456.87i 2.40806i
$$606$$ 0 0
$$607$$ 730.000i 1.20264i 0.799010 + 0.601318i $$0.205357\pi$$
−0.799010 + 0.601318i $$0.794643\pi$$
$$608$$ 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
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −739.382 −1.18301
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 406.868i 0.644799i 0.946604 + 0.322399i $$0.104489\pi$$
−0.946604 + 0.322399i $$0.895511\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −1381.50 −2.17558
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ −2170.23 −3.34397
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 454.808i 0.696491i 0.937403 + 0.348245i $$0.113222\pi$$
−0.937403 + 0.348245i $$0.886778\pi$$
$$654$$ 0 0
$$655$$ 1695.54i 2.58861i
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −83.5128 −0.126727 −0.0633633 0.997991i $$-0.520183\pi$$
−0.0633633 + 0.997991i $$0.520183\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −1249.00 −1.85587 −0.927933 0.372746i $$-0.878416\pi$$
−0.927933 + 0.372746i $$0.878416\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 146.969i 0.217089i 0.994092 + 0.108545i $$0.0346190\pi$$
−0.994092 + 0.108545i $$0.965381\pi$$
$$678$$ 0 0
$$679$$ − 1738.81i − 2.56084i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 293.939 0.430364 0.215182 0.976574i $$-0.430965\pi$$
0.215182 + 0.976574i $$0.430965\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 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
$$701$$ − 657.253i − 0.937593i −0.883306 0.468797i $$-0.844688\pi$$
0.883306 0.468797i $$-0.155312\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −1756.48 −2.48442
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ −134.853 −0.187036
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ − 557.618i − 0.769129i
$$726$$ 0 0
$$727$$ − 631.662i − 0.868861i −0.900705 0.434430i $$-0.856950\pi$$
0.900705 0.434430i $$-0.143050\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 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
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 1892.76 2.54062
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 316.923i 0.423128i
$$750$$ 0 0
$$751$$ 1167.69i 1.55485i 0.628977 + 0.777424i $$0.283474\pi$$
−0.628977 + 0.777424i $$0.716526\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 706.693 0.936017
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 1534.09 1.99491 0.997456 0.0712917i $$-0.0227121\pi$$
0.997456 + 0.0712917i $$0.0227121\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1028.79i 1.33090i 0.746442 + 0.665450i $$0.231760\pi$$
−0.746442 + 0.665450i $$0.768240\pi$$
$$774$$ 0 0
$$775$$ 1165.29i 1.50360i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 0 0
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1586.04i 1.99001i 0.0998129 + 0.995006i $$0.468176\pi$$
−0.0998129 + 0.995006i $$0.531824\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 2654.18 3.30533
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1499.09i 1.82593i 0.408039 + 0.912965i $$0.366213\pi$$
−0.408039 + 0.912965i $$0.633787\pi$$
$$822$$ 0 0
$$823$$ 1601.13i 1.94548i 0.231898 + 0.972740i $$0.425507\pi$$
−0.231898 + 0.972740i $$0.574493\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 587.878 0.710856 0.355428 0.934704i $$-0.384335\pi$$
0.355428 + 0.934704i $$0.384335\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ −23.0000 −0.0273484
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1120.64i 1.32621i
$$846$$ 0 0
$$847$$ − 2962.79i − 3.49798i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ −1401.15 −1.61982
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ − 1070.58i − 1.23196i
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 539.160 0.616183
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 2809.50 3.16029
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 2230.40i 2.49206i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1805.56 2.00841
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ 3046.29 3.33657
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 3448.16i − 3.76026i
$$918$$ 0 0
$$919$$ − 1153.92i − 1.25563i −0.778362 0.627815i $$-0.783949\pi$$
0.778362 0.627815i $$-0.216051\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 1729.29 1.84556 0.922782 0.385323i $$-0.125910\pi$$
0.922782 + 0.385323i $$0.125910\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 543.718i 0.577809i 0.957358 + 0.288904i $$0.0932910\pi$$
−0.957358 + 0.288904i $$0.906709\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1816.89 −1.91857 −0.959285 0.282439i $$-0.908857\pi$$
−0.959285 + 0.282439i $$0.908857\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −2812.20 −2.92633
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 2424.42i − 2.51235i
$$966$$ 0 0
$$967$$ − 691.956i − 0.715570i −0.933804 0.357785i $$-0.883532\pi$$
0.933804 0.357785i $$-0.116468\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 1563.65 1.61035 0.805176 0.593036i $$-0.202071\pi$$
0.805176 + 0.593036i $$0.202071\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ −22.9126 −0.0232615
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 539.366i 0.544264i 0.962260 + 0.272132i $$0.0877288\pi$$
−0.962260 + 0.272132i $$0.912271\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 1299.58 1.30611
$$996$$ 0 0
$$997$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$998$$ 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 1728.3.b.h.1567.7 yes 8
3.2 odd 2 inner 1728.3.b.h.1567.1 8
4.3 odd 2 inner 1728.3.b.h.1567.8 yes 8
8.3 odd 2 inner 1728.3.b.h.1567.2 yes 8
8.5 even 2 inner 1728.3.b.h.1567.1 8
12.11 even 2 inner 1728.3.b.h.1567.2 yes 8
24.5 odd 2 CM 1728.3.b.h.1567.7 yes 8
24.11 even 2 inner 1728.3.b.h.1567.8 yes 8

By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.b.h.1567.1 8 3.2 odd 2 inner
1728.3.b.h.1567.1 8 8.5 even 2 inner
1728.3.b.h.1567.2 yes 8 8.3 odd 2 inner
1728.3.b.h.1567.2 yes 8 12.11 even 2 inner
1728.3.b.h.1567.7 yes 8 1.1 even 1 trivial
1728.3.b.h.1567.7 yes 8 24.5 odd 2 CM
1728.3.b.h.1567.8 yes 8 4.3 odd 2 inner
1728.3.b.h.1567.8 yes 8 24.11 even 2 inner