# Properties

 Label 384.2.f.a.191.2 Level $384$ Weight $2$ Character 384.191 Analytic conductor $3.066$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 384.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.06625543762$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 191.2 Root $$-0.707107 + 1.22474i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.191 Dual form 384.2.f.a.191.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.73205i q^{3} +2.82843 q^{5} -4.89898i q^{7} -3.00000 q^{9} +O(q^{10})$$ $$q-1.73205i q^{3} +2.82843 q^{5} -4.89898i q^{7} -3.00000 q^{9} +3.46410i q^{11} -4.89898i q^{15} -8.48528 q^{21} +3.00000 q^{25} +5.19615i q^{27} +2.82843 q^{29} -4.89898i q^{31} +6.00000 q^{33} -13.8564i q^{35} -8.48528 q^{45} -17.0000 q^{49} +14.1421 q^{53} +9.79796i q^{55} +10.3923i q^{59} +14.6969i q^{63} +14.0000 q^{73} -5.19615i q^{75} +16.9706 q^{77} +14.6969i q^{79} +9.00000 q^{81} +17.3205i q^{83} -4.89898i q^{87} -8.48528 q^{93} +2.00000 q^{97} -10.3923i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} + 12q^{25} + 24q^{33} - 68q^{49} + 56q^{73} + 36q^{81} + 8q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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
$$3$$ − 1.73205i − 1.00000i
$$4$$ 0 0
$$5$$ 2.82843 1.26491 0.632456 0.774597i $$-0.282047\pi$$
0.632456 + 0.774597i $$0.282047\pi$$
$$6$$ 0 0
$$7$$ − 4.89898i − 1.85164i −0.377964 0.925820i $$-0.623376\pi$$
0.377964 0.925820i $$-0.376624\pi$$
$$8$$ 0 0
$$9$$ −3.00000 −1.00000
$$10$$ 0 0
$$11$$ 3.46410i 1.04447i 0.852803 + 0.522233i $$0.174901\pi$$
−0.852803 + 0.522233i $$0.825099\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ − 4.89898i − 1.26491i
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ −8.48528 −1.85164
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ 3.00000 0.600000
$$26$$ 0 0
$$27$$ 5.19615i 1.00000i
$$28$$ 0 0
$$29$$ 2.82843 0.525226 0.262613 0.964901i $$-0.415416\pi$$
0.262613 + 0.964901i $$0.415416\pi$$
$$30$$ 0 0
$$31$$ − 4.89898i − 0.879883i −0.898027 0.439941i $$-0.854999\pi$$
0.898027 0.439941i $$-0.145001\pi$$
$$32$$ 0 0
$$33$$ 6.00000 1.04447
$$34$$ 0 0
$$35$$ − 13.8564i − 2.34216i
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ −8.48528 −1.26491
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −17.0000 −2.42857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 14.1421 1.94257 0.971286 0.237915i $$-0.0764641\pi$$
0.971286 + 0.237915i $$0.0764641\pi$$
$$54$$ 0 0
$$55$$ 9.79796i 1.32116i
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.3923i 1.35296i 0.736460 + 0.676481i $$0.236496\pi$$
−0.736460 + 0.676481i $$0.763504\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 14.6969i 1.85164i
$$64$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 0 0
$$75$$ − 5.19615i − 0.600000i
$$76$$ 0 0
$$77$$ 16.9706 1.93398
$$78$$ 0 0
$$79$$ 14.6969i 1.65353i 0.562544 + 0.826767i $$0.309823\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 0 0
$$81$$ 9.00000 1.00000
$$82$$ 0 0
$$83$$ 17.3205i 1.90117i 0.310460 + 0.950586i $$0.399517\pi$$
−0.310460 + 0.950586i $$0.600483\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 4.89898i − 0.525226i
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −8.48528 −0.879883
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.00000 0.203069 0.101535 0.994832i $$-0.467625\pi$$
0.101535 + 0.994832i $$0.467625\pi$$
$$98$$ 0 0
$$99$$ − 10.3923i − 1.04447i
$$100$$ 0 0
$$101$$ −19.7990 −1.97007 −0.985037 0.172345i $$-0.944865\pi$$
−0.985037 + 0.172345i $$0.944865\pi$$
$$102$$ 0 0
$$103$$ 14.6969i 1.44813i 0.689730 + 0.724066i $$0.257729\pi$$
−0.689730 + 0.724066i $$0.742271\pi$$
