# Properties

 Label 688.3.b.a.257.1 Level 688 Weight 3 Character 688.257 Self dual yes Analytic conductor 18.747 Analytic rank 0 Dimension 1 CM discriminant -43 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$688 = 2^{4} \cdot 43$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 688.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.7466421880$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 43) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 257.1 Character $$\chi$$ $$=$$ 688.257

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+9.00000 q^{9} +O(q^{10})$$ $$q+9.00000 q^{9} +21.0000 q^{11} -17.0000 q^{13} -9.00000 q^{17} -3.00000 q^{23} +25.0000 q^{25} -19.0000 q^{31} +39.0000 q^{41} +43.0000 q^{43} +78.0000 q^{47} +49.0000 q^{49} +63.0000 q^{53} +54.0000 q^{59} -91.0000 q^{67} +14.0000 q^{79} +81.0000 q^{81} -123.000 q^{83} -193.000 q^{97} +189.000 q^{99} +O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/688\mathbb{Z}\right)^\times$$.

 $$n$$ $$431$$ $$433$$ $$517$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$4$$ 0 0
$$5$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$8$$ 0 0
$$9$$ 9.00000 1.00000
$$10$$ 0 0
$$11$$ 21.0000 1.90909 0.954545 0.298065i $$-0.0963413\pi$$
0.954545 + 0.298065i $$0.0963413\pi$$
$$12$$ 0 0
$$13$$ −17.0000 −1.30769 −0.653846 0.756628i $$-0.726846\pi$$
−0.653846 + 0.756628i $$0.726846\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −9.00000 −0.529412 −0.264706 0.964329i $$-0.585275\pi$$
−0.264706 + 0.964329i $$0.585275\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −3.00000 −0.130435 −0.0652174 0.997871i $$-0.520774\pi$$
−0.0652174 + 0.997871i $$0.520774\pi$$
$$24$$ 0 0
$$25$$ 25.0000 1.00000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$30$$ 0 0
$$31$$ −19.0000 −0.612903 −0.306452 0.951886i $$-0.599142\pi$$
−0.306452 + 0.951886i $$0.599142\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 39.0000 0.951220 0.475610 0.879656i $$-0.342227\pi$$
0.475610 + 0.879656i $$0.342227\pi$$
$$42$$ 0 0
$$43$$ 43.0000 1.00000
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 78.0000 1.65957 0.829787 0.558080i $$-0.188462\pi$$
0.829787 + 0.558080i $$0.188462\pi$$
$$48$$ 0 0
$$49$$ 49.0000 1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 63.0000 1.18868 0.594340 0.804214i $$-0.297414\pi$$
0.594340 + 0.804214i $$0.297414\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 54.0000 0.915254 0.457627 0.889144i $$-0.348699\pi$$
0.457627 + 0.889144i $$0.348699\pi$$
$$60$$ 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
$$67$$ −91.0000 −1.35821 −0.679104 0.734042i $$-0.737632\pi$$
−0.679104 + 0.734042i $$0.737632\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 14.0000 0.177215 0.0886076 0.996067i $$-0.471758\pi$$
0.0886076 + 0.996067i $$0.471758\pi$$
$$80$$ 0 0
$$81$$ 81.0000 1.00000
$$82$$ 0 0
$$83$$ −123.000 −1.48193 −0.740964 0.671545i $$-0.765631\pi$$
−0.740964 + 0.671545i $$0.765631\pi$$
$$84$$ 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
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −193.000 −1.98969 −0.994845 0.101404i $$-0.967667\pi$$
−0.994845 + 0.101404i $$0.967667\pi$$
$$98$$ 0 0
$$99$$ 189.000 1.90909
$$100$$ 0 0
$$101$$ 159.000 1.57426 0.787129 0.616789i $$-0.211567\pi$$
0.787129 + 0.616789i $$0.211567\pi$$
$$102$$ 0 0
$$103$$ 181.000 1.75728 0.878641 0.477483i $$-0.158451\pi$$
0.878641 + 0.477483i $$0.158451\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −42.0000 −0.392523 −0.196262 0.980552i $$-0.562880\pi$$
−0.196262 + 0.980552i $$0.562880\pi$$
$$108$$ 0 0
$$109$$ −169.000 −1.55046 −0.775229 0.631680i $$-0.782366\pi$$
−0.775229 + 0.631680i $$0.782366\pi$$
$$110$$ 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
$$117$$ −153.000 −1.30769
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 320.000 2.64463
