# Properties

 Label 432.2.c.b.431.1 Level $432$ Weight $2$ Character 432.431 Analytic conductor $3.450$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$432 = 2^{4} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 432.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.44953736732$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 431.1 Root $$0.500000 + 0.866025i$$ of defining polynomial Character $$\chi$$ $$=$$ 432.431 Dual form 432.2.c.b.431.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205i q^{7} +O(q^{10})$$ $$q-1.73205i q^{7} +7.00000 q^{13} -8.66025i q^{19} +5.00000 q^{25} +10.3923i q^{31} -1.00000 q^{37} -10.3923i q^{43} +4.00000 q^{49} -13.0000 q^{61} +12.1244i q^{67} -17.0000 q^{73} +12.1244i q^{79} -12.1244i q^{91} -5.00000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 14 q^{13} + 10 q^{25} - 2 q^{37} + 8 q^{49} - 26 q^{61} - 34 q^{73} - 10 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$$.

 $$n$$ $$271$$ $$325$$ $$353$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 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
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ 7.00000 1.94145 0.970725 0.240192i $$-0.0772105\pi$$
0.970725 + 0.240192i $$0.0772105\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ − 8.66025i − 1.98680i −0.114708 0.993399i $$-0.536593\pi$$
0.114708 0.993399i $$-0.463407\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 5.00000 1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 10.3923i 1.86651i 0.359211 + 0.933257i $$0.383046\pi$$
−0.359211 + 0.933257i $$0.616954\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −1.00000 −0.164399 −0.0821995 0.996616i $$-0.526194\pi$$
−0.0821995 + 0.996616i $$0.526194\pi$$
$$38$$ 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 $$-0.708828\pi$$
0.609994 0.792406i $$-0.291172\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ 4.00000 0.571429
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −13.0000 −1.66448 −0.832240 0.554416i $$-0.812942\pi$$
−0.832240 + 0.554416i $$0.812942\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.1244i 1.48123i 0.671932 + 0.740613i $$0.265465\pi$$
−0.671932 + 0.740613i $$0.734535\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −17.0000 −1.98970 −0.994850 0.101361i $$-0.967680\pi$$
−0.994850 + 0.101361i $$0.967680\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 12.1244i 1.36410i 0.731307 + 0.682048i $$0.238911\pi$$
−0.731307 + 0.682048i $$0.761089\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ − 12.1244i − 1.27098i
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 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
$$101$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$102$$ 0 0
$$103$$ 19.0526i 1.87730i 0.344865 + 0.938652i $$0.387925\pi$$
−0.344865 + 0.938652i $$0.612075\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 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
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −11.0000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 10.3923i 0.922168i 0.887357 + 0.461084i $$0.152539\pi$$
−0.887357 + 0.461084i $$0.847461\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ −15.0000 −1.30066
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ − 22.5167i − 1.90984i −0.296866 0.954919i $$-0.595942\pi$$
0.296866 0.954919i $$-0.404058\pi$$
$$140$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$150$$ 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
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 14.0000 1.11732 0.558661 0.829396i $$-0.311315\pi$$
0.558661 + 0.829396i $$0.311315\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 19.0526i 1.49231i 0.665771 + 0.746156i $$0.268103\pi$$
−0.665771 + 0.746156i $$0.731897\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ 36.0000 2.76923
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ − 8.66025i − 0.654654i
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 19.0000 1.41226 0.706129 0.708083i $$-0.250440\pi$$
0.706129 + 0.708083i $$0.250440\pi$$
$$182$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −25.0000 −1.79954 −0.899770 0.436365i $$-0.856266\pi$$
−0.899770 + 0.436365i $$0.856266\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$198$$ 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
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ − 1.73205i − 0.119239i −0.998221 0.0596196i $$-0.981011\pi$$
0.998221 0.0596196i $$-0.0189888\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 18.0000 1.22192
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 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
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 22.0000 1.45380 0.726900 0.686743i $$-0.240960\pi$$
0.726900 + 0.686743i $$0.240960\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 31.0000 1.99689 0.998443 0.0557856i $$-0.0177663\pi$$
0.998443 + 0.0557856i $$0.0177663\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 60.6218i − 3.85727i
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$258$$ 0 0
$$259$$ 1.73205i 0.107624i
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$270$$ 0 0
$$271$$ 32.9090i 1.99908i 0.0303728 + 0.999539i $$0.490331\pi$$
−0.0303728 + 0.999539i $$0.509669\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −26.0000 −1.56219 −0.781094 0.624413i $$-0.785338\pi$$
