Properties

 Label 3024.2.b.q.1567.2 Level $3024$ Weight $2$ Character 3024.1567 Analytic conductor $24.147$ Analytic rank $0$ Dimension $4$ CM discriminant -84 Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3024 = 2^{4} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3024.b (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$24.1467615712$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-6}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 24 x^{2} + 81$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

 Embedding label 1567.2 Root $$-2.01563i$$ of defining polynomial Character $$\chi$$ $$=$$ 3024.1567 Dual form 3024.2.b.q.1567.3

$q$-expansion

 $$f(q)$$ $$=$$ $$q-2.01563i q^{5} -2.64575 q^{7} +O(q^{10})$$ $$q-2.01563i q^{5} -2.64575 q^{7} +2.01563i q^{11} +7.34847i q^{17} -3.35425 q^{19} -9.36409i q^{23} +0.937254 q^{25} +2.29150 q^{31} +5.33284i q^{35} +11.9373 q^{37} +9.36409i q^{41} +7.00000 q^{49} +4.06275 q^{55} +16.7126i q^{71} -5.33284i q^{77} +14.8118 q^{85} -17.4266i q^{89} +6.76091i q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 24q^{19} - 28q^{25} - 12q^{31} + 16q^{37} + 28q^{49} + 48q^{55} - 36q^{85} + O(q^{100})$$

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
$$5$$ − 2.01563i − 0.901415i −0.892672 0.450708i $$-0.851172\pi$$
0.892672 0.450708i $$-0.148828\pi$$
$$6$$ 0 0
$$7$$ −2.64575 −1.00000
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 2.01563i 0.607734i 0.952714 + 0.303867i $$0.0982778\pi$$
−0.952714 + 0.303867i $$0.901722\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 7.34847i 1.78227i 0.453743 + 0.891133i $$0.350089\pi$$
−0.453743 + 0.891133i $$0.649911\pi$$
$$18$$ 0 0
$$19$$ −3.35425 −0.769517 −0.384759 0.923017i $$-0.625715\pi$$
−0.384759 + 0.923017i $$0.625715\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ − 9.36409i − 1.95255i −0.216537 0.976274i $$-0.569476\pi$$
0.216537 0.976274i $$-0.430524\pi$$
$$24$$ 0 0
$$25$$ 0.937254 0.187451
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$30$$ 0 0
$$31$$ 2.29150 0.411566 0.205783 0.978598i $$-0.434026\pi$$
0.205783 + 0.978598i $$0.434026\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 5.33284i 0.901415i
$$36$$ 0 0
$$37$$ 11.9373 1.96247 0.981236 0.192809i $$-0.0617599\pi$$
0.981236 + 0.192809i $$0.0617599\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 9.36409i 1.46243i 0.682149 + 0.731213i $$0.261045\pi$$
−0.682149 + 0.731213i $$0.738955\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 7.00000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$54$$ 0 0
$$55$$ 4.06275 0.547821
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 16.7126i 1.98342i 0.128510 + 0.991708i $$0.458981\pi$$
−0.128510 + 0.991708i $$0.541019\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 5.33284i − 0.607734i
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 14.8118 1.60656
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ − 17.4266i − 1.84722i −0.383339 0.923608i $$-0.625226\pi$$
0.383339 0.923608i $$-0.374774\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 6.76091i 0.693655i
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ − 7.34847i − 0.731200i −0.930772 0.365600i $$-0.880864\pi$$
0.930772 0.365600i $$-0.119136\pi$$
$$102$$ 0 0
$$103$$ 20.2915 1.99938 0.999691 0.0248745i $$-0.00791862\pi$$
0.999691 + 0.0248745i $$0.00791862\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 19.4422i − 1.87955i −0.341793 0.939775i $$-0.611034\pi$$
0.341793 0.939775i $$-0.388966\pi$$
$$108$$ 0 0
$$109$$ 20.8745 1.99942 0.999708 0.0241802i $$-0.00769755\pi$$
0.999708 + 0.0241802i $$0.00769755\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ −18.8745 −1.76006
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ − 19.4422i − 1.78227i
$$120$$ 0 0
$$121$$ 6.93725 0.630659
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ − 11.9673i − 1.07039i
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 8.87451 0.769517
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$138$$ 0 0
$$139$$ 21.1660 1.79528 0.897639 0.440732i $$-0.145281\pi$$
0.897639 + 0.440732i $$0.145281\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 4.61881i − 0.370992i
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 24.7751i 1.95255i
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 13.0000 1.00000
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 21.4578i 1.63141i 0.578468 + 0.815705i $$0.303651\pi$$
−0.578468 + 0.815705i $$0.696349\pi$$
$$174$$ 0 0
$$175$$ −2.47974 −0.187451
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 19.4422i 1.45318i 0.687071 + 0.726590i $$0.258896\pi$$
−0.687071 + 0.726590i $$0.741104\pi$$
$$180$$ 0 0
