# Properties

 Label 2304.3.h.d.2177.1 Level $2304$ Weight $3$ Character 2304.2177 Analytic conductor $62.779$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2304.h (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$62.7794529086$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 288) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## Embedding invariants

 Embedding label 2177.1 Root $$-0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 2304.2177 Dual form 2304.3.h.d.2177.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.41421 q^{5} +O(q^{10})$$ $$q-1.41421 q^{5} -24.0000i q^{13} +32.5269i q^{17} -23.0000 q^{25} -1.41421 q^{29} +70.0000i q^{37} -69.2965i q^{41} -49.0000 q^{49} +103.238 q^{53} +22.0000i q^{61} +33.9411i q^{65} +96.0000 q^{73} -46.0000i q^{85} +168.291i q^{89} -144.000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 92q^{25} - 196q^{49} + 384q^{73} - 576q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times$$.

 $$n$$ $$1279$$ $$1793$$ $$2053$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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
$$5$$ −1.41421 −0.282843 −0.141421 0.989949i $$-0.545167\pi$$
−0.141421 + 0.989949i $$0.545167\pi$$
$$6$$ 0 0
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$12$$ 0 0
$$13$$ − 24.0000i − 1.84615i −0.384615 0.923077i $$-0.625666\pi$$
0.384615 0.923077i $$-0.374334\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 32.5269i 1.91335i 0.291162 + 0.956674i $$0.405958\pi$$
−0.291162 + 0.956674i $$0.594042\pi$$
$$18$$ 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
$$25$$ −23.0000 −0.920000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −1.41421 −0.0487660 −0.0243830 0.999703i $$-0.507762\pi$$
−0.0243830 + 0.999703i $$0.507762\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 70.0000i 1.89189i 0.324324 + 0.945946i $$0.394863\pi$$
−0.324324 + 0.945946i $$0.605137\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ − 69.2965i − 1.69016i −0.534642 0.845079i $$-0.679553\pi$$
0.534642 0.845079i $$-0.320447\pi$$
$$42$$ 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
$$49$$ −49.0000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 103.238 1.94788 0.973940 0.226808i $$-0.0728289\pi$$
0.973940 + 0.226808i $$0.0728289\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 22.0000i 0.360656i 0.983607 + 0.180328i $$0.0577159\pi$$
−0.983607 + 0.180328i $$0.942284\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ 33.9411i 0.522171i
$$66$$ 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
$$73$$ 96.0000 1.31507 0.657534 0.753425i $$-0.271599\pi$$
0.657534 + 0.753425i $$0.271599\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ − 46.0000i − 0.541176i
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 168.291i 1.89091i 0.325746 + 0.945457i $$0.394385\pi$$
−0.325746 + 0.945457i $$0.605615\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −144.000 −1.48454 −0.742268 0.670103i $$-0.766250\pi$$
−0.742268 + 0.670103i $$0.766250\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 168.291 1.66625 0.833126 0.553084i $$-0.186549\pi$$
0.833126 + 0.553084i $$0.186549\pi$$
$$102$$ 0 0
$$103$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$108$$ 0 0
$$109$$ 120.000i 1.10092i 0.834862 + 0.550459i $$0.185547\pi$$
−0.834862 + 0.550459i $$0.814453\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 137.179i 1.21397i 0.794713 + 0.606985i $$0.207621\pi$$
−0.794713 + 0.606985i $$0.792379\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −121.000 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 67.8823 0.543058
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 272.943i 1.99229i 0.0877432 + 0.996143i $$0.472035\pi$$
−0.0877432 + 0.996143i $$0.527965\pi$$
$$138$$ 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
$$145$$ 2.00000 0.0137931
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −270.115 −1.81285 −0.906425 0.422366i $$-0.861200\pi$$
−0.906425 + 0.422366i $$0.861200\pi$$
$$150$$ 0 0
$$151$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 170.000i 1.08280i 0.840764 + 0.541401i $$0.182106\pi$$
−0.840764 + 0.541401i $$0.817894\pi$$
$$158$$ 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
$$169$$ −407.000 −2.40828
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 306.884 1.77390 0.886949 0.461867i $$-0.152820\pi$$
0.886949 + 0.461867i $$0.152820\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ 360.000i 1.98895i 0.104972 + 0.994475i $$0.466525\pi$$
−0.104972 + 0.994475i $$0.533475\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ − 98.9949i − 0.535108i
$$186$$ 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
$$193$$ 190.000 0.984456 0.492228 0.870466i $$-0.336183\pi$$
