# Properties

 Label 864.3.h.a.593.1 Level $864$ Weight $3$ Character 864.593 Self dual yes Analytic conductor $23.542$ Analytic rank $0$ Dimension $2$ CM discriminant -24 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$864 = 2^{5} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 864.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$23.5422948407$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: no (minimal twist has level 216) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 593.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 864.593

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-9.48528 q^{5} +3.48528 q^{7} +O(q^{10})$$ $$q-9.48528 q^{5} +3.48528 q^{7} -21.9706 q^{11} +64.9706 q^{25} +50.0000 q^{29} -23.4264 q^{31} -33.0589 q^{35} -36.8528 q^{49} +89.4264 q^{53} +208.397 q^{55} +10.0000 q^{59} +93.7939 q^{73} -76.5736 q^{77} +58.0000 q^{79} +151.853 q^{83} +61.0589 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{5} - 10q^{7} + O(q^{10})$$ $$2q - 2q^{5} - 10q^{7} - 10q^{11} + 96q^{25} + 100q^{29} + 38q^{31} - 134q^{35} + 96q^{49} + 94q^{53} + 298q^{55} + 20q^{59} - 50q^{73} - 238q^{77} + 116q^{79} + 134q^{83} + 190q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$353$$ $$703$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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
$$5$$ −9.48528 −1.89706 −0.948528 0.316693i $$-0.897428\pi$$
−0.948528 + 0.316693i $$0.897428\pi$$
$$6$$ 0 0
$$7$$ 3.48528 0.497897 0.248949 0.968517i $$-0.419915\pi$$
0.248949 + 0.968517i $$0.419915\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −21.9706 −1.99732 −0.998662 0.0517139i $$-0.983532\pi$$
−0.998662 + 0.0517139i $$0.983532\pi$$
$$12$$ 0 0
$$13$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$24$$ 0 0
$$25$$ 64.9706 2.59882
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 50.0000 1.72414 0.862069 0.506791i $$-0.169168\pi$$
0.862069 + 0.506791i $$0.169168\pi$$
$$30$$ 0 0
$$31$$ −23.4264 −0.755691 −0.377845 0.925869i $$-0.623335\pi$$
−0.377845 + 0.925869i $$0.623335\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −33.0589 −0.944539
$$36$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$48$$ 0 0
$$49$$ −36.8528 −0.752098
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 89.4264 1.68729 0.843645 0.536901i $$-0.180405\pi$$
0.843645 + 0.536901i $$0.180405\pi$$
$$54$$ 0 0
$$55$$ 208.397 3.78904
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 10.0000 0.169492 0.0847458 0.996403i $$-0.472992\pi$$
0.0847458 + 0.996403i $$0.472992\pi$$
$$60$$ 0 0
$$61$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$72$$ 0 0
$$73$$ 93.7939 1.28485 0.642424 0.766349i $$-0.277929\pi$$
0.642424 + 0.766349i $$0.277929\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −76.5736 −0.994462
$$78$$ 0 0
$$79$$ 58.0000 0.734177 0.367089 0.930186i $$-0.380355\pi$$
0.367089 + 0.930186i $$0.380355\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 151.853 1.82955 0.914776 0.403962i $$-0.132367\pi$$
0.914776 + 0.403962i $$0.132367\pi$$
$$84$$ 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
$$97$$ 61.0589 0.629473 0.314736 0.949179i $$-0.398084\pi$$
0.314736 + 0.949179i $$0.398084\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 35.6030 0.352505 0.176253 0.984345i $$-0.443602\pi$$
0.176253 + 0.984345i $$0.443602\pi$$
$$102$$ 0 0
$$103$$ 10.0000 0.0970874 0.0485437 0.998821i $$-0.484542\pi$$
0.0485437 + 0.998821i $$0.484542\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −126.706 −1.18416 −0.592082 0.805877i $$-0.701694\pi$$
−0.592082 + 0.805877i $$0.701694\pi$$
$$108$$ 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
$$121$$ 361.706 2.98930
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −379.132 −3.03306
$$126$$ 0 0
$$127$$ 21.6619 0.170566 0.0852831 0.996357i $$-0.472821\pi$$
0.0852831 + 0.996357i $$0.472821\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −192.882 −1.47238 −0.736192 0.676773i $$-0.763378\pi$$
−0.736192 + 0.676773i $$0.763378\pi$$
$$132$$ 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
$$139$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ −474.264 −3.27079
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −204.397 −1.37179 −0.685896 0.727700i $$-0.740589\pi$$
−0.685896 + 0.727700i $$0.740589\pi$$
$$150$$ 0 0
$$151$$ −191.426 −1.26772 −0.633862 0.773446i $$-0.718531\pi$$
−0.633862 + 0.773446i $$0.718531\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 222.206 1.43359
$$156$$ 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
$$167$$ 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