$$181$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 24.0610i − 1.76900i
$$186$$ 0 0
$$187$$ −14.8118 −1.08314
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 20.0298i 1.44930i 0.689115 + 0.724652i $$0.258000\pi$$
−0.689115 + 0.724652i $$0.742000\pi$$
$$192$$ 0 0
$$193$$ 4.00000 0.287926 0.143963 0.989583i $$-0.454015\pi$$
0.143963 + 0.989583i $$0.454015\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$198$$ 0 0
$$199$$ −21.3542 −1.51376 −0.756881 0.653552i $$-0.773278\pi$$
−0.756881 + 0.653552i $$0.773278\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 18.8745 1.31825
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ − 6.76091i − 0.467662i
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −6.06275 −0.411566
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −1.22876 −0.0822836 −0.0411418 0.999153i $$-0.513100\pi$$
−0.0411418 + 0.999153i $$0.513100\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 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
$$239$$ − 19.4422i − 1.25761i −0.777562 0.628806i $$-0.783544\pi$$
0.777562 0.628806i $$-0.216456\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ − 14.1094i − 0.901415i
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 18.8745 1.18663
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 12.6813i 0.791039i 0.918458 + 0.395519i $$0.129435\pi$$
−0.918458 + 0.395519i $$0.870565\pi$$
$$258$$ 0 0
$$259$$ −31.5830 −1.96247
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 17.4266i 1.07457i 0.843401 + 0.537285i $$0.180550\pi$$
−0.843401 + 0.537285i $$0.819450\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ − 2.72966i − 0.166430i −0.996532 0.0832151i $$-0.973481\pi$$
0.996532 0.0832151i $$-0.0265189\pi$$
$$270$$ 0 0
$$271$$ −10.5830 −0.642872 −0.321436 0.946931i $$-0.604165\pi$$
−0.321436 + 0.946931i $$0.604165\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 1.88915i 0.113920i
$$276$$ 0 0
$$277$$ 23.9373 1.43825 0.719125 0.694881i $$-0.244543\pi$$
0.719125 + 0.694881i $$0.244543\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 26.4575 1.57274 0.786368 0.617758i $$-0.211959\pi$$
0.786368 + 0.617758i $$0.211959\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 24.7751i − 1.46243i
$$288$$ 0 0
$$289$$ −37.0000 −2.17647
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 22.0454i 1.28791i 0.765065 + 0.643953i $$0.222707\pi$$
−0.765065 + 0.643953i $$0.777293\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 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
$$307$$ 32.6458 1.86319 0.931596 0.363496i $$-0.118417\pi$$
0.931596 + 0.363496i $$0.118417\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ − 24.6486i − 1.37148i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −32.7490 −1.78395 −0.891976 0.452082i $$-0.850681\pi$$
−0.891976 + 0.452082i $$0.850681\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 4.61881i 0.250123i
$$342$$ 0 0
$$343$$ −18.5203 −1.00000
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 32.1235i 1.72448i 0.506498 + 0.862241i $$0.330940\pi$$
−0.506498 + 0.862241i $$0.669060\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 36.1548i 1.92433i 0.272476 + 0.962163i $$0.412157\pi$$
−0.272476 + 0.962163i $$0.587843\pi$$
$$354$$ 0 0
$$355$$ 33.6863 1.78788
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 19.4422i 1.02612i 0.858352 + 0.513061i $$0.171488\pi$$
−0.858352 + 0.513061i $$0.828512\pi$$
$$360$$ 0 0
$$361$$ −7.74902 −0.407843
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −37.2288 −1.94333 −0.971663 0.236372i $$-0.924042\pi$$
−0.971663 + 0.236372i $$0.924042\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 32.8745 1.70218 0.851089 0.525022i $$-0.175943\pi$$
0.851089 + 0.525022i $$0.175943\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ −10.7490 −0.547821
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$390$$ 0 0
$$391$$ 68.8118 3.47996
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 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
$$407$$ 24.0610i 1.19266i
$$408$$ 0 0
$$409$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ −12.0627 −0.587902 −0.293951 0.955820i $$-0.594970\pi$$
−0.293951 + 0.955820i $$0.594970\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 6.88738i 0.334087i
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ − 38.7580i − 1.86691i −0.358700 0.933453i $$-0.616780\pi$$
0.358700 0.933453i $$-0.383220\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 31.4095i 1.50252i
$$438$$ 0 0
$$439$$ −5.29150 −0.252550 −0.126275 0.991995i $$-0.540302\pi$$
−0.126275 + 0.991995i $$0.540302\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ − 39.4720i − 1.87537i −0.347484 0.937686i $$-0.612964\pi$$
