# Properties

 Label 576.4.a.p.1.1 Level $576$ Weight $4$ Character 576.1 Self dual yes Analytic conductor $33.985$ Analytic rank $1$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$576 = 2^{6} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 576.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$33.9851001633$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 288) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 576.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+4.00000 q^{5} +O(q^{10})$$ $$q+4.00000 q^{5} -18.0000 q^{13} -104.000 q^{17} -109.000 q^{25} +284.000 q^{29} -214.000 q^{37} -472.000 q^{41} -343.000 q^{49} +572.000 q^{53} -830.000 q^{61} -72.0000 q^{65} -1098.00 q^{73} -416.000 q^{85} +176.000 q^{89} -594.000 q^{97} +O(q^{100})$$

## 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
$$5$$ 4.00000 0.357771 0.178885 0.983870i $$-0.442751\pi$$
0.178885 + 0.983870i $$0.442751\pi$$
$$6$$ 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
$$13$$ −18.0000 −0.384023 −0.192012 0.981393i $$-0.561501\pi$$
−0.192012 + 0.981393i $$0.561501\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −104.000 −1.48375 −0.741874 0.670540i $$-0.766063\pi$$
−0.741874 + 0.670540i $$0.766063\pi$$
$$18$$ 0 0
$$19$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ 0 0
$$25$$ −109.000 −0.872000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 284.000 1.81853 0.909267 0.416214i $$-0.136643\pi$$
0.909267 + 0.416214i $$0.136643\pi$$
$$30$$ 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
$$37$$ −214.000 −0.950848 −0.475424 0.879757i $$-0.657705\pi$$
−0.475424 + 0.879757i $$0.657705\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −472.000 −1.79790 −0.898951 0.438048i $$-0.855670\pi$$
−0.898951 + 0.438048i $$0.855670\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 0 0
$$49$$ −343.000 −1.00000
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 572.000 1.48246 0.741229 0.671253i $$-0.234243\pi$$
0.741229 + 0.671253i $$0.234243\pi$$
$$54$$ 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
$$61$$ −830.000 −1.74214 −0.871071 0.491158i $$-0.836574\pi$$
−0.871071 + 0.491158i $$0.836574\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −72.0000 −0.137392
$$66$$ 0 0
$$67$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ −1098.00 −1.76043 −0.880214 0.474578i $$-0.842601\pi$$
−0.880214 + 0.474578i $$0.842601\pi$$
$$74$$ 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
$$85$$ −416.000 −0.530842
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ 176.000 0.209618 0.104809 0.994492i $$-0.466577\pi$$
0.104809 + 0.994492i $$0.466577\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −594.000 −0.621769 −0.310884 0.950448i $$-0.600625\pi$$
−0.310884 + 0.950448i $$0.600625\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1940.00 −1.91126 −0.955630 0.294570i $$-0.904823\pi$$
−0.955630 + 0.294570i $$0.904823\pi$$
$$102$$ 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
$$109$$ −1746.00 −1.53428 −0.767140 0.641480i $$-0.778321\pi$$
−0.767140 + 0.641480i $$0.778321\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1328.00 1.10556 0.552778 0.833329i $$-0.313568\pi$$
0.552778 + 0.833329i $$0.313568\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1331.00 −1.00000
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −936.000 −0.669747
$$126$$ 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
$$137$$ 2776.00 1.73117 0.865583 0.500766i $$-0.166948\pi$$
0.865583 + 0.500766i $$0.166948\pi$$
$$138$$ 0 0
$$139$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1136.00 0.650618
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −940.000 −0.516831 −0.258415 0.966034i $$-0.583200\pi$$
−0.258415 + 0.966034i $$0.583200\pi$$
$$150$$ 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
$$157$$ −286.000 −0.145384 −0.0726920 0.997354i $$-0.523159\pi$$
−0.0726920 + 0.997354i $$0.523159\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −1873.00 −0.852526
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ −2012.00 −0.884217 −0.442108 0.896962i $$-0.645769\pi$$
−0.442108 + 0.896962i $$0.645769\pi$$
$$174$$ 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
$$181$$ 3942.00 1.61882 0.809410 0.587243i $$-0.199787\pi$$
0.809410 + 0.587243i $$0.199787\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −856.000 −0.340186
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 0 0
$$193$$ 5362.00 1.99982 0.999910 0.0134266i $$-0.00427395\pi$$