$$173$$ 319.985 1.84962 0.924812 0.380425i $$-0.124222\pi$$
0.924812 + 0.380425i $$0.124222\pi$$
$$174$$ 0 0
$$175$$ 226.441 1.29395
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 352.588 1.96976 0.984882 0.173225i $$-0.0554187\pi$$
0.984882 + 0.173225i $$0.0554187\pi$$
$$180$$ 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
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$192$$ 0 0
$$193$$ 75.6173 0.391800 0.195900 0.980624i $$-0.437237\pi$$
0.195900 + 0.980624i $$0.437237\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 199.985 1.01515 0.507576 0.861607i $$-0.330542\pi$$
0.507576 + 0.861607i $$0.330542\pi$$
$$198$$ 0 0
$$199$$ −397.985 −1.99992 −0.999962 0.00872575i $$-0.997222\pi$$
−0.999962 + 0.00872575i $$0.997222\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 174.264 0.858444
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −81.6476 −0.376256
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 0 0
$$222$$ 0 0
$$223$$ −230.000 −1.03139 −0.515695 0.856772i $$-0.672466\pi$$
−0.515695 + 0.856772i $$0.672466\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 346.000 1.52423 0.762115 0.647442i $$-0.224161\pi$$
0.762115 + 0.647442i $$0.224161\pi$$
$$228$$ 0 0
$$229$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$240$$ 0 0
$$241$$ −382.000 −1.58506 −0.792531 0.609831i $$-0.791237\pi$$
−0.792531 + 0.609831i $$0.791237\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 349.559 1.42677
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −470.000 −1.87251 −0.936255 0.351321i $$-0.885733\pi$$
−0.936255 + 0.351321i $$0.885733\pi$$
$$252$$ 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
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$264$$ 0 0
$$265$$ −848.235 −3.20089
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −430.000 −1.59851 −0.799257 0.600990i $$-0.794773\pi$$
−0.799257 + 0.600990i $$0.794773\pi$$
$$270$$ 0 0
$$271$$ 437.690 1.61509 0.807547 0.589803i $$-0.200795\pi$$
0.807547 + 0.589803i $$0.200795\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −1427.44 −5.19069
$$276$$ 0 0
$$277$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 289.000 1.00000
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 386.000 1.31741 0.658703 0.752403i $$-0.271105\pi$$
0.658703 + 0.752403i $$0.271105\pi$$
$$294$$ 0 0
$$295$$ −94.8528 −0.321535
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$312$$ 0 0
$$313$$ −553.500 −1.76837 −0.884185 0.467138i $$-0.845285\pi$$
−0.884185 + 0.467138i $$0.845285\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −299.690 −0.945396 −0.472698 0.881225i $$-0.656720\pi$$
−0.472698 + 0.881225i $$0.656720\pi$$
$$318$$ 0 0
$$319$$ −1098.53 −3.44366
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −190.000 −0.563798 −0.281899 0.959444i $$-0.590964\pi$$
−0.281899 + 0.959444i $$0.590964\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 514.691 1.50936
$$342$$ 0 0
$$343$$ −299.221 −0.872365
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 540.970 1.55899 0.779495 0.626408i $$-0.215476\pi$$
0.779495 + 0.626408i $$0.215476\pi$$
$$348$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$360$$ 0 0
$$361$$ 361.000 1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −889.662 −2.43743
$$366$$ 0 0
$$367$$ 516.220 1.40659 0.703297 0.710896i $$-0.251710\pi$$
0.703297 + 0.710896i $$0.251710\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 311.676 0.840098
$$372$$ 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
$$385$$ 726.322 1.88655
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 748.367 1.92382 0.961911 0.273363i $$-0.0881361\pi$$
0.961911 + 0.273363i $$0.0881361\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −550.146 −1.39278
$$396$$ 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
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 19.5887 0.0478942 0.0239471 0.999713i $$-0.492377\pi$$
0.0239471 + 0.999713i $$0.492377\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 34.8528 0.0843894
$$414$$ 0 0
$$415$$ −1440.37 −3.47076
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 730.000 1.74224 0.871122 0.491067i $$-0.163393\pi$$
0.871122 + 0.491067i $$0.163393\pi$$
$$420$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$432$$ 0 0
$$433$$ 810.176 1.87108 0.935538 0.353227i $$-0.114916\pi$$
0.935538 + 0.353227i $$0.114916\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 860.249 1.95956 0.979782 0.200066i $$-0.0641157\pi$$
0.979782 + 0.200066i $$0.0641157\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −86.0000 −0.194131 −0.0970655 0.995278i $$-0.530946\pi$$
−0.0970655 + 0.995278i $$0.530946\pi$$