0.347484 0.937686i $$-0.387036\pi$$
$$444$$ 0 0
$$445$$ −35.1255 −1.66511
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$450$$ 0 0
$$451$$ −18.8745 −0.888766
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −20.7490 −0.970598 −0.485299 0.874348i $$-0.661289\pi$$
−0.485299 + 0.874348i $$0.661289\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ − 29.5203i − 1.37490i −0.726232 0.687450i $$-0.758730\pi$$
0.726232 0.687450i $$-0.241270\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −3.14378 −0.144247
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 34.7267i 1.56719i 0.621269 + 0.783597i $$0.286617\pi$$
−0.621269 + 0.783597i $$0.713383\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 44.2173i − 1.98342i
$$498$$ 0 0
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ −14.8118 −0.659115
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ − 36.7423i − 1.62858i −0.580461 0.814288i $$-0.697128\pi$$
0.580461 0.814288i $$-0.302872\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ − 40.9001i − 1.80227i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 4.61881i − 0.202354i −0.994868 0.101177i $$-0.967739\pi$$
0.994868 0.101177i $$-0.0322608\pi$$
$$522$$ 0 0
$$523$$ 36.1660 1.58143 0.790715 0.612185i $$-0.209709\pi$$
0.790715 + 0.612185i $$0.209709\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 16.8390i 0.733520i
$$528$$ 0 0
$$529$$ −64.6863 −2.81245
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ −39.1882 −1.69426
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 14.1094i 0.607734i
$$540$$ 0 0
$$541$$ −35.6863 −1.53427 −0.767136 0.641484i $$-0.778319\pi$$
−0.767136 + 0.641484i $$0.778319\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ − 42.0752i − 1.80230i
$$546$$ 0 0
$$547$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ − 8.77653i − 0.366007i
$$576$$ 0 0
$$577$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ −7.68627 −0.316707
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 40.9001i − 1.67956i −0.542923 0.839782i $$-0.682683\pi$$
0.542923 0.839782i $$-0.317317\pi$$
$$594$$ 0 0
$$595$$ −39.1882 −1.60656
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ − 6.76091i − 0.276243i −0.990415 0.138122i $$-0.955894\pi$$
0.990415 0.138122i $$-0.0441065\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ − 13.9829i − 0.568486i
$$606$$ 0 0
$$607$$ 26.4575 1.07388 0.536939 0.843621i $$-0.319581\pi$$
0.536939 + 0.843621i $$0.319581\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ 38.8745 1.57013 0.785063 0.619416i $$-0.212630\pi$$
0.785063 + 0.619416i $$0.212630\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$618$$ 0 0
$$619$$ −49.5830 −1.99291 −0.996455 0.0841320i $$-0.973188\pi$$
−0.996455 + 0.0841320i $$0.973188\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 46.1064i 1.84722i
$$624$$ 0 0
$$625$$ −19.4353 −0.777411
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 87.7205i 3.49765i
$$630$$ 0 0
$$631$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 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
$$643$$ 48.5203 1.91345 0.956726 0.290990i $$-0.0939846\pi$$
0.956726 + 0.290990i $$0.0939846\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 43.5033i 1.69465i 0.531078 + 0.847323i $$0.321787\pi$$
−0.531078 + 0.847323i $$0.678213\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ − 17.8877i − 0.693655i
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −44.0000 −1.69608 −0.848038 0.529936i $$-0.822216\pi$$
−0.848038 + 0.529936i $$0.822216\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 51.5658i 1.98183i 0.134478 + 0.990917i $$0.457064\pi$$
−0.134478 + 0.990917i $$0.542936\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 19.3157i 0.739097i 0.929212 + 0.369548i $$0.120488\pi$$
−0.929212 + 0.369548i $$0.879512\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ −42.3320 −1.61039 −0.805193 0.593013i $$-0.797938\pi$$
−0.805193 + 0.593013i $$0.797938\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ − 42.6628i − 1.61829i
$$696$$ 0 0
$$697$$ −68.8118 −2.60643
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$702$$ 0 0
$$703$$ −40.0405 −1.51016
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 19.4422i 0.731200i
$$708$$ 0 0
$$709$$ −9.12549 −0.342715 −0.171358 0.985209i $$-0.554815\pi$$
−0.171358 + 0.985209i $$0.554815\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ − 21.4578i − 0.803603i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −53.6863 −1.99938
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 52.9150 1.96251 0.981255 0.192715i $$-0.0617292\pi$$
0.981255 + 0.192715i $$0.0617292\pi$$