$$104$$ 0 0
$$105$$ −24.0000 −2.34216
$$106$$ 0 0
$$107$$ − 17.3205i − 1.67444i −0.546869 0.837218i $$-0.684180\pi$$
0.546869 0.837218i $$-0.315820\pi$$
$$108$$ 0 0
$$109$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1.00000 −0.0909091
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −5.65685 −0.505964
$$126$$ 0 0
$$127$$ − 4.89898i − 0.434714i −0.976092 0.217357i $$-0.930256\pi$$
0.976092 0.217357i $$-0.0697436\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ − 3.46410i − 0.302660i −0.988483 0.151330i $$-0.951644\pi$$
0.988483 0.151330i $$-0.0483556\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 14.6969i 1.26491i
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 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
$$145$$ 8.00000 0.664364
$$146$$ 0 0
$$147$$ 29.4449i 2.42857i
$$148$$ 0 0
$$149$$ 2.82843 0.231714 0.115857 0.993266i $$-0.463039\pi$$
0.115857 + 0.993266i $$0.463039\pi$$
$$150$$ 0 0
$$151$$ − 24.4949i − 1.99337i −0.0813788 0.996683i $$-0.525932\pi$$
0.0813788 0.996683i $$-0.474068\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ − 13.8564i − 1.11297i
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ − 24.4949i − 1.94257i
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 16.9706 1.32116
$$166$$ 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
$$173$$ −19.7990 −1.50529 −0.752645 0.658427i $$-0.771222\pi$$
−0.752645 + 0.658427i $$0.771222\pi$$
$$174$$ 0 0
$$175$$ − 14.6969i − 1.11098i
$$176$$ 0 0
$$177$$ 18.0000 1.35296
$$178$$ 0 0
$$179$$ 24.2487i 1.81243i 0.422813 + 0.906217i $$0.361043\pi$$
−0.422813 + 0.906217i $$0.638957\pi$$
$$180$$ 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
$$189$$ 25.4558 1.85164
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ −26.0000 −1.87152 −0.935760 0.352636i $$-0.885285\pi$$
−0.935760 + 0.352636i $$0.885285\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 14.1421 1.00759 0.503793 0.863825i $$-0.331938\pi$$
0.503793 + 0.863825i $$0.331938\pi$$
$$198$$ 0 0
$$199$$ − 24.4949i − 1.73640i −0.496217 0.868199i $$-0.665278\pi$$
0.496217 0.868199i $$-0.334722\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ − 13.8564i − 0.972529i
$$204$$ 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
$$217$$ −24.0000 −1.62923
$$218$$ 0 0
$$219$$ − 24.2487i − 1.63858i
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 14.6969i 0.984180i 0.870544 + 0.492090i $$0.163767\pi$$
−0.870544 + 0.492090i $$0.836233\pi$$
$$224$$ 0 0
$$225$$ −9.00000 −0.600000
$$226$$ 0 0
$$227$$ − 10.3923i − 0.689761i −0.938647 0.344881i $$-0.887919\pi$$
0.938647 0.344881i $$-0.112081\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ − 29.3939i − 1.93398i
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 25.4558 1.65353
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ −10.0000 −0.644157 −0.322078 0.946713i $$-0.604381\pi$$
−0.322078 + 0.946713i $$0.604381\pi$$
$$242$$ 0 0
$$243$$ − 15.5885i − 1.00000i
$$244$$ 0 0
$$245$$ −48.0833 −3.07193
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 30.0000 1.90117
$$250$$ 0 0
$$251$$ 31.1769i 1.96787i 0.178529 + 0.983935i $$0.442866\pi$$
−0.178529 + 0.983935i $$0.557134\pi$$
$$252$$ 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
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −8.48528 −0.525226
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 40.0000 2.45718
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −31.1127 −1.89697 −0.948487 0.316815i $$-0.897387\pi$$
−0.948487 + 0.316815i $$0.897387\pi$$
$$270$$ 0 0
$$271$$ − 24.4949i − 1.48796i −0.668202 0.743980i $$-0.732936\pi$$