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 133.000 1.04724 0.523622 0.851951i $$-0.324580\pi$$
0.523622 + 0.851951i $$0.324580\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$138$$ 0 0
$$139$$ 109.000 0.784173 0.392086 0.919928i $$-0.371753\pi$$
0.392086 + 0.919928i $$0.371753\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −357.000 −2.49650
$$144$$ 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
$$151$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$152$$ 0 0
$$153$$ −81.0000 −0.529412
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −291.000 −1.74251 −0.871257 0.490826i $$-0.836695\pi$$
−0.871257 + 0.490826i $$0.836695\pi$$
$$168$$ 0 0
$$169$$ 120.000 0.710059
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −342.000 −1.97688 −0.988439 0.151617i $$-0.951552\pi$$
−0.988439 + 0.151617i $$0.951552\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$180$$ 0 0
$$181$$ −326.000 −1.80110 −0.900552 0.434747i $$-0.856838\pi$$
−0.900552 + 0.434747i $$0.856838\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −189.000 −1.01070
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ 343.000 1.77720 0.888601 0.458681i $$-0.151678\pi$$
0.888601 + 0.458681i $$0.151678\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −294.000 −1.49239 −0.746193 0.665730i $$-0.768120\pi$$
−0.746193 + 0.665730i $$0.768120\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −27.0000 −0.130435
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 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
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 153.000 0.692308
$$222$$ 0 0
$$223$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$224$$ 0 0
$$225$$ 225.000 1.00000
$$226$$ 0 0
$$227$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$228$$ 0 0
$$229$$ 71.0000 0.310044 0.155022 0.987911i $$-0.450455\pi$$
0.155022 + 0.987911i $$0.450455\pi$$
$$230$$ 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
$$239$$ −306.000 −1.28033 −0.640167 0.768235i $$-0.721135\pi$$
−0.640167 + 0.768235i $$0.721135\pi$$
$$240$$ 0 0
$$241$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −459.000 −1.82869 −0.914343 0.404941i $$-0.867292\pi$$
−0.914343 + 0.404941i $$0.867292\pi$$
$$252$$ 0 0
$$253$$ −63.0000 −0.249012
$$254$$ 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
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −537.000 −1.99628 −0.998141 0.0609427i $$-0.980589\pi$$
−0.998141 + 0.0609427i $$0.980589\pi$$
$$270$$ 0 0
$$271$$ 533.000 1.96679 0.983395 0.181479i $$-0.0580884\pi$$
0.983395 + 0.181479i $$0.0580884\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 525.000 1.90909
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ −171.000 −0.612903
$$280$$ 0 0
$$281$$ −513.000 −1.82562 −0.912811 0.408381i $$-0.866093\pi$$
−0.912811 + 0.408381i $$0.866093\pi$$
$$282$$ 0 0
$$283$$ −523.000 −1.84806 −0.924028 0.382324i $$-0.875124\pi$$
−0.924028 + 0.382324i $$0.875124\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −208.000 −0.719723
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ −102.000 −0.348123 −0.174061 0.984735i $$-0.555689\pi$$
−0.174061 + 0.984735i $$0.555689\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 51.0000 0.170569
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −227.000 −0.739414 −0.369707 0.929148i $$-0.620542\pi$$
−0.369707 + 0.929148i $$0.620542\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 453.000 1.45659 0.728296 0.685263i $$-0.240313\pi$$
0.728296 + 0.685263i $$0.240313\pi$$
$$312$$ 0 0
$$313$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 591.000 1.86435 0.932177 0.362004i $$-0.117907\pi$$
0.932177 + 0.362004i $$0.117907\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ −425.000 −1.30769
$$326$$ 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