$$728$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ − 54.1689i − 1.98727i −0.112666 0.993633i $$-0.535939\pi$$
0.112666 0.993633i $$-0.464061\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 51.4393i 1.87955i
$$750$$ 0 0
$$751$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 36.7423i 1.33191i 0.745992 + 0.665955i $$0.231976\pi$$
−0.745992 + 0.665955i $$0.768024\pi$$
$$762$$ 0 0
$$763$$ −55.2288 −1.99942
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ − 55.5970i − 1.99969i −0.0177365 0.999843i $$-0.505646\pi$$
0.0177365 0.999843i $$-0.494354\pi$$
$$774$$ 0 0
$$775$$ 2.14772 0.0771484
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ − 31.4095i − 1.12536i
$$780$$ 0 0
$$781$$ −33.6863 −1.20539
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 21.1660 0.754487 0.377243 0.926114i $$-0.376872\pi$$
0.377243 + 0.926114i $$0.376872\pi$$
$$788$$ 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
$$797$$ − 31.4095i − 1.11258i −0.830988 0.556291i $$-0.812224\pi$$
0.830988 0.556291i $$-0.187776\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 49.9373 1.76006
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$810$$ 0 0
$$811$$ 54.1660 1.90203 0.951013 0.309151i $$-0.100045\pi$$
0.951013 + 0.309151i $$0.100045\pi$$
$$812$$ 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
$$821$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7.93603i 0.275963i 0.990435 + 0.137981i $$0.0440614\pi$$
−0.990435 + 0.137981i $$0.955939\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 51.4393i 1.78227i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −29.0000 −1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ − 26.2031i − 0.901415i
$$846$$ 0 0
$$847$$ −18.3542 −0.630659
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ − 111.782i − 3.83182i
$$852$$ 0 0
$$853$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 14.8234i − 0.506358i −0.967419 0.253179i $$-0.918524\pi$$
0.967419 0.253179i $$-0.0814762\pi$$
$$858$$ 0 0
$$859$$ −35.1033 −1.19771 −0.598854 0.800858i $$-0.704377\pi$$
−0.598854 + 0.800858i $$0.704377\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 58.3267i 1.98546i 0.120351 + 0.992731i $$0.461598\pi$$
−0.120351 + 0.992731i $$0.538402\pi$$
$$864$$ 0 0
$$865$$ 43.2510 1.47058
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 31.6624i 1.07039i
$$876$$ 0 0
$$877$$ −22.0000 −0.742887 −0.371444 0.928456i $$-0.621137\pi$$
−0.371444 + 0.928456i $$0.621137\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 11.9673i 0.403188i 0.979469 + 0.201594i $$0.0646121\pi$$
−0.979469 + 0.201594i $$0.935388\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 39.1882 1.30992
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 0 0
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ − 58.3267i − 1.93245i −0.257702 0.966224i $$-0.582965\pi$$
0.257702 0.966224i $$-0.417035\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 11.1882 0.367867
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 36.7423i 1.20548i 0.797939 + 0.602739i $$0.205924\pi$$
−0.797939 + 0.602739i $$0.794076\pi$$
$$930$$ 0 0
$$931$$ −23.4797 −0.769517
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 29.8550i 0.976362i
$$936$$ 0 0
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 27.3783i 0.892505i 0.894907 + 0.446253i $$0.147242\pi$$
−0.894907 + 0.446253i $$0.852758\pi$$
$$942$$ 0 0
$$943$$ 87.6863 2.85546
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 61.5174i 1.99905i 0.0308624 + 0.999524i $$0.490175\pi$$
−0.0308624 + 0.999524i $$0.509825\pi$$
$$948$$ 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
$$955$$ 40.3725 1.30642
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −25.7490 −0.830613
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ − 8.06250i − 0.259541i
$$966$$ 0 0
$$967$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ −56.0000 −1.79528
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$978$$ 0 0
$$979$$ 35.1255 1.12262
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 43.0422i 1.36453i
$$996$$ 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 3024.2.b.q.1567.2 4
3.2 odd 2 inner 3024.2.b.q.1567.3 yes 4
4.3 odd 2 3024.2.b.t.1567.2 yes 4
7.6 odd 2 3024.2.b.t.1567.3 yes 4
12.11 even 2 3024.2.b.t.1567.3 yes 4
21.20 even 2 3024.2.b.t.1567.2 yes 4
28.27 even 2 inner 3024.2.b.q.1567.3 yes 4
84.83 odd 2 CM 3024.2.b.q.1567.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
3024.2.b.q.1567.2 4 1.1 even 1 trivial
3024.2.b.q.1567.2 4 84.83 odd 2 CM
3024.2.b.q.1567.3 yes 4 3.2 odd 2 inner
3024.2.b.q.1567.3 yes 4 28.27 even 2 inner
3024.2.b.t.1567.2 yes 4 4.3 odd 2
3024.2.b.t.1567.2 yes 4 21.20 even 2
3024.2.b.t.1567.3 yes 4 7.6 odd 2
3024.2.b.t.1567.3 yes 4 12.11 even 2