$$444$$ 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
$$457$$ −909.881 −1.99099 −0.995494 0.0948261i $$-0.969771\pi$$
−0.995494 + 0.0948261i $$0.969771\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 388.367 0.842444 0.421222 0.906958i $$-0.361601\pi$$
0.421222 + 0.906958i $$0.361601\pi$$
$$462$$ 0 0
$$463$$ 647.161 1.39776 0.698878 0.715241i $$-0.253683\pi$$
0.698878 + 0.715241i $$0.253683\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 276.970 0.593083 0.296541 0.955020i $$-0.404167\pi$$
0.296541 + 0.955020i $$0.404167\pi$$
$$468$$ 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
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$480$$ 0 0
$$481$$ 0 0
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −579.161 −1.19415
$$486$$ 0 0
$$487$$ 970.000 1.99179 0.995893 0.0905356i $$-0.0288579\pi$$
0.995893 + 0.0905356i $$0.0288579\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 981.705 1.99940 0.999699 0.0245196i $$-0.00780562\pi$$
0.999699 + 0.0245196i $$0.00780562\pi$$
$$492$$ 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
$$505$$ −337.705 −0.668722
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −615.309 −1.20886 −0.604429 0.796659i $$-0.706599\pi$$
−0.604429 + 0.796659i $$0.706599\pi$$
$$510$$ 0 0
$$511$$ 326.898 0.639723
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −94.8528 −0.184180
$$516$$ 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
$$523$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 529.000 1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 0 0
$$534$$ 0 0
$$535$$ 1201.84 2.24643
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 809.677 1.50218
$$540$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 202.146 0.365545
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 94.5433 0.169737 0.0848683 0.996392i $$-0.472953\pi$$
0.0848683 + 0.996392i $$0.472953\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ −770.381 −1.36835 −0.684175 0.729318i $$-0.739838\pi$$
−0.684175 + 0.729318i $$0.739838\pi$$
$$564$$ 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
$$571$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 290.000 0.502600 0.251300 0.967909i $$-0.419142\pi$$
0.251300 + 0.967909i $$0.419142\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 529.250 0.910929
$$582$$ 0 0
$$583$$ −1964.75 −3.37007
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −1115.82 −1.90089 −0.950445 0.310893i $$-0.899372\pi$$
−0.950445 + 0.310893i $$0.899372\pi$$
$$588$$ 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
$$599$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$600$$ 0 0
$$601$$ 514.147 0.855486 0.427743 0.903900i $$-0.359309\pi$$
0.427743 + 0.903900i $$0.359309\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −3430.88 −5.67088
$$606$$ 0 0
$$607$$ 730.000 1.20264 0.601318 0.799010i $$-0.294643\pi$$
0.601318 + 0.799010i $$0.294643\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1971.91 3.15506
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 0 0
$$630$$ 0 0
$$631$$ 831.132 1.31717 0.658583 0.752508i $$-0.271156\pi$$
0.658583 + 0.752508i $$0.271156\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ −205.469 −0.323574
$$636$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$648$$ 0 0
$$649$$ −219.706 −0.338529
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −218.249 −0.334225 −0.167112 0.985938i $$-0.553444\pi$$
−0.167112 + 0.985938i $$0.553444\pi$$
$$654$$ 0 0
$$655$$ 1829.54 2.79319
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 585.352 0.888242 0.444121 0.895967i $$-0.353516\pi$$
0.444121 + 0.895967i $$0.353516\pi$$
$$660$$ 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
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ −1059.00 −1.57355 −0.786774 0.617241i $$-0.788250\pi$$
−0.786774 + 0.617241i $$0.788250\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 1346.00 1.98818 0.994092 0.108545i $$-0.0346190\pi$$
0.994092 + 0.108545i $$0.0346190\pi$$
$$678$$ 0 0
$$679$$ 212.807 0.313413
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −1334.00 −1.95315 −0.976574 0.215182i $$-0.930965\pi$$
−0.976574 + 0.215182i $$0.930965\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 0 0
$$690$$ 0 0
$$691$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$692$$ 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
$$701$$ 1188.40 1.69529 0.847643 0.530567i $$-0.178021\pi$$
0.847643 + 0.530567i $$0.178021\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 124.087 0.175511
$$708$$ 0 0
$$709$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$710$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$720$$ 0 0