0.668202 0.743980i $$-0.267064\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 10.3923i 0.626680i
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 14.6969i 0.879883i
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 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
$$289$$ 17.0000 1.00000
$$290$$ 0 0
$$291$$ − 3.46410i − 0.203069i
$$292$$ 0 0
$$293$$ 14.1421 0.826192 0.413096 0.910687i $$-0.364447\pi$$
0.413096 + 0.910687i $$0.364447\pi$$
$$294$$ 0 0
$$295$$ 29.3939i 1.71138i
$$296$$ 0 0
$$297$$ −18.0000 −1.04447
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 34.2929i 1.97007i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 25.4558 1.44813
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −34.0000 −1.92179 −0.960897 0.276907i $$-0.910691\pi$$
−0.960897 + 0.276907i $$0.910691\pi$$
$$314$$ 0 0
$$315$$ 41.5692i 2.34216i
$$316$$ 0 0
$$317$$ −31.1127 −1.74746 −0.873732 0.486408i $$-0.838307\pi$$
−0.873732 + 0.486408i $$0.838307\pi$$
$$318$$ 0 0
$$319$$ 9.79796i 0.548580i
$$320$$ 0 0
$$321$$ −30.0000 −1.67444
$$322$$ 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
$$337$$ 22.0000 1.19842 0.599208 0.800593i $$-0.295482\pi$$
0.599208 + 0.800593i $$0.295482\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 16.9706 0.919007
$$342$$ 0 0
$$343$$ 48.9898i 2.64520i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 24.2487i − 1.30174i −0.759190 0.650870i $$-0.774404\pi$$
0.759190 0.650870i $$-0.225596\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 0 0 1.00000 $$0$$
−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
$$361$$ −19.0000 −1.00000
$$362$$ 0 0
$$363$$ 1.73205i 0.0909091i
$$364$$ 0 0
$$365$$ 39.5980 2.07265
$$366$$ 0 0
$$367$$ − 4.89898i − 0.255725i −0.991792 0.127862i $$-0.959188\pi$$
0.991792 0.127862i $$-0.0408116\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ − 69.2820i − 3.59694i
$$372$$ 0 0
$$373$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$374$$ 0 0
$$375$$ 9.79796i 0.505964i
$$376$$ 0 0
$$377$$ 0 0
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ −8.48528 −0.434714
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 48.0000 2.44631
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −31.1127 −1.57748 −0.788738 0.614729i $$-0.789265\pi$$
−0.788738 + 0.614729i $$0.789265\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ −6.00000 −0.302660
$$394$$ 0 0
$$395$$ 41.5692i 2.09157i
$$396$$ 0 0
$$397$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 25.4558 1.26491
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 50.9117 2.50520
$$414$$ 0 0
$$415$$ 48.9898i 2.40481i
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ − 10.3923i − 0.507697i −0.967244 0.253849i $$-0.918303\pi$$
0.967244 0.253849i $$-0.0816965\pi$$
$$420$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −14.0000 −0.672797 −0.336399 0.941720i $$-0.609209\pi$$
−0.336399 + 0.941720i $$0.609209\pi$$
$$434$$ 0 0
$$435$$ − 13.8564i − 0.664364i
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ − 24.4949i − 1.16908i −0.811366 0.584539i $$-0.801275\pi$$
0.811366 0.584539i $$-0.198725\pi$$
$$440$$ 0 0
$$441$$ 51.0000 2.42857
$$442$$ 0 0
$$443$$ 31.1769i 1.48126i 0.671913 + 0.740630i $$0.265473\pi$$
−0.671913 + 0.740630i $$0.734527\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 4.89898i − 0.231714i
$$448$$ 0 0
$$449$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ −42.4264 −1.99337
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −38.0000 −1.77757 −0.888783 0.458329i $$-0.848448\pi$$
−0.888783 + 0.458329i $$0.848448\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −19.7990 −0.922131 −0.461065 0.887366i $$-0.652533\pi$$