0.999910 + 0.0134266i $$0.00427395\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 5404.00 1.95441 0.977206 0.212295i $$-0.0680936\pi$$
0.977206 + 0.212295i $$0.0680936\pi$$
$$198$$ 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
$$205$$ −1888.00 −0.643237
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1872.00 0.569793
$$222$$ 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
$$229$$ 6390.00 1.84394 0.921972 0.387257i $$-0.126577\pi$$
0.921972 + 0.387257i $$0.126577\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −7088.00 −1.99292 −0.996460 0.0840693i $$-0.973208\pi$$
−0.996460 + 0.0840693i $$0.973208\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 0 0
$$241$$ 5310.00 1.41928 0.709641 0.704563i $$-0.248857\pi$$
0.709641 + 0.704563i $$0.248857\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ −1372.00 −0.357771
$$246$$ 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
$$257$$ −8096.00 −1.96504 −0.982519 0.186164i $$-0.940394\pi$$
−0.982519 + 0.186164i $$0.940394\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ 2288.00 0.530380
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ 8140.00 1.84500 0.922499 0.385999i $$-0.126143\pi$$
0.922499 + 0.385999i $$0.126143\pi$$
$$270$$ 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
$$277$$ 9126.00 1.97952 0.989762 0.142727i $$-0.0455871\pi$$
0.989762 + 0.142727i $$0.0455871\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 5792.00 1.22961 0.614807 0.788677i $$-0.289234\pi$$
0.614807 + 0.788677i $$0.289234\pi$$
$$282$$ 0 0
$$283$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 5903.00 1.20151
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 3452.00 0.688287 0.344143 0.938917i $$-0.388169\pi$$
0.344143 + 0.938917i $$0.388169\pi$$
$$294$$ 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
$$305$$ −3320.00 −0.623287
$$306$$ 0 0
$$307$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$312$$ 0 0
$$313$$ −6838.00 −1.23485 −0.617423 0.786632i $$-0.711823\pi$$
−0.617423 + 0.786632i $$0.711823\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −4676.00 −0.828487 −0.414243 0.910166i $$-0.635954\pi$$
−0.414243 + 0.910166i $$0.635954\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 0 0
$$324$$ 0 0
$$325$$ 1962.00 0.334868
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 12366.0 1.99887 0.999435 0.0336216i $$-0.0107041\pi$$
0.999435 + 0.0336216i $$0.0107041\pi$$
$$338$$ 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
$$349$$ −9470.00 −1.45249 −0.726243 0.687438i $$-0.758735\pi$$
−0.726243 + 0.687438i $$0.758735\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ 12848.0 1.93720 0.968598 0.248633i $$-0.0799813\pi$$
0.968598 + 0.248633i $$0.0799813\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$360$$ 0 0
$$361$$ −6859.00 −1.00000
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −4392.00 −0.629830
$$366$$ 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
$$373$$ 12922.0 1.79377 0.896884 0.442265i $$-0.145825\pi$$
0.896884 + 0.442265i $$0.145825\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −5112.00 −0.698359
$$378$$ 0 0
$$379$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −15340.0 −1.99941 −0.999703 0.0243735i $$-0.992241\pi$$
−0.999703 + 0.0243735i $$0.992241\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −9614.00 −1.21540 −0.607699 0.794168i $$-0.707907\pi$$
−0.607699 + 0.794168i $$0.707907\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 15880.0 1.97758 0.988790 0.149315i $$-0.0477068\pi$$
0.988790 + 0.149315i $$0.0477068\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −7146.00 −0.863929 −0.431964 0.901891i $$-0.642179\pi$$
−0.431964 + 0.901891i $$0.642179\pi$$
$$410$$ 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
$$421$$ −10890.0 −1.26068 −0.630340 0.776319i $$-0.717084\pi$$
−0.630340 + 0.776319i $$0.717084\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ 11336.0 1.29383
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 0 0
$$433$$ −4862.00 −0.539614 −0.269807 0.962914i $$-0.586960\pi$$
−0.269807 + 0.962914i $$0.586960\pi$$
$$434$$ 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
$$445$$ 704.000 0.0749951
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −10120.0 −1.06368 −0.531840 0.846845i $$-0.678499\pi$$
−0.531840 + 0.846845i $$0.678499\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −16506.0 −1.68954 −0.844768 0.535132i $$-0.820262\pi$$