$$337$$ 287.000 0.851632 0.425816 0.904810i $$-0.359987\pi$$
0.425816 + 0.904810i $$0.359987\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −399.000 −1.17009
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$348$$ 0 0
$$349$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 663.000 1.87819 0.939093 0.343662i $$-0.111667\pi$$
0.939093 + 0.343662i $$0.111667\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 357.000 0.994429 0.497214 0.867628i $$-0.334356\pi$$
0.497214 + 0.867628i $$0.334356\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −562.000 −1.53134 −0.765668 0.643236i $$-0.777591\pi$$
−0.765668 + 0.643236i $$0.777591\pi$$
$$368$$ 0 0
$$369$$ 351.000 0.951220
$$370$$ 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
$$379$$ 317.000 0.836412 0.418206 0.908352i $$-0.362659\pi$$
0.418206 + 0.908352i $$0.362659\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 387.000 1.00000
$$388$$ 0 0
$$389$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$390$$ 0 0
$$391$$ 27.0000 0.0690537
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 106.000 0.267003 0.133501 0.991049i $$-0.457378\pi$$
0.133501 + 0.991049i $$0.457378\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −273.000 −0.680798 −0.340399 0.940281i $$-0.610562\pi$$
−0.340399 + 0.940281i $$0.610562\pi$$
$$402$$ 0 0
$$403$$ 323.000 0.801489
$$404$$ 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
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$420$$ 0 0
$$421$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$422$$ 0 0
$$423$$ 702.000 1.65957
$$424$$ 0 0
$$425$$ −225.000 −0.529412
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −819.000 −1.90023 −0.950116 0.311897i $$-0.899036\pi$$
−0.950116 + 0.311897i $$0.899036\pi$$
$$432$$ 0 0
$$433$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ −491.000 −1.11845 −0.559226 0.829016i $$-0.688901\pi$$
−0.559226 + 0.829016i $$0.688901\pi$$
$$440$$ 0 0
$$441$$ 441.000 1.00000
$$442$$ 0 0
$$443$$ −714.000 −1.61174 −0.805869 0.592094i $$-0.798302\pi$$
−0.805869 + 0.592094i $$0.798302\pi$$
$$444$$ 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
$$451$$ 819.000 1.81596
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 234.000 0.507592 0.253796 0.967258i $$-0.418321\pi$$
0.253796 + 0.967258i $$0.418321\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 903.000 1.90909
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 567.000 1.18868
$$478$$ 0 0
$$479$$ 117.000 0.244259 0.122129 0.992514i $$-0.461028\pi$$
0.122129 + 0.992514i $$0.461028\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 574.000 1.17864 0.589322 0.807898i $$-0.299395\pi$$
0.589322 + 0.807898i $$0.299395\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$492$$ 0 0
$$493$$ 0 0
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −57.0000 −0.111984 −0.0559921 0.998431i $$-0.517832\pi$$
−0.0559921 + 0.998431i $$0.517832\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1638.00 3.16828
$$518$$ 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
$$527$$ 171.000 0.324478
$$528$$ 0 0
$$529$$ −520.000 −0.982987
$$530$$ 0 0
$$531$$ 486.000 0.915254
$$532$$ 0 0
$$533$$ −663.000 −1.24390
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 1029.00 1.90909
$$540$$ 0 0
$$541$$ 7.00000 0.0129390 0.00646950 0.999979i $$-0.497941\pi$$
0.00646950 + 0.999979i $$0.497941\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 1013.00 1.85192 0.925960 0.377622i $$-0.123258\pi$$
0.925960 + 0.377622i $$0.123258\pi$$
$$548$$ 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
$$557$$ −993.000 −1.78276 −0.891382 0.453252i $$-0.850264\pi$$
−0.891382 + 0.453252i $$0.850264\pi$$
$$558$$ 0 0
$$559$$ −731.000 −1.30769
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −1083.00 −1.92362 −0.961812 0.273712i $$-0.911749\pi$$