−0.781094 + 0.624413i $$0.785338\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ − 10.3923i − 0.617758i −0.951101 0.308879i $$-0.900046\pi$$
0.951101 0.308879i $$-0.0999539\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 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
$$305$$ 0 0
$$306$$ 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
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −13.0000 −0.734803 −0.367402 0.930062i $$-0.619753\pi$$
−0.367402 + 0.930062i $$0.619753\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 35.0000 1.94145
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 19.0526i 1.04722i 0.851957 + 0.523612i $$0.175416\pi$$
−0.851957 + 0.523612i $$0.824584\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −29.0000 −1.57973 −0.789865 0.613280i $$-0.789850\pi$$
−0.789865 + 0.613280i $$0.789850\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ − 19.0526i − 1.02874i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 23.0000 1.23116 0.615581 0.788074i $$-0.288921\pi$$
0.615581 + 0.788074i $$0.288921\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −56.0000 −2.94737
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 22.5167i − 1.17536i −0.809093 0.587680i $$-0.800041\pi$$
0.809093 0.587680i $$-0.199959\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −25.0000 −1.29445 −0.647225 0.762299i $$-0.724071\pi$$
−0.647225 + 0.762299i $$0.724071\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 12.1244i 0.622786i 0.950281 + 0.311393i $$0.100796\pi$$
−0.950281 + 0.311393i $$0.899204\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −34.0000 −1.70641 −0.853206 0.521575i $$-0.825345\pi$$
−0.853206 + 0.521575i $$0.825345\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 72.7461i 3.62374i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 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
$$413$$ 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
$$421$$ −41.0000 −1.99822 −0.999109 0.0422075i $$-0.986561\pi$$
−0.999109 + 0.0422075i $$0.986561\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 22.5167i 1.08966i
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ 2.00000 0.0961139 0.0480569 0.998845i $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ − 31.1769i − 1.48799i −0.668184 0.743996i $$-0.732928\pi$$
0.668184 0.743996i $$-0.267072\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$462$$ 0 0
$$463$$ − 1.73205i − 0.0804952i −0.999190 0.0402476i $$-0.987185\pi$$
0.999190 0.0402476i $$-0.0128147\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 21.0000 0.969690
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ − 43.3013i − 1.98680i
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −7.00000 −0.319173
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 39.8372i 1.80519i 0.430486 + 0.902597i $$0.358342\pi$$
−0.430486 + 0.902597i $$0.641658\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 31.1769i 1.39567i 0.716258 + 0.697835i $$0.245853\pi$$
−0.716258 + 0.697835i $$0.754147\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ 29.4449i 1.30256i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ − 29.4449i − 1.28753i −0.765222 0.643767i $$-0.777371\pi$$
0.765222 0.643767i $$-0.222629\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −17.0000 −0.730887 −0.365444 0.930834i $$-0.619083\pi$$
−0.365444 + 0.930834i $$0.619083\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 22.5167i − 0.962743i −0.876517 0.481371i $$-0.840139\pi$$
0.876517 0.481371i $$-0.159861\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 21.0000 0.893011
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$558$$ 0 0
$$559$$ − 72.7461i − 3.07683i
$$560$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 0 0
$$571$$ − 8.66025i − 0.362420i −0.983444 0.181210i $$-0.941999\pi$$
0.983444 0.181210i $$-0.0580014\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 35.0000 1.45707 0.728535 0.685009i $$-0.240202\pi$$
0.728535 + 0.685009i $$0.240202\pi$$
$$578$$ 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
$$589$$ 90.0000 3.70839
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 39.8372i 1.61694i 0.588537 + 0.808470i $$0.299704\pi$$
−0.588537 + 0.808470i $$0.700296\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −37.0000 −1.49442 −0.747208 0.664590i $$-0.768606\pi$$
−0.747208 + 0.664590i $$0.768606\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 0 0
$$619$$ − 8.66025i − 0.348085i −0.984738 0.174042i $$-0.944317\pi$$
0.984738 0.174042i $$-0.0556830\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 25.0000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ − 50.2295i − 1.99960i −0.0199047 0.999802i $$-0.506336\pi$$
0.0199047 0.999802i $$-0.493664\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 28.0000 1.10940
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 31.1769i 1.22950i 0.788723 + 0.614749i $$0.210743\pi$$
−0.788723 + 0.614749i $$0.789257\pi$$
$$644$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ 11.0000 0.427850 0.213925 0.976850i $$-0.431375\pi$$
0.213925 + 0.976850i $$0.431375\pi$$
$$662$$ 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
$$673$$ −13.0000 −0.501113 −0.250557 0.968102i $$-0.580614\pi$$
−0.250557 + 0.968102i $$0.580614\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 8.66025i 0.332350i
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ − 51.9615i − 1.97671i −0.152167 0.988355i $$-0.548625\pi$$