−0.844768 + 0.535132i $$0.820262\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −19660.0 −1.98624 −0.993121 0.117093i $$-0.962642\pi$$
−0.993121 + 0.117093i $$0.962642\pi$$
$$462$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ 3852.00 0.365148
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −2376.00 −0.222451
$$486$$ 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
$$493$$ −29536.0 −2.69824
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ −7760.00 −0.683793
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ 17996.0 1.56711 0.783555 0.621323i $$-0.213404\pi$$
0.783555 + 0.621323i $$0.213404\pi$$
$$510$$ 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
$$521$$ 1480.00 0.124453 0.0622265 0.998062i $$-0.480180\pi$$
0.0622265 + 0.998062i $$0.480180\pi$$
$$522$$ 0 0
$$523$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −12167.0 −1.00000
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 8496.00 0.690436
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −5922.00 −0.470622 −0.235311 0.971920i $$-0.575611\pi$$
−0.235311 + 0.971920i $$0.575611\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −6984.00 −0.548921
$$546$$ 0 0
$$547$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 0 0
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 24836.0 1.88929 0.944646 0.328093i $$-0.106406\pi$$
0.944646 + 0.328093i $$0.106406\pi$$
$$558$$ 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
$$565$$ 5312.00 0.395535
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −4280.00 −0.315337 −0.157669 0.987492i $$-0.550398\pi$$
−0.157669 + 0.987492i $$0.550398\pi$$
$$570$$ 0 0
$$571$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −3454.00 −0.249206 −0.124603 0.992207i $$-0.539766\pi$$
−0.124603 + 0.992207i $$0.539766\pi$$
$$578$$ 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
$$593$$ −24368.0 −1.68748 −0.843738 0.536755i $$-0.819650\pi$$
−0.843738 + 0.536755i $$0.819650\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −17030.0 −1.15585 −0.577927 0.816089i $$-0.696138\pi$$
−0.577927 + 0.816089i $$0.696138\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ −5324.00 −0.357771
$$606$$ 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
$$613$$ −23222.0 −1.53006 −0.765031 0.643994i $$-0.777276\pi$$
−0.765031 + 0.643994i $$0.777276\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −26464.0 −1.72674 −0.863372 0.504569i $$-0.831652\pi$$
−0.863372 + 0.504569i $$0.831652\pi$$
$$618$$ 0 0
$$619$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 9881.00 0.632384
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 22256.0 1.41082
$$630$$ 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
$$637$$ 6174.00 0.384023
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 14872.0 0.916394 0.458197 0.888851i $$-0.348495\pi$$
0.458197 + 0.888851i $$0.348495\pi$$
$$642$$ 0 0
$$643$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 1012.00 0.0606472 0.0303236 0.999540i $$-0.490346\pi$$
0.0303236 + 0.999540i $$0.490346\pi$$
$$654$$ 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
$$661$$ 25850.0 1.52110 0.760551 0.649278i $$-0.224929\pi$$
0.760551 + 0.649278i $$0.224929\pi$$
$$662$$ 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
$$673$$ −4462.00 −0.255568 −0.127784 0.991802i $$-0.540786\pi$$
−0.127784 + 0.991802i $$0.540786\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 34996.0 1.98671 0.993357 0.115072i $$-0.0367100\pi$$
0.993357 + 0.115072i $$0.0367100\pi$$
$$678$$ 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
$$685$$ 11104.0 0.619361
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −10296.0 −0.569298
$$690$$ 0 0
$$691$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 49088.0 2.66763
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −31252.0 −1.68384 −0.841920 0.539602i $$-0.818575\pi$$
−0.841920 + 0.539602i $$0.818575\pi$$
$$702$$ 0 0
$$703$$ 0 0
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −36810.0 −1.94983 −0.974914 0.222580i $$-0.928552\pi$$
−0.974914 + 0.222580i $$0.928552\pi$$
$$710$$ 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 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ −30956.0 −1.58576
$$726$$ 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
$$733$$ 38718.0 1.95100 0.975499 0.220003i $$-0.0706066\pi$$
0.975499 + 0.220003i $$0.0706066\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ 0 0
$$745$$ −3760.00 −0.184907
$$746$$ 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
$$757$$ 35046.0 1.68265 0.841327 0.540527i $$-0.181775\pi$$