−0.961812 + 0.273712i $$0.911749\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −969.000 −1.70299 −0.851494 0.524365i $$-0.824303\pi$$
−0.851494 + 0.524365i $$0.824303\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −75.0000 −0.130435
$$576$$ 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
$$583$$ 1323.00 2.26930
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 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
$$599$$ 909.000 1.51753 0.758765 0.651365i $$-0.225803\pi$$
0.758765 + 0.651365i $$0.225803\pi$$
$$600$$ 0 0
$$601$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$602$$ 0 0
$$603$$ −819.000 −1.35821
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −1326.00 −2.17021
$$612$$ 0 0
$$613$$ 538.000 0.877651 0.438825 0.898572i $$-0.355395\pi$$
0.438825 + 0.898572i $$0.355395\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −873.000 −1.41491 −0.707455 0.706758i $$-0.750157\pi$$
−0.707455 + 0.706758i $$0.750157\pi$$
$$618$$ 0 0
$$619$$ −1066.00 −1.72213 −0.861066 0.508493i $$-0.830203\pi$$
−0.861066 + 0.508493i $$0.830203\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 625.000 1.00000
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 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
$$637$$ −833.000 −1.30769
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$642$$ 0 0
$$643$$ 262.000 0.407465 0.203733 0.979027i $$-0.434693\pi$$
0.203733 + 0.979027i $$0.434693\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ 1134.00 1.74730
$$650$$ 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
$$659$$ 789.000 1.19727 0.598634 0.801022i $$-0.295710\pi$$
0.598634 + 0.801022i $$0.295710\pi$$
$$660$$ 0 0
$$661$$ 1279.00 1.93495 0.967474 0.252972i $$-0.0814081\pi$$
0.967474 + 0.252972i $$0.0814081\pi$$
$$662$$ 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
$$673$$ 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
$$683$$ 741.000 1.08492 0.542460 0.840082i $$-0.317493\pi$$
0.542460 + 0.840082i $$0.317493\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1071.00 −1.55443
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −351.000 −0.503587
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 714.000 1.01854 0.509272 0.860605i $$-0.329915\pi$$
0.509272 + 0.860605i $$0.329915\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −689.000 −0.971791 −0.485896 0.874017i $$-0.661506\pi$$
−0.485896 + 0.874017i $$0.661506\pi$$
$$710$$ 0 0
$$711$$ 126.000 0.177215
$$712$$ 0 0
$$713$$ 57.0000 0.0799439
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −1266.00 −1.76078 −0.880389 0.474251i $$-0.842719\pi$$
−0.880389 + 0.474251i $$0.842719\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$728$$ 0 0
$$729$$ 729.000 1.00000
$$730$$ 0 0
$$731$$ −387.000 −0.529412
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −1911.00 −2.59294
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −1107.00 −1.48193
$$748$$ 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
$$757$$ 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
$$767$$ −918.000 −1.19687
$$768$$ 0 0
$$769$$ −1214.00 −1.57867 −0.789337 0.613960i $$-0.789575\pi$$
−0.789337 + 0.613960i $$0.789575\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$774$$ 0 0
$$775$$ −475.000 −0.612903
$$776$$ 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
$$787$$ −26.0000 −0.0330368 −0.0165184 0.999864i $$-0.505258\pi$$
−0.0165184 + 0.999864i $$0.505258\pi$$
$$788$$ 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
$$797$$ 906.000 1.13676 0.568381 0.822765i $$-0.307570\pi$$
0.568381 + 0.822765i $$0.307570\pi$$
$$798$$ 0 0
$$799$$ −702.000 −0.878598
$$800$$ 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
$$809$$ −1134.00 −1.40173 −0.700865 0.713294i $$-0.747203\pi$$
−0.700865 + 0.713294i $$0.747203\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$812$$ 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