−0.461065 + 0.887366i $$0.652533\pi$$
$$462$$ 0 0
$$463$$ 34.2929i 1.59372i 0.604161 + 0.796862i $$0.293508\pi$$
−0.604161 + 0.796862i $$0.706492\pi$$
$$464$$ 0 0
$$465$$ −24.0000 −1.11297
$$466$$ 0 0
$$467$$ 17.3205i 0.801498i 0.916188 + 0.400749i $$0.131250\pi$$
−0.916188 + 0.400749i $$0.868750\pi$$
$$468$$ 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
$$477$$ −42.4264 −1.94257
$$478$$ 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
$$485$$ 5.65685 0.256865
$$486$$ 0 0
$$487$$ − 44.0908i − 1.99795i −0.0453143 0.998973i $$-0.514429\pi$$
0.0453143 0.998973i $$-0.485571\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 38.1051i 1.71966i 0.510581 + 0.859830i $$0.329431\pi$$
−0.510581 + 0.859830i $$0.670569\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ − 29.3939i − 1.32116i
$$496$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ −56.0000 −2.49197
$$506$$ 0 0
$$507$$ − 22.5167i − 1.00000i
$$508$$ 0 0
$$509$$ 2.82843 0.125368 0.0626839 0.998033i $$-0.480034\pi$$
0.0626839 + 0.998033i $$0.480034\pi$$
$$510$$ 0 0
$$511$$ − 68.5857i − 3.03405i
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 41.5692i 1.83176i
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 34.2929i 1.50529i
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ −25.4558 −1.11098
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ − 31.1769i − 1.35296i
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ − 48.9898i − 2.11801i
$$536$$ 0 0
$$537$$ 42.0000 1.81243
$$538$$ 0 0
$$539$$ − 58.8897i − 2.53656i
$$540$$ 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
$$553$$ 72.0000 3.06175
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 14.1421 0.599222 0.299611 0.954062i $$-0.403143\pi$$
0.299611 + 0.954062i $$0.403143\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ − 38.1051i − 1.60594i −0.596020 0.802970i $$-0.703252\pi$$
0.596020 0.802970i $$-0.296748\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 44.0908i − 1.85164i
$$568$$ 0 0
$$569$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$570$$ 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
$$577$$ 38.0000 1.58196 0.790980 0.611842i $$-0.209571\pi$$
0.790980 + 0.611842i $$0.209571\pi$$
$$578$$ 0 0
$$579$$ 45.0333i 1.87152i
$$580$$ 0 0
$$581$$ 84.8528 3.52029
$$582$$ 0 0
$$583$$ 48.9898i 2.02895i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 17.3205i − 0.714894i −0.933933 0.357447i $$-0.883647\pi$$
0.933933 0.357447i $$-0.116353\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ − 24.4949i − 1.00759i
$$592$$ 0 0
$$593$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −42.4264 −1.73640
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −2.00000 −0.0815817 −0.0407909 0.999168i $$-0.512988\pi$$
−0.0407909 + 0.999168i $$0.512988\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −2.82843 −0.114992
$$606$$ 0 0
$$607$$ − 44.0908i − 1.78959i −0.446476 0.894795i $$-0.647321\pi$$
0.446476 0.894795i $$-0.352679\pi$$
$$608$$ 0 0
$$609$$ −24.0000 −0.972529
$$610$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$618$$ 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
$$625$$ −31.0000 −1.24000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ − 4.89898i − 0.195025i −0.995234 0.0975126i $$-0.968911\pi$$
0.995234 0.0975126i $$-0.0310886\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ − 13.8564i − 0.549875i
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ −36.0000 −1.41312
$$650$$ 0 0
$$651$$ 41.5692i 1.62923i
$$652$$ 0 0
$$653$$ 48.0833 1.88164 0.940822 0.338902i $$-0.110055\pi$$