0.492228 + 0.870466i $$0.336183\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 236.174 1.19885 0.599426 0.800431i $$-0.295396\pi$$
0.599426 + 0.800431i $$0.295396\pi$$
$$198$$ 0 0
$$199$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 98.0000i 0.478049i
$$206$$ 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
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 780.646 3.53233
$$222$$ 0 0
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$228$$ 0 0
$$229$$ 120.000i 0.524017i 0.965066 + 0.262009i $$0.0843849\pi$$
−0.965066 + 0.262009i $$0.915615\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 442.649i 1.89978i 0.312584 + 0.949890i $$0.398806\pi$$
−0.312584 + 0.949890i $$0.601194\pi$$
$$234$$ 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
$$241$$ 240.000 0.995851 0.497925 0.867220i $$-0.334095\pi$$
0.497925 + 0.867220i $$0.334095\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 69.2965 0.282843
$$246$$ 0 0
$$247$$ 0 0
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 405.879i 1.57930i 0.613560 + 0.789648i $$0.289737\pi$$
−0.613560 + 0.789648i $$0.710263\pi$$
$$258$$ 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
$$265$$ −146.000 −0.550943
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −270.115 −1.00414 −0.502072 0.864826i $$-0.667429\pi$$
−0.502072 + 0.864826i $$0.667429\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 504.000i − 1.81949i −0.415162 0.909747i $$-0.636275\pi$$
0.415162 0.909747i $$-0.363725\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 100.409i 0.357328i 0.983910 + 0.178664i $$0.0571775\pi$$
−0.983910 + 0.178664i $$0.942822\pi$$
$$282$$ 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
$$289$$ −769.000 −2.66090
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 306.884 1.04739 0.523693 0.851907i $$-0.324554\pi$$
0.523693 + 0.851907i $$0.324554\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 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
$$305$$ − 31.1127i − 0.102009i
$$306$$ 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
$$313$$ −50.0000 −0.159744 −0.0798722 0.996805i $$-0.525451\pi$$
−0.0798722 + 0.996805i $$0.525451\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −541.644 −1.70866 −0.854328 0.519735i $$-0.826031\pi$$
−0.854328 + 0.519735i $$0.826031\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 552.000i 1.69846i
$$326$$ 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
$$337$$ −576.000 −1.70920 −0.854599 0.519288i $$-0.826197\pi$$
−0.854599 + 0.519288i $$0.826197\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ 598.000i 1.71347i 0.515759 + 0.856734i $$0.327510\pi$$
−0.515759 + 0.856734i $$0.672490\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ − 66.4680i − 0.188295i −0.995558 0.0941474i $$-0.969988\pi$$
0.995558 0.0941474i $$-0.0300125\pi$$
$$354$$ 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
$$365$$ −135.765 −0.371958
$$366$$ 0 0
$$367$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ − 550.000i − 1.47453i −0.675603 0.737265i $$-0.736117\pi$$
0.675603 0.737265i $$-0.263883\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 33.9411i 0.0900295i
$$378$$ 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
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 748.119 1.92319 0.961593 0.274481i $$-0.0885061\pi$$
0.961593 + 0.274481i $$0.0885061\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 650.000i − 1.63728i −0.574307 0.818640i $$-0.694729\pi$$
0.574307 0.818640i $$-0.305271\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 507.703i 1.26609i 0.774114 + 0.633046i $$0.218195\pi$$
−0.774114 + 0.633046i $$0.781805\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ 240.000 0.586797 0.293399 0.955990i $$-0.405214\pi$$
0.293399 + 0.955990i $$0.405214\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$420$$ 0 0
$$421$$ 840.000i 1.99525i 0.0688836 + 0.997625i $$0.478056\pi$$
−0.0688836 + 0.997625i $$0.521944\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ − 748.119i − 1.76028i
$$426$$ 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
$$433$$ −290.000 −0.669746 −0.334873 0.942263i $$-0.608693\pi$$
−0.334873 + 0.942263i $$0.608693\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 0 0
$$438$$ 0 0
$$439$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$444$$ 0 0
$$445$$ − 238.000i − 0.534831i
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ − 100.409i − 0.223628i −0.993729 0.111814i $$-0.964334\pi$$
0.993729 0.111814i $$-0.0356662\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 336.000 0.735230 0.367615 0.929978i $$-0.380174\pi$$
0.367615 + 0.929978i $$0.380174\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 168.291 0.365057 0.182529 0.983201i $$-0.441572\pi$$