0.152167 0.988355i $$-0.451375\pi$$
$$692$$ 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
$$701$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$702$$ 0 0
$$703$$ 8.66025i 0.326628i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 31.0000 1.16423 0.582115 0.813107i $$-0.302225\pi$$
0.582115 + 0.813107i $$0.302225\pi$$
$$710$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 33.0000 1.22898
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 31.1769i − 1.15629i −0.815935 0.578144i $$-0.803777\pi$$
0.815935 0.578144i $$-0.196223\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −50.0000 −1.84679 −0.923396 0.383849i $$-0.874598\pi$$
−0.923396 + 0.383849i $$0.874598\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ − 51.9615i − 1.91144i −0.294285 0.955718i $$-0.595081\pi$$
0.294285 0.955718i $$-0.404919\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 53.6936i 1.95931i 0.200698 + 0.979653i $$0.435679\pi$$
−0.200698 + 0.979653i $$0.564321\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 55.0000 1.99901 0.999505 0.0314762i $$-0.0100208\pi$$
0.999505 + 0.0314762i $$0.0100208\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 3.46410i 0.125409i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 47.0000 1.69486 0.847432 0.530904i $$-0.178148\pi$$
0.847432 + 0.530904i $$0.178148\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 51.9615i 1.86651i
$$776$$ 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
$$787$$ − 50.2295i − 1.79049i −0.445577 0.895244i $$-0.647001\pi$$
0.445577 0.895244i $$-0.352999\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −91.0000 −3.23150
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ − 10.3923i − 0.364923i −0.983213 0.182462i $$-0.941593\pi$$
0.983213 0.182462i $$-0.0584065\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −90.0000 −3.14870
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$822$$ 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
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ −53.0000 −1.84077 −0.920383 0.391018i $$-0.872123\pi$$
−0.920383 + 0.391018i $$0.872123\pi$$
$$830$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 29.0000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 19.0526i 0.654654i
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 23.0000 0.787505 0.393753 0.919216i $$-0.371177\pi$$
0.393753 + 0.919216i $$0.371177\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 39.8372i 1.35923i 0.733571 + 0.679613i $$0.237852\pi$$
−0.733571 + 0.679613i $$0.762148\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 84.8705i 2.87573i
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −25.0000 −0.844190 −0.422095 0.906552i $$-0.638705\pi$$
−0.422095 + 0.906552i $$0.638705\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ − 22.5167i − 0.757746i −0.925449 0.378873i $$-0.876312\pi$$
0.925449 0.378873i $$-0.123688\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ 18.0000 0.603701
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 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
$$907$$ 12.1244i 0.402583i 0.979531 + 0.201291i $$0.0645138\pi$$
−0.979531 + 0.201291i $$0.935486\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ − 31.1769i − 1.02843i −0.857661 0.514216i $$-0.828083\pi$$
0.857661 0.514216i $$-0.171917\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −5.00000 −0.164399
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ − 34.6410i − 1.13531i
$$932$$ 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
$$941$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ −119.000 −3.86290
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −77.0000 −2.48387
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 12.1244i 0.389893i 0.980814 + 0.194946i $$0.0624533\pi$$
−0.980814 + 0.194946i $$0.937547\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ −39.0000 −1.25028
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 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
$$991$$ 60.6218i 1.92571i 0.270011 + 0.962857i $$0.412973\pi$$
−0.270011 + 0.962857i $$0.587027\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −10.0000 −0.316703 −0.158352 0.987383i $$-0.550618\pi$$
−0.158352 + 0.987383i $$0.550618\pi$$
$$998$$ 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 432.2.c.b.431.1 2
3.2 odd 2 CM 432.2.c.b.431.1 2
4.3 odd 2 inner 432.2.c.b.431.2 yes 2
8.3 odd 2 1728.2.c.a.1727.2 2
8.5 even 2 1728.2.c.a.1727.1 2
9.2 odd 6 1296.2.s.c.863.1 2
9.4 even 3 1296.2.s.d.431.1 2
9.5 odd 6 1296.2.s.d.431.1 2
9.7 even 3 1296.2.s.c.863.1 2
12.11 even 2 inner 432.2.c.b.431.2 yes 2
24.5 odd 2 1728.2.c.a.1727.1 2
24.11 even 2 1728.2.c.a.1727.2 2
36.7 odd 6 1296.2.s.d.863.1 2
36.11 even 6 1296.2.s.d.863.1 2
36.23 even 6 1296.2.s.c.431.1 2
36.31 odd 6 1296.2.s.c.431.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
432.2.c.b.431.1 2 1.1 even 1 trivial
432.2.c.b.431.1 2 3.2 odd 2 CM
432.2.c.b.431.2 yes 2 4.3 odd 2 inner
432.2.c.b.431.2 yes 2 12.11 even 2 inner
1296.2.s.c.431.1 2 36.23 even 6
1296.2.s.c.431.1 2 36.31 odd 6
1296.2.s.c.863.1 2 9.2 odd 6
1296.2.s.c.863.1 2 9.7 even 3
1296.2.s.d.431.1 2 9.4 even 3
1296.2.s.d.431.1 2 9.5 odd 6
1296.2.s.d.863.1 2 36.7 odd 6
1296.2.s.d.863.1 2 36.11 even 6
1728.2.c.a.1727.1 2 8.5 even 2
1728.2.c.a.1727.1 2 24.5 odd 2
1728.2.c.a.1727.2 2 8.3 odd 2
1728.2.c.a.1727.2 2 24.11 even 2