0.940822 + 0.338902i $$0.110055\pi$$
$$654$$ 0 0
$$655$$ − 9.79796i − 0.382838i
$$656$$ 0 0
$$657$$ −42.0000 −1.63858
$$658$$ 0 0
$$659$$ 24.2487i 0.944596i 0.881439 + 0.472298i $$0.156575\pi$$
−0.881439 + 0.472298i $$0.843425\pi$$
$$660$$ 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
$$669$$ 25.4558 0.984180
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 34.0000 1.31060 0.655302 0.755367i $$-0.272541\pi$$
0.655302 + 0.755367i $$0.272541\pi$$
$$674$$ 0 0
$$675$$ 15.5885i 0.600000i
$$676$$ 0 0
$$677$$ 2.82843 0.108705 0.0543526 0.998522i $$-0.482690\pi$$
0.0543526 + 0.998522i $$0.482690\pi$$
$$678$$ 0 0
$$679$$ − 9.79796i − 0.376011i
$$680$$ 0 0
$$681$$ −18.0000 −0.689761
$$682$$ 0 0
$$683$$ − 51.9615i − 1.98825i −0.108227 0.994126i $$-0.534517\pi$$
0.108227 0.994126i $$-0.465483\pi$$
$$684$$ 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
$$693$$ −50.9117 −1.93398
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 36.7696 1.38877 0.694383 0.719605i $$-0.255677\pi$$
0.694383 + 0.719605i $$0.255677\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 96.9948i 3.64787i
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 0 0
$$711$$ − 44.0908i − 1.65353i
$$712$$ 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
$$721$$ 72.0000 2.68142
$$722$$ 0 0
$$723$$ 17.3205i 0.644157i
$$724$$ 0 0
$$725$$ 8.48528 0.315135
$$726$$ 0 0
$$727$$ 53.8888i 1.99862i 0.0370879 + 0.999312i $$0.488192\pi$$
−0.0370879 + 0.999312i $$0.511808\pi$$
$$728$$ 0 0
$$729$$ −27.0000 −1.00000
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 83.2827i 3.07193i
$$736$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ 8.00000 0.293097
$$746$$ 0 0
$$747$$ − 51.9615i − 1.90117i
$$748$$ 0 0
$$749$$ −84.8528 −3.10045
$$750$$ 0 0
$$751$$ 53.8888i 1.96643i 0.182453 + 0.983215i $$0.441596\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ 0 0
$$753$$ 54.0000 1.96787
$$754$$ 0 0
$$755$$ − 69.2820i − 2.52143i
$$756$$ 0 0
$$757$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −26.0000 −0.937584 −0.468792 0.883309i $$-0.655311\pi$$
−0.468792 + 0.883309i $$0.655311\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −19.7990 −0.712120 −0.356060 0.934463i $$-0.615880\pi$$
−0.356060 + 0.934463i $$0.615880\pi$$
$$774$$ 0 0
$$775$$ − 14.6969i − 0.527930i
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 0 0
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 14.6969i 0.525226i
$$784$$ 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
$$795$$ − 69.2820i − 2.45718i
$$796$$ 0 0
$$797$$ −53.7401 −1.90357 −0.951786 0.306762i $$-0.900754\pi$$
−0.951786 + 0.306762i $$0.900754\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 48.4974i 1.71144i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 53.8888i 1.89697i
$$808$$ 0 0
$$809$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 0 0
$$813$$ −42.4264 −1.48796
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 48.0833 1.67812 0.839059 0.544041i $$-0.183106\pi$$
0.839059 + 0.544041i $$0.183106\pi$$
$$822$$ 0 0
$$823$$ 34.2929i 1.19537i 0.801730 + 0.597687i $$0.203913\pi$$
−0.801730 + 0.597687i $$0.796087\pi$$
$$824$$ 0 0
$$825$$ 18.0000 0.626680
$$826$$ 0 0
$$827$$ 10.3923i 0.361376i 0.983540 + 0.180688i $$0.0578324\pi$$
−0.983540 + 0.180688i $$0.942168\pi$$
$$828$$ 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
$$837$$ 25.4558 0.879883
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ −21.0000 −0.724138
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 36.7696 1.26491
$$846$$ 0 0
$$847$$ 4.89898i 0.168331i
$$848$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ −56.0000 −1.90406
