# Properties

 Label 684.3.h.a.37.2 Level $684$ Weight $3$ Character 684.37 Self dual yes Analytic conductor $18.638$ Analytic rank $0$ Dimension $2$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{57})$$ Defining polynomial: $$x^{2} - x - 14$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 37.2 Root $$-3.27492$$ of defining polynomial Character $$\chi$$ $$=$$ 684.37

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.725083 q^{5} +13.8248 q^{7} +O(q^{10})$$ $$q-0.725083 q^{5} +13.8248 q^{7} +20.3746 q^{11} -18.9244 q^{17} -19.0000 q^{19} +30.0000 q^{23} -24.4743 q^{25} -10.0241 q^{35} +53.8248 q^{43} +86.5739 q^{47} +142.124 q^{49} -14.7733 q^{55} +5.12376 q^{61} -112.072 q^{73} +281.674 q^{77} -90.0000 q^{83} +13.7218 q^{85} +13.7766 q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{5} + 5q^{7} + O(q^{10})$$ $$2q - 9q^{5} + 5q^{7} + 3q^{11} + 15q^{17} - 38q^{19} + 60q^{23} + 19q^{25} + 63q^{35} + 85q^{43} + 75q^{47} + 171q^{49} + 129q^{55} - 103q^{61} + 25q^{73} + 435q^{77} - 180q^{83} - 267q^{85} + 171q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$343$$ $$533$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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
$$5$$ −0.725083 −0.145017 −0.0725083 0.997368i $$-0.523100\pi$$
−0.0725083 + 0.997368i $$0.523100\pi$$
$$6$$ 0 0
$$7$$ 13.8248 1.97496 0.987482 0.157730i $$-0.0504176\pi$$
0.987482 + 0.157730i $$0.0504176\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 20.3746 1.85224 0.926118 0.377235i $$-0.123125\pi$$
0.926118 + 0.377235i $$0.123125\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −18.9244 −1.11320 −0.556601 0.830780i $$-0.687895\pi$$
−0.556601 + 0.830780i $$0.687895\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −1.00000
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 30.0000 1.30435 0.652174 0.758069i $$-0.273857\pi$$
0.652174 + 0.758069i $$0.273857\pi$$
$$24$$ 0 0
$$25$$ −24.4743 −0.978970
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −10.0241 −0.286403
$$36$$ 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
$$43$$ 53.8248 1.25174 0.625869 0.779928i $$-0.284744\pi$$
0.625869 + 0.779928i $$0.284744\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 86.5739 1.84200 0.920999 0.389564i $$-0.127374\pi$$
0.920999 + 0.389564i $$0.127374\pi$$
$$48$$ 0 0
$$49$$ 142.124 2.90048
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$54$$ 0 0
$$55$$ −14.7733 −0.268605
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$60$$ 0 0
$$61$$ 5.12376 0.0839960 0.0419980 0.999118i $$-0.486628\pi$$
0.0419980 + 0.999118i $$0.486628\pi$$
$$62$$ 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
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ −112.072 −1.53524 −0.767618 0.640907i $$-0.778558\pi$$
−0.767618 + 0.640907i $$0.778558\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 281.674 3.65810
$$78$$ 0 0
$$79$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −90.0000 −1.08434 −0.542169 0.840270i $$-0.682397\pi$$
−0.542169 + 0.840270i $$0.682397\pi$$
$$84$$ 0 0
$$85$$ 13.7218 0.161433
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 13.7766 0.145017
$$96$$ 0 0
$$97$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 102.000 1.00990 0.504950 0.863148i $$-0.331511\pi$$
0.504950 + 0.863148i $$0.331511\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$108$$ 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
$$115$$ −21.7525 −0.189152
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −261.625 −2.19853
$$120$$ 0 0
$$121$$ 294.124 2.43077
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 35.8729 0.286983
$$126$$ 0 0
$$127$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −238.622 −1.82154 −0.910771 0.412911i $$-0.864512\pi$$
−0.910771 + 0.412911i $$0.864512\pi$$
$$132$$ 0 0
$$133$$ −262.670 −1.97496
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 40.6769 0.296912 0.148456 0.988919i $$-0.452570\pi$$
0.148456 + 0.988919i $$0.452570\pi$$
$$138$$ 0 0
$$139$$ −71.3713 −0.513462 −0.256731 0.966483i $$-0.582646\pi$$