0.182529 + 0.983201i $$0.441572\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$468$$ 0 0
$$469$$ 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
$$481$$ 1680.00 3.49272
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 203.647 0.419890
$$486$$ 0 0
$$487$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 0 0
$$493$$ − 46.0000i − 0.0933063i
$$494$$ 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
$$505$$ −238.000 −0.471287
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 337.997 0.664041 0.332021 0.943272i $$-0.392270\pi$$
0.332021 + 0.943272i $$0.392270\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ − 1016.82i − 1.95167i −0.218511 0.975835i $$-0.570120\pi$$
0.218511 0.975835i $$-0.429880\pi$$
$$522$$ 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
$$533$$ −1663.12 −3.12029
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ − 840.000i − 1.55268i −0.630314 0.776340i $$-0.717074\pi$$
0.630314 0.776340i $$-0.282926\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ − 169.706i − 0.311386i
$$546$$ 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
$$557$$ −985.707 −1.76967 −0.884836 0.465903i $$-0.845729\pi$$
−0.884836 + 0.465903i $$0.845729\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$564$$ 0 0
$$565$$ − 194.000i − 0.343363i
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 408.708i 0.718291i 0.933282 + 0.359146i $$0.116932\pi$$
−0.933282 + 0.359146i $$0.883068\pi$$
$$570$$ 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
$$577$$ −1150.00 −1.99307 −0.996534 0.0831889i $$-0.973490\pi$$
−0.996534 + 0.0831889i $$0.973490\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ − 137.179i − 0.231330i −0.993288 0.115665i $$-0.963100\pi$$
0.993288 0.115665i $$-0.0368999\pi$$
$$594$$ 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
$$601$$ −1102.00 −1.83361 −0.916805 0.399334i $$-0.869241\pi$$
−0.916805 + 0.399334i $$0.869241\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 171.120 0.282843
$$606$$ 0 0
$$607$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ 0 0
$$613$$ − 70.0000i − 0.114192i −0.998369 0.0570962i $$-0.981816\pi$$
0.998369 0.0570962i $$-0.0181842\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 711.349i 1.15292i 0.817127 + 0.576458i $$0.195566\pi$$
−0.817127 + 0.576458i $$0.804434\pi$$
$$618$$ 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
$$625$$ 479.000 0.766400
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ −2276.88 −3.61985
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1176.00i 1.84615i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ − 578.413i − 0.902361i −0.892433 0.451180i $$-0.851003\pi$$
0.892433 0.451180i $$-0.148997\pi$$
$$642$$ 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
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1254.41 −1.92099 −0.960496 0.278295i $$-0.910231\pi$$
−0.960496 + 0.278295i $$0.910231\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$660$$ 0 0
$$661$$ − 1178.00i − 1.78215i −0.453858 0.891074i $$-0.649953\pi$$
0.453858 0.891074i $$-0.350047\pi$$
$$662$$ 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
$$673$$ 770.000 1.14413 0.572065 0.820208i $$-0.306142\pi$$
0.572065 + 0.820208i $$0.306142\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 881.055 1.30141 0.650705 0.759330i $$-0.274473\pi$$
0.650705 + 0.759330i $$0.274473\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$684$$ 0 0
$$685$$ − 386.000i − 0.563504i
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ − 2477.70i − 3.59608i
$$690$$ 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
$$697$$ 2254.00 3.23386
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 1288.35 1.83787 0.918936 0.394406i $$-0.129050\pi$$
0.918936 + 0.394406i $$0.129050\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ − 1320.00i − 1.86178i −0.365303 0.930889i $$-0.619035\pi$$
0.365303 0.930889i $$-0.380965\pi$$
$$710$$ 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
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 32.5269 0.0448647
$$726$$ 0 0
$$727$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ − 216.000i − 0.294679i −0.989086 0.147340i $$-0.952929\pi$$
0.989086 0.147340i $$-0.0470711\pi$$
$$734$$ 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
$$745$$ 382.000 0.512752
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 936.000i − 1.23646i −0.785997 0.618230i $$-0.787850\pi$$
0.785997 0.618230i $$-0.212150\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ − 1019.65i − 1.33988i −0.742416 0.669940i $$-0.766320\pi$$
0.742416 0.669940i $$-0.233680\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 962.000 1.25098 0.625488 0.780234i $$-0.284900\pi$$