$$721$$ 34.8528 0.0483395
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 3248.53 4.48073
$$726$$ 0 0
$$727$$ −818.338 −1.12564 −0.562818 0.826581i $$-0.690283\pi$$
−0.562818 + 0.826581i $$0.690283\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$744$$ 0 0
$$745$$ 1938.76 2.60237
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ −441.605 −0.589592
$$750$$ 0 0
$$751$$ −234.310 −0.311997 −0.155998 0.987757i $$-0.549859\pi$$
−0.155998 + 0.987757i $$0.549859\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 1815.73 2.40494
$$756$$ 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
$$769$$ −672.087 −0.873975 −0.436987 0.899468i $$-0.643955\pi$$
−0.436987 + 0.899468i $$0.643955\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 1154.00 1.49288 0.746442 0.665450i $$-0.231760\pi$$
0.746442 + 0.665450i $$0.231760\pi$$
$$774$$ 0 0
$$775$$ −1522.03 −1.96391
$$776$$ 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
$$787$$ 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
$$797$$ 1453.10 1.82321 0.911607 0.411063i $$-0.134842\pi$$
0.911607 + 0.411063i $$0.134842\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −2060.71 −2.56626
$$804$$ 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
$$811$$ 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
$$821$$ −670.000 −0.816078 −0.408039 0.912965i $$-0.633787\pi$$
−0.408039 + 0.912965i $$0.633787\pi$$
$$822$$ 0 0
$$823$$ −1131.13 −1.37440 −0.687199 0.726469i $$-0.741160\pi$$
−0.687199 + 0.726469i $$0.741160\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 1546.00 1.86941 0.934704 0.355428i $$-0.115665\pi$$
0.934704 + 0.355428i $$0.115665\pi$$
$$828$$ 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
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$840$$ 0 0
$$841$$ 1659.00 1.97265
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −1603.01 −1.89706
$$846$$ 0 0
$$847$$ 1260.65 1.48837
$$848$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$864$$ 0 0
$$865$$ −3035.15 −3.50884
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −1274.29 −1.46639
$$870$$ 0 0
$$871$$ 0 0
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −1321.38 −1.51015
$$876$$ 0 0
$$877$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$888$$ 0 0
$$889$$ 75.4978 0.0849244
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ −3344.40 −3.73675
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ −1171.32 −1.30291
$$900$$ 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
$$913$$ −3336.29 −3.65421
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −672.249 −0.733096
$$918$$ 0 0
$$919$$ 1815.92 1.97598 0.987989 0.154523i $$-0.0493840\pi$$
0.987989 + 0.154523i $$0.0493840\pi$$
$$920$$ 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
$$929$$ 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
$$937$$ 239.293 0.255382 0.127691 0.991814i $$-0.459243\pi$$
0.127691 + 0.991814i $$0.459243\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1371.75 −1.45776 −0.728878 0.684644i $$-0.759958\pi$$
−0.728878 + 0.684644i $$0.759958\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −1840.94 −1.94397 −0.971985 0.235043i $$-0.924477\pi$$
−0.971985 + 0.235043i $$0.924477\pi$$
$$948$$ 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
$$961$$ −412.203 −0.428932
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −717.252 −0.743266
$$966$$ 0 0
$$967$$ 1218.04 1.25961 0.629805 0.776753i $$-0.283135\pi$$
0.629805 + 0.776753i $$0.283135\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −778.324 −0.801569 −0.400785 0.916172i $$-0.631262\pi$$
−0.400785 + 0.916172i $$0.631262\pi$$
$$972$$ 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 1.00000 $$0$$
−1.00000 $$\pi$$
$$984$$ 0 0
$$985$$ −1896.91 −1.92580
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 1921.37 1.93882 0.969408 0.245457i $$-0.0789379\pi$$
0.969408 + 0.245457i $$0.0789379\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 3775.00 3.79397
$$996$$ 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 864.3.h.a.593.1 2
3.2 odd 2 864.3.h.b.593.2 2
4.3 odd 2 216.3.h.a.53.1 2
8.3 odd 2 216.3.h.b.53.2 yes 2
8.5 even 2 864.3.h.b.593.2 2
12.11 even 2 216.3.h.b.53.2 yes 2
24.5 odd 2 CM 864.3.h.a.593.1 2
24.11 even 2 216.3.h.a.53.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.h.a.53.1 2 4.3 odd 2
216.3.h.a.53.1 2 24.11 even 2
216.3.h.b.53.2 yes 2 8.3 odd 2
216.3.h.b.53.2 yes 2 12.11 even 2
864.3.h.a.593.1 2 1.1 even 1 trivial
864.3.h.a.593.1 2 24.5 odd 2 CM
864.3.h.b.593.2 2 3.2 odd 2
864.3.h.b.593.2 2 8.5 even 2