−0.256731 + 0.966483i $$0.582646\pi$$
$$140$$ 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
$$149$$ −296.120 −1.98739 −0.993693 0.112137i $$-0.964231\pi$$
−0.993693 + 0.112137i $$0.964231\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 10.0000 0.0636943 0.0318471 0.999493i $$-0.489861\pi$$
0.0318471 + 0.999493i $$0.489861\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 414.743 2.57604
$$162$$ 0 0
$$163$$ 250.000 1.53374 0.766871 0.641801i $$-0.221813\pi$$
0.766871 + 0.641801i $$0.221813\pi$$
$$164$$ 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
$$173$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$174$$ 0 0
$$175$$ −338.350 −1.93343
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 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
$$187$$ −385.577 −2.06191
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 274.368 1.43648 0.718241 0.695795i $$-0.244948\pi$$
0.718241 + 0.695795i $$0.244948\pi$$
$$192$$ 0 0
$$193$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −90.0000 −0.456853 −0.228426 0.973561i $$-0.573358\pi$$
−0.228426 + 0.973561i $$0.573358\pi$$
$$198$$ 0 0
$$199$$ 169.619 0.852356 0.426178 0.904639i $$-0.359860\pi$$
0.426178 + 0.904639i $$0.359860\pi$$
$$200$$ 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
$$209$$ −387.117 −1.85224
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −39.0274 −0.181523
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ −387.866 −1.69374 −0.846870 0.531800i $$-0.821516\pi$$
−0.846870 + 0.531800i $$0.821516\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −258.924 −1.11126 −0.555632 0.831428i $$-0.687524\pi$$
−0.555632 + 0.831428i $$0.687524\pi$$
$$234$$ 0 0
$$235$$ −62.7733 −0.267120
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −358.622 −1.50051 −0.750255 0.661148i $$-0.770070\pi$$
−0.750255 + 0.661148i $$0.770070\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −103.051 −0.420618
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −420.615 −1.67576 −0.837879 0.545855i $$-0.816205\pi$$
−0.837879 + 0.545855i $$0.816205\pi$$
$$252$$ 0 0
$$253$$ 611.238 2.41596
$$254$$ 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
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 88.1686 0.335242 0.167621 0.985852i $$-0.446392\pi$$
0.167621 + 0.985852i $$0.446392\pi$$
$$264$$ 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
$$271$$ −142.000 −0.523985 −0.261993 0.965070i $$-0.584380\pi$$
−0.261993 + 0.965070i $$0.584380\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −498.653 −1.81328
$$276$$ 0 0
$$277$$ −392.072 −1.41542 −0.707712 0.706501i $$-0.750272\pi$$
−0.707712 + 0.706501i $$0.750272\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ −548.567 −1.93840 −0.969200 0.246274i $$-0.920794\pi$$
−0.969200 + 0.246274i $$0.920794\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 69.1337 0.239217
$$290$$ 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
$$301$$ 744.114 2.47214
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −3.71515 −0.0121808
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 169.378 0.544623 0.272312 0.962209i $$-0.412212\pi$$
0.272312 + 0.962209i $$0.412212\pi$$
$$312$$ 0 0
$$313$$ −590.000 −1.88498 −0.942492 0.334229i $$-0.891524\pi$$
−0.942492 + 0.334229i $$0.891524\pi$$
$$314$$ 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
$$323$$ 359.564 1.11320
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 1196.86 3.63788
$$330$$ 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
$$337$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 1287.41 3.75339
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 477.172 1.37514 0.687568 0.726120i $$-0.258678\pi$$
0.687568 + 0.726120i $$0.258678\pi$$
$$348$$ 0 0
$$349$$ −659.866 −1.89073 −0.945367 0.326007i $$-0.894297\pi$$
−0.945367 + 0.326007i $$0.894297\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 510.000 1.44476 0.722380 0.691497i $$-0.243048\pi$$
0.722380 + 0.691497i $$0.243048\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −463.612 −1.29140 −0.645699 0.763592i $$-0.723434\pi$$