$$821$$ 567.000 0.690621 0.345311 0.938488i $$-0.387774\pi$$
0.345311 + 0.938488i $$0.387774\pi$$
$$822$$ 0 0
$$823$$ −1603.00 −1.94775 −0.973876 0.227080i $$-0.927082\pi$$
−0.973876 + 0.227080i $$0.927082\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1482.00 −1.79202 −0.896010 0.444035i $$-0.853547\pi$$
−0.896010 + 0.444035i $$0.853547\pi$$
$$828$$ 0 0
$$829$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −441.000 −0.529412
$$834$$ 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
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 1319.00 1.54631 0.773154 0.634219i $$-0.218678\pi$$
0.773154 + 0.634219i $$0.218678\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −1038.00 −1.21120 −0.605601 0.795769i $$-0.707067\pi$$
−0.605601 + 0.795769i $$0.707067\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 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
$$869$$ 294.000 0.338320
$$870$$ 0 0
$$871$$ 1547.00 1.77612
$$872$$ 0 0
$$873$$ −1737.00 −1.98969
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −1729.00 −1.97149 −0.985747 0.168235i $$-0.946193\pi$$
−0.985747 + 0.168235i $$0.946193\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 1719.00 1.95119 0.975596 0.219574i $$-0.0704666\pi$$
0.975596 + 0.219574i $$0.0704666\pi$$
$$882$$ 0 0
$$883$$ 1717.00 1.94451 0.972254 0.233929i $$-0.0751583\pi$$
0.972254 + 0.233929i $$0.0751583\pi$$
$$884$$ 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
$$891$$ 1701.00 1.90909
$$892$$ 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
$$901$$ −567.000 −0.629301
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 293.000 0.323043 0.161521 0.986869i $$-0.448360\pi$$
0.161521 + 0.986869i $$0.448360\pi$$
$$908$$ 0 0
$$909$$ 1431.00 1.57426
$$910$$ 0 0
$$911$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$912$$ 0 0
$$913$$ −2583.00 −2.82913
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −763.000 −0.830250 −0.415125 0.909764i $$-0.636262\pi$$
−0.415125 + 0.909764i $$0.636262\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 1629.00 1.75728
$$928$$ 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
$$937$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 1839.00 1.95430 0.977152 0.212542i $$-0.0681742\pi$$
0.977152 + 0.212542i $$0.0681742\pi$$
$$942$$ 0 0
$$943$$ −117.000 −0.124072
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 213.000 0.224921 0.112460 0.993656i $$-0.464127\pi$$
0.112460 + 0.993656i $$0.464127\pi$$
$$948$$ 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
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −600.000 −0.624350
$$962$$ 0 0
$$963$$ −378.000 −0.392523
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −1547.00 −1.59979 −0.799897 0.600138i $$-0.795112\pi$$
−0.799897 + 0.600138i $$0.795112\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −867.000 −0.892894 −0.446447 0.894810i $$-0.647311\pi$$
−0.446447 + 0.894810i $$0.647311\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −798.000 −0.816786 −0.408393 0.912806i $$-0.633911\pi$$
−0.408393 + 0.912806i $$0.633911\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ −1521.00 −1.55046
$$982$$ 0 0
$$983$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −129.000 −0.130435
$$990$$ 0 0
$$991$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 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 688.3.b.a.257.1 1
4.3 odd 2 43.3.b.a.42.1 1
12.11 even 2 387.3.b.a.343.1 1
43.42 odd 2 CM 688.3.b.a.257.1 1
172.171 even 2 43.3.b.a.42.1 1
516.515 odd 2 387.3.b.a.343.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
43.3.b.a.42.1 1 4.3 odd 2
43.3.b.a.42.1 1 172.171 even 2
387.3.b.a.343.1 1 12.11 even 2
387.3.b.a.343.1 1 516.515 odd 2
688.3.b.a.257.1 1 1.1 even 1 trivial
688.3.b.a.257.1 1 43.42 odd 2 CM