0.625488 + 0.780234i $$0.284900\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ 782.060 1.01172 0.505860 0.862615i $$-0.331175\pi$$
0.505860 + 0.862615i $$0.331175\pi$$
$$774$$ 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
$$785$$ − 240.416i − 0.306263i
$$786$$ 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
$$793$$ 528.000 0.665826
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −1593.82 −1.99977 −0.999886 0.0150826i $$-0.995199\pi$$
−0.999886 + 0.0150826i $$0.995199\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 677.408i 0.837340i 0.908138 + 0.418670i $$0.137504\pi$$
−0.908138 + 0.418670i $$0.862496\pi$$
$$810$$ 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
$$821$$ −1596.65 −1.94476 −0.972379 0.233406i $$-0.925013\pi$$
−0.972379 + 0.233406i $$0.925013\pi$$
$$822$$ 0 0
$$823$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$828$$ 0 0
$$829$$ 1080.00i 1.30277i 0.758745 + 0.651387i $$0.225813\pi$$
−0.758745 + 0.651387i $$0.774187\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ − 1593.82i − 1.91335i
$$834$$ 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
$$841$$ −839.000 −0.997622
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 575.585 0.681166
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ − 410.000i − 0.480657i −0.970692 0.240328i $$-0.922745\pi$$
0.970692 0.240328i $$-0.0772551\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 1494.82i − 1.74425i −0.489282 0.872126i $$-0.662741\pi$$
0.489282 0.872126i $$-0.337259\pi$$
$$858$$ 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
$$865$$ −434.000 −0.501734
$$866$$ 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
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 1610.00i − 1.83580i −0.396807 0.917902i $$-0.629882\pi$$
0.396807 0.917902i $$-0.370118\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 609.526i 0.691857i 0.938261 + 0.345929i $$0.112436\pi$$
−0.938261 + 0.345929i $$0.887564\pi$$
$$882$$ 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
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 0 0
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 3358.00i 3.72697i
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ − 509.117i − 0.562560i
$$906$$ 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
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ − 1610.00i − 1.74054i
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 1118.64i 1.20414i 0.798445 + 0.602068i $$0.205657\pi$$
−0.798445 + 0.602068i $$0.794343\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −430.000 −0.458911 −0.229456 0.973319i $$-0.573695\pi$$
−0.229456 + 0.973319i $$0.573695\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −1868.18 −1.98531 −0.992655 0.120982i $$-0.961396\pi$$
−0.992655 + 0.120982i $$0.961396\pi$$
$$942$$ 0 0
$$943$$ 0 0
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$948$$ 0 0
$$949$$ − 2304.00i − 2.42782i
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ − 1899.29i − 1.99296i −0.0838437 0.996479i $$-0.526720\pi$$
0.0838437 0.996479i $$-0.473280\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −961.000 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ −268.701 −0.278446
$$966$$ 0 0
$$967$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 985.707i 1.00891i 0.863437 + 0.504456i $$0.168307\pi$$
−0.863437 + 0.504456i $$0.831693\pi$$
$$978$$ 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
$$985$$ −334.000 −0.339086
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 1850.00i 1.85557i 0.373119 + 0.927783i $$0.378288\pi$$
−0.373119 + 0.927783i $$0.621712\pi$$
$$998$$ 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 2304.3.h.d.2177.1 4
3.2 odd 2 inner 2304.3.h.d.2177.3 4
4.3 odd 2 CM 2304.3.h.d.2177.1 4
8.3 odd 2 inner 2304.3.h.d.2177.4 4
8.5 even 2 inner 2304.3.h.d.2177.4 4
12.11 even 2 inner 2304.3.h.d.2177.3 4
16.3 odd 4 288.3.e.c.161.2 yes 2
16.5 even 4 576.3.e.d.449.1 2
16.11 odd 4 576.3.e.d.449.1 2
16.13 even 4 288.3.e.c.161.2 yes 2
24.5 odd 2 inner 2304.3.h.d.2177.2 4
24.11 even 2 inner 2304.3.h.d.2177.2 4
48.5 odd 4 576.3.e.d.449.2 2
48.11 even 4 576.3.e.d.449.2 2
48.29 odd 4 288.3.e.c.161.1 2
48.35 even 4 288.3.e.c.161.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.e.c.161.1 2 48.29 odd 4
288.3.e.c.161.1 2 48.35 even 4
288.3.e.c.161.2 yes 2 16.3 odd 4
288.3.e.c.161.2 yes 2 16.13 even 4
576.3.e.d.449.1 2 16.5 even 4
576.3.e.d.449.1 2 16.11 odd 4
576.3.e.d.449.2 2 48.5 odd 4
576.3.e.d.449.2 2 48.11 even 4
2304.3.h.d.2177.1 4 1.1 even 1 trivial
2304.3.h.d.2177.1 4 4.3 odd 2 CM
2304.3.h.d.2177.2 4 24.5 odd 2 inner
2304.3.h.d.2177.2 4 24.11 even 2 inner
2304.3.h.d.2177.3 4 3.2 odd 2 inner
2304.3.h.d.2177.3 4 12.11 even 2 inner
2304.3.h.d.2177.4 4 8.3 odd 2 inner
2304.3.h.d.2177.4 4 8.5 even 2 inner