−0.645699 + 0.763592i $$0.723434\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 81.2617 0.222635
$$366$$ 0 0
$$367$$ 50.0000 0.136240 0.0681199 0.997677i $$-0.478300\pi$$
0.0681199 + 0.997677i $$0.478300\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ −204.237 −0.530485
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −737.111 −1.89489 −0.947443 0.319925i $$-0.896342\pi$$
−0.947443 + 0.319925i $$0.896342\pi$$
$$390$$ 0 0
$$391$$ −567.733 −1.45200
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 610.320 1.53733 0.768665 0.639652i $$-0.220921\pi$$
0.768665 + 0.639652i $$0.220921\pi$$
$$398$$ 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
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 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
$$415$$ 65.2575 0.157247
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ −762.000 −1.81862 −0.909308 0.416124i $$-0.863388\pi$$
−0.909308 + 0.416124i $$0.863388\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 463.161 1.08979
$$426$$ 0 0
$$427$$ 70.8347 0.165889
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −570.000 −1.30435
$$438$$ 0 0
$$439$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −788.808 −1.78061 −0.890303 0.455369i $$-0.849507\pi$$
−0.890303 + 0.455369i $$0.849507\pi$$
$$444$$ 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
$$457$$ −265.062 −0.580005 −0.290003 0.957026i $$-0.593656\pi$$
−0.290003 + 0.957026i $$0.593656\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 921.860 1.99970 0.999848 0.0174455i $$-0.00555337\pi$$
0.999848 + 0.0174455i $$0.00555337\pi$$
$$462$$ 0 0
$$463$$ 86.8148 0.187505 0.0937525 0.995596i $$-0.470114\pi$$
0.0937525 + 0.995596i $$0.470114\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 295.179 0.632074 0.316037 0.948747i $$-0.397648\pi$$
0.316037 + 0.948747i $$0.397648\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 1096.66 2.31851
$$474$$ 0 0
$$475$$ 465.011 0.978970
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 942.000 1.96660 0.983299 0.182000i $$-0.0582571\pi$$
0.983299 + 0.182000i $$0.0582571\pi$$
$$480$$ 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
$$491$$ 918.000 1.86965 0.934827 0.355104i $$-0.115554\pi$$
0.934827 + 0.355104i $$0.115554\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 474.609 0.951120 0.475560 0.879683i $$-0.342245\pi$$
0.475560 + 0.879683i $$0.342245\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −930.000 −1.84891 −0.924453 0.381295i $$-0.875478\pi$$
−0.924453 + 0.381295i $$0.875478\pi$$
$$504$$ 0 0
$$505$$ −73.9584 −0.146452
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$510$$ 0 0
$$511$$ −1549.37 −3.03204
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1763.91 3.41181
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 371.000 0.701323
$$530$$ 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
$$539$$ 2895.71 5.37238
$$540$$ 0 0
$$541$$ −620.856 −1.14761 −0.573804 0.818992i $$-0.694533\pi$$
−0.573804 + 0.818992i $$0.694533\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 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
$$557$$ 370.079 0.664415 0.332207 0.943206i $$-0.392207\pi$$
0.332207 + 0.943206i $$0.392207\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$564$$ 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
$$571$$ 458.000 0.802102 0.401051 0.916056i $$-0.368645\pi$$
0.401051 + 0.916056i $$0.368645\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −734.228 −1.27692
$$576$$ 0 0
$$577$$ 447.928 0.776305 0.388152 0.921595i $$-0.373113\pi$$
0.388152 + 0.921595i $$0.373113\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −1244.23 −2.14153
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −271.831 −0.463086 −0.231543 0.972825i $$-0.574377\pi$$
−0.231543 + 0.972825i $$0.574377\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 30.0000 0.0505902 0.0252951 0.999680i $$-0.491947\pi$$
0.0252951 + 0.999680i $$0.491947\pi$$
$$594$$ 0 0
$$595$$ 189.700 0.318824