0.841327 + 0.540527i $$0.181775\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 27320.0 1.30138 0.650689 0.759344i $$-0.274480\pi$$
0.650689 + 0.759344i $$0.274480\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 0 0
$$768$$ 0 0
$$769$$ 9650.00 0.452520 0.226260 0.974067i $$-0.427350\pi$$
0.226260 + 0.974067i $$0.427350\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −16852.0 −0.784119 −0.392060 0.919940i $$-0.628237\pi$$
−0.392060 + 0.919940i $$0.628237\pi$$
$$774$$ 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
$$785$$ −1144.00 −0.0520142
$$786$$ 0 0
$$787$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 14940.0 0.669023
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 16276.0 0.723370 0.361685 0.932300i $$-0.382202\pi$$
0.361685 + 0.932300i $$0.382202\pi$$
$$798$$ 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
$$809$$ 39704.0 1.72549 0.862743 0.505643i $$-0.168745\pi$$
0.862743 + 0.505643i $$0.168745\pi$$
$$810$$ 0 0
$$811$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$812$$ 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
$$821$$ −47012.0 −1.99845 −0.999227 0.0393212i $$-0.987480\pi$$
−0.999227 + 0.0393212i $$0.987480\pi$$
$$822$$ 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
$$829$$ 23166.0 0.970553 0.485276 0.874361i $$-0.338719\pi$$
0.485276 + 0.874361i $$0.338719\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ 35672.0 1.48375
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$840$$ 0 0
$$841$$ 56267.0 2.30706
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ −7492.00 −0.305009
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 20378.0 0.817971 0.408986 0.912541i $$-0.365883\pi$$
0.408986 + 0.912541i $$0.365883\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −20056.0 −0.799416 −0.399708 0.916642i $$-0.630889\pi$$
−0.399708 + 0.916642i $$0.630889\pi$$
$$858$$ 0 0
$$859$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$864$$ 0 0
$$865$$ −8048.00 −0.316347
$$866$$ 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
$$877$$ 42514.0 1.63694 0.818470 0.574550i $$-0.194823\pi$$
0.818470 + 0.574550i $$0.194823\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 51808.0 1.98122 0.990611 0.136714i $$-0.0436541\pi$$
0.990611 + 0.136714i $$0.0436541\pi$$
$$882$$ 0 0
$$883$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$888$$ 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
$$901$$ −59488.0 −2.19959
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 15768.0 0.579167
$$906$$ 0 0
$$907$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ 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
$$925$$ 23326.0 0.829140
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −47480.0 −1.67682 −0.838411 0.545038i $$-0.816515\pi$$
−0.838411 + 0.545038i $$0.816515\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 51946.0 1.81110 0.905551 0.424238i $$-0.139458\pi$$
0.905551 + 0.424238i $$0.139458\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −48460.0 −1.67880 −0.839400 0.543514i $$-0.817093\pi$$
−0.839400 + 0.543514i $$0.817093\pi$$
$$942$$ 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
$$949$$ 19764.0 0.676045
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −15512.0 −0.527264 −0.263632 0.964623i $$-0.584921\pi$$
−0.263632 + 0.964623i $$0.584921\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −29791.0 −1.00000
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 21448.0 0.715477
$$966$$ 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
$$977$$ −22936.0 −0.751062 −0.375531 0.926810i $$-0.622540\pi$$
−0.375531 + 0.926810i $$0.622540\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$984$$ 0 0
$$985$$ 21616.0 0.699232
$$986$$ 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
$$997$$ −52886.0 −1.67996 −0.839978 0.542620i $$-0.817432\pi$$
−0.839978 + 0.542620i $$0.817432\pi$$
$$998$$ 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 576.4.a.p.1.1 1
3.2 odd 2 576.4.a.i.1.1 1
4.3 odd 2 CM 576.4.a.p.1.1 1
8.3 odd 2 288.4.a.d.1.1 1
8.5 even 2 288.4.a.d.1.1 1
12.11 even 2 576.4.a.i.1.1 1
24.5 odd 2 288.4.a.g.1.1 yes 1
24.11 even 2 288.4.a.g.1.1 yes 1

By twisted newform
Twist Min Dim Char Parity Ord Type
288.4.a.d.1.1 1 8.3 odd 2
288.4.a.d.1.1 1 8.5 even 2
288.4.a.g.1.1 yes 1 24.5 odd 2
288.4.a.g.1.1 yes 1 24.11 even 2
576.4.a.i.1.1 1 3.2 odd 2
576.4.a.i.1.1 1 12.11 even 2
576.4.a.p.1.1 1 1.1 even 1 trivial
576.4.a.p.1.1 1 4.3 odd 2 CM