$$866$$ 0 0
$$867$$ − 29.4449i − 1.00000i
$$868$$ 0 0
$$869$$ −50.9117 −1.72706
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ −6.00000 −0.203069
$$874$$ 0 0
$$875$$ 27.7128i 0.936864i
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ − 24.4949i − 0.826192i
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 50.9117 1.71138
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 0 0
$$889$$ −24.0000 −0.804934
$$890$$ 0 0
$$891$$ 31.1769i 1.04447i
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 68.5857i 2.29257i
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ − 13.8564i − 0.462137i
$$900$$ 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
$$909$$ 59.3970 1.97007
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 0 0
$$913$$ −60.0000 −1.98571
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −16.9706 −0.560417
$$918$$ 0 0
$$919$$ 34.2929i 1.13122i 0.824674 + 0.565608i $$0.191359\pi$$
−0.824674 + 0.565608i $$0.808641\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 44.0908i − 1.44813i
$$928$$ 0 0
$$929$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 58.0000 1.89478 0.947389 0.320085i $$-0.103712\pi$$
0.947389 + 0.320085i $$0.103712\pi$$
$$938$$ 0 0
$$939$$ 58.8897i 1.92179i
$$940$$ 0 0
$$941$$ 48.0833 1.56747 0.783735 0.621096i $$-0.213312\pi$$
0.783735 + 0.621096i $$0.213312\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 72.0000 2.34216
$$946$$ 0 0
$$947$$ 24.2487i 0.787977i 0.919115 + 0.393989i $$0.128905\pi$$
−0.919115 + 0.393989i $$0.871095\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 53.8888i 1.74746i
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 16.9706 0.548580
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 7.00000 0.225806
$$962$$ 0 0
$$963$$ 51.9615i 1.67444i
$$964$$ 0 0
$$965$$ −73.5391 −2.36731
$$966$$ 0 0
$$967$$ − 4.89898i − 0.157541i −0.996893 0.0787703i $$-0.974901\pi$$
0.996893 0.0787703i $$-0.0250994\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 3.46410i 0.111168i 0.998454 + 0.0555842i $$0.0177021\pi$$
−0.998454 + 0.0555842i $$0.982298\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 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
$$985$$ 40.0000 1.27451
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ − 24.4949i − 0.778106i −0.921215 0.389053i $$-0.872802\pi$$
0.921215 0.389053i $$-0.127198\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ − 69.2820i − 2.19639i
$$996$$ 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 384.2.f.a.191.2 yes 4
3.2 odd 2 inner 384.2.f.a.191.3 yes 4
4.3 odd 2 inner 384.2.f.a.191.4 yes 4
8.3 odd 2 inner 384.2.f.a.191.1 4
8.5 even 2 inner 384.2.f.a.191.3 yes 4
12.11 even 2 inner 384.2.f.a.191.1 4
16.3 odd 4 768.2.c.j.767.1 4
16.5 even 4 768.2.c.j.767.2 4
16.11 odd 4 768.2.c.j.767.4 4
16.13 even 4 768.2.c.j.767.3 4
24.5 odd 2 CM 384.2.f.a.191.2 yes 4
24.11 even 2 inner 384.2.f.a.191.4 yes 4
48.5 odd 4 768.2.c.j.767.3 4
48.11 even 4 768.2.c.j.767.1 4
48.29 odd 4 768.2.c.j.767.2 4
48.35 even 4 768.2.c.j.767.4 4

By twisted newform
Twist Min Dim Char Parity Ord Type
384.2.f.a.191.1 4 8.3 odd 2 inner
384.2.f.a.191.1 4 12.11 even 2 inner
384.2.f.a.191.2 yes 4 1.1 even 1 trivial
384.2.f.a.191.2 yes 4 24.5 odd 2 CM
384.2.f.a.191.3 yes 4 3.2 odd 2 inner
384.2.f.a.191.3 yes 4 8.5 even 2 inner
384.2.f.a.191.4 yes 4 4.3 odd 2 inner
384.2.f.a.191.4 yes 4 24.11 even 2 inner
768.2.c.j.767.1 4 16.3 odd 4
768.2.c.j.767.1 4 48.11 even 4
768.2.c.j.767.2 4 16.5 even 4
768.2.c.j.767.2 4 48.29 odd 4
768.2.c.j.767.3 4 16.13 even 4
768.2.c.j.767.3 4 48.5 odd 4
768.2.c.j.767.4 4 16.11 odd 4
768.2.c.j.767.4 4 48.35 even 4