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −213.264 −0.352503
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ −1178.05 −1.92178 −0.960891 0.276927i $$-0.910684\pi$$
−0.960891 + 0.276927i $$0.910684\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −1072.31 −1.73795 −0.868973 0.494859i $$-0.835220\pi$$
−0.868973 + 0.494859i $$0.835220\pi$$
$$618$$ 0 0
$$619$$ −662.000 −1.06947 −0.534733 0.845021i $$-0.679588\pi$$
−0.534733 + 0.845021i $$0.679588\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 585.846 0.937353
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ −104.361 −0.165390 −0.0826952 0.996575i $$-0.526353\pi$$
−0.0826952 + 0.996575i $$0.526353\pi$$
$$632$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ −2.58717 −0.00402359 −0.00201180 0.999998i $$-0.500640\pi$$
−0.00201180 + 0.999998i $$0.500640\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −1208.41 −1.86771 −0.933856 0.357650i $$-0.883578\pi$$
−0.933856 + 0.357650i $$0.883578\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −895.901 −1.37198 −0.685989 0.727612i $$-0.740630\pi$$
−0.685989 + 0.727612i $$0.740630\pi$$
$$654$$ 0 0
$$655$$ 173.021 0.264154
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$660$$ 0 0
$$661$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 190.458 0.286403
$$666$$ 0 0
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 104.394 0.155580
$$672$$ 0 0
$$673$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$684$$ 0 0
$$685$$ −29.4941 −0.0430571
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 1267.60 1.83444 0.917221 0.398380i $$-0.130427\pi$$
0.917221 + 0.398380i $$0.130427\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 51.7501 0.0744606
$$696$$ 0 0
$$697$$ 0 0
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1098.00 −1.56633 −0.783167 0.621812i $$-0.786397\pi$$
−0.783167 + 0.621812i $$0.786397\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1410.12 1.99452
$$708$$ 0 0
$$709$$ −1318.00 −1.85896 −0.929478 0.368877i $$-0.879742\pi$$
−0.929478 + 0.368877i $$0.879742\pi$$
$$710$$ 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
$$719$$ 1406.35 1.95599 0.977994 0.208635i $$-0.0669021\pi$$
0.977994 + 0.208635i $$0.0669021\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −1214.55 −1.67063 −0.835315 0.549772i $$-0.814715\pi$$
−0.835315 + 0.549772i $$0.814715\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −1018.60 −1.39344
$$732$$ 0 0
$$733$$ −1270.00 −1.73261 −0.866303 0.499519i $$-0.833510\pi$$
−0.866303 + 0.499519i $$0.833510\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 915.599 1.23897 0.619485 0.785008i $$-0.287341\pi$$
0.619485 + 0.785008i $$0.287341\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 214.712 0.288204
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 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
$$757$$ −728.650 −0.962550 −0.481275 0.876570i $$-0.659826\pi$$
−0.481275 + 0.876570i $$0.659826\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 543.880 0.714691 0.357345 0.933972i $$-0.383682\pi$$
0.357345 + 0.933972i $$0.383682\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 431.104 0.560603 0.280302 0.959912i $$-0.409566\pi$$
0.280302 + 0.959912i $$0.409566\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 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
$$785$$ −7.25083 −0.00923672
$$786$$ 0 0
$$787$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$788$$ 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
$$797$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$798$$ 0 0
$$799$$ −1638.36 −2.05051
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −2283.43 −2.84362
$$804$$ 0 0
$$805$$ −300.723 −0.373569
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −551.130 −0.681249 −0.340624 0.940199i $$-0.610638\pi$$
−0.340624 + 0.940199i $$0.610638\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −181.271 −0.222418
$$816$$ 0 0
$$817$$ −1022.67 −1.25174
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 1583.85 1.92918 0.964588 0.263762i $$-0.0849634\pi$$
0.964588 + 0.263762i $$0.0849634\pi$$
$$822$$ 0 0
$$823$$ 340.835 0.414137 0.207068 0.978326i $$-0.433608\pi$$
0.207068 + 0.978326i $$0.433608\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −2689.61 −3.22882
$$834$$ 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
$$841$$ 841.000 1.00000
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −122.539 −0.145017
$$846$$ 0 0
$$847$$ 4066.19 4.80069
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ −1030.00 −1.20750 −0.603751 0.797173i $$-0.706328\pi$$
−0.603751 + 0.797173i $$0.706328\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$858$$ 0 0
$$859$$ 1482.61 1.72597 0.862985 0.505229i $$-0.168592\pi$$
0.862985 + 0.505229i $$0.168592\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 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
$$875$$ 495.934 0.566782
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1184.84 1.34488 0.672442 0.740150i $$-0.265245\pi$$
0.672442 + 0.740150i $$0.265245\pi$$
$$882$$ 0 0
$$883$$ −1765.15 −1.99903 −0.999516 0.0311055i $$-0.990097\pi$$
−0.999516 + 0.0311055i $$0.990097\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −1644.90 −1.84200
$$894$$ 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
$$907$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ −1833.71 −2.00845
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −3298.89 −3.59748
$$918$$ 0 0
$$919$$ 1762.00 1.91730 0.958651 0.284585i $$-0.0918559\pi$$
0.958651 + 0.284585i $$0.0918559\pi$$
$$920$$ 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
$$929$$ −642.000 −0.691066 −0.345533 0.938407i $$-0.612302\pi$$
−0.345533 + 0.938407i $$0.612302\pi$$
$$930$$ 0 0
$$931$$ −2700.35 −2.90048
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 279.575 0.299011
$$936$$ 0 0
$$937$$ 1429.29 1.52539 0.762695 0.646759i $$-0.223876\pi$$
0.762695 + 0.646759i $$0.223876\pi$$
$$938$$ 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
$$947$$ 1830.00 1.93242 0.966209 0.257760i $$-0.0829843\pi$$
0.966209 + 0.257760i $$0.0829843\pi$$
$$948$$ 0 0
$$949$$ 0 0
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$954$$ 0 0
$$955$$ −198.939 −0.208314
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 562.348 0.586390
$$960$$ 0 0
$$961$$ 961.000 1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1790.00 −1.85109 −0.925543 0.378643i $$-0.876391\pi$$
−0.925543 + 0.378643i $$0.876391\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$972$$ 0 0
$$973$$ −986.690 −1.01407
$$974$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 65.2575 0.0662512
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1614.74 1.63270
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −122.988 −0.123606
$$996$$ 0 0
$$997$$ −749.680 −0.751936 −0.375968 0.926633i $$-0.622690\pi$$
−0.375968 + 0.926633i $$0.622690\pi$$
$$998$$ 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 684.3.h.a.37.2 2
3.2 odd 2 76.3.c.b.37.1 2
4.3 odd 2 2736.3.o.c.721.2 2
12.11 even 2 304.3.e.e.113.1 2
15.2 even 4 1900.3.g.a.949.4 4
15.8 even 4 1900.3.g.a.949.1 4
15.14 odd 2 1900.3.e.a.1101.1 2
19.18 odd 2 CM 684.3.h.a.37.2 2
24.5 odd 2 1216.3.e.f.1025.2 2
24.11 even 2 1216.3.e.e.1025.2 2
57.56 even 2 76.3.c.b.37.1 2
76.75 even 2 2736.3.o.c.721.2 2
228.227 odd 2 304.3.e.e.113.1 2
285.113 odd 4 1900.3.g.a.949.1 4
285.227 odd 4 1900.3.g.a.949.4 4
285.284 even 2 1900.3.e.a.1101.1 2
456.227 odd 2 1216.3.e.e.1025.2 2
456.341 even 2 1216.3.e.f.1025.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
76.3.c.b.37.1 2 3.2 odd 2
76.3.c.b.37.1 2 57.56 even 2
304.3.e.e.113.1 2 12.11 even 2
304.3.e.e.113.1 2 228.227 odd 2
684.3.h.a.37.2 2 1.1 even 1 trivial
684.3.h.a.37.2 2 19.18 odd 2 CM
1216.3.e.e.1025.2 2 24.11 even 2
1216.3.e.e.1025.2 2 456.227 odd 2
1216.3.e.f.1025.2 2 24.5 odd 2
1216.3.e.f.1025.2 2 456.341 even 2
1900.3.e.a.1101.1 2 15.14 odd 2
1900.3.e.a.1101.1 2 285.284 even 2
1900.3.g.a.949.1 4 15.8 even 4
1900.3.g.a.949.1 4 285.113 odd 4
1900.3.g.a.949.4 4 15.2 even 4
1900.3.g.a.949.4 4 285.227 odd 4
2736.3.o.c.721.2 2 4.3 odd 2
2736.3.o.c.721.2 2 76.75 even 2