LMFDB - Embedded newform 1764.3.d.b.685.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,3,Mod(685,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.685"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			685.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1764.685
Dual form			1764.3.d.b.685.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$-1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 25.9808i	− 1.99852i	−0.0384615	−	0.999260i	$-0.512246\pi$
$13$	− 25.9808i	− 1.99852i	0.0384615	−	0.999260i	$-0.487754\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	8.66025i	0.455803i	0.973684	+	0.227901i	$0.0731864\pi$
$19$	8.66025i	0.455803i	−0.973684	+	0.227901i	$0.926814\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	25.0000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	19.0526i	0.614599i	0.951613	+	0.307299i	$0.0994253\pi$
$31$	19.0526i	0.614599i	−0.951613	+	0.307299i	$0.900575\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−47.0000	−1.27027	−0.635135	−	0.772401i	$-0.719056\pi$
$37$	−47.0000	−1.27027	−0.635135	+	0.772401i	$0.719056\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−83.0000	−1.93023	−0.965116	−	0.261822i	$-0.915677\pi$
$43$	−83.0000	−1.93023	−0.965116	+	0.261822i	$0.915677\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	− 96.9948i	− 1.59008i	−0.606557	−	0.795040i	$-0.707450\pi$
$61$	− 96.9948i	− 1.59008i	0.606557	−	0.795040i	$-0.292550\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−109.000	−1.62687	−0.813433	−	0.581659i	$-0.802404\pi$
$67$	−109.000	−1.62687	−0.813433	+	0.581659i	$0.802404\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	29.4449i	0.403354i	0.979452	+	0.201677i	$0.0646392\pi$
$73$	29.4449i	0.403354i	−0.979452	+	0.201677i	$0.935361\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	131.000	1.65823	0.829114	−	0.559080i	$-0.188845\pi$
$79$	131.000	1.65823	0.829114	+	0.559080i	$0.188845\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	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
$97$	− 193.990i	− 1.99989i	−0.0103093	−	0.999947i	$-0.503282\pi$
$97$	− 193.990i	− 1.99989i	0.0103093	−	0.999947i	$-0.496718\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 202.650i	− 1.96748i	−0.179612	−	0.983738i	$-0.557484\pi$
$103$	− 202.650i	− 1.96748i	0.179612	−	0.983738i	$-0.442516\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−143.000	−1.31193	−0.655963	−	0.754793i	$-0.727737\pi$
$109$	−143.000	−1.31193	−0.655963	+	0.754793i	$0.727737\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	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
$125$	0	0
$126$	0	0
$127$	−253.000	−1.99213	−0.996063	−	0.0886483i	$-0.971745\pi$
$127$	−253.000	−1.99213	−0.996063	+	0.0886483i	$0.971745\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	− 119.512i	− 0.859795i	−0.902878	−	0.429898i	$-0.858550\pi$
$139$	− 119.512i	− 0.859795i	0.902878	−	0.429898i	$-0.141450\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	286.000	1.89404	0.947020	−	0.321175i	$-0.104078\pi$
$151$	286.000	1.89404	0.947020	+	0.321175i	$0.104078\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	290.985i	1.85340i	0.375796	+	0.926702i	$0.377369\pi$
$157$	290.985i	1.85340i	−0.375796	+	0.926702i	$0.622631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−262.000	−1.60736	−0.803681	−	0.595060i	$-0.797128\pi$
$163$	−262.000	−1.60736	−0.803681	+	0.595060i	$0.797128\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−506.000	−2.99408
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	− 361.999i	− 1.99999i	−0.00276243	−	0.999996i	$-0.500879\pi$
$181$	− 361.999i	− 1.99999i	0.00276243	−	0.999996i	$-0.499121\pi$
$182$	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		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−143.000	−0.740933	−0.370466	−	0.928846i	$-0.620802\pi$
$193$	−143.000	−0.740933	−0.370466	+	0.928846i	$0.620802\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	96.9948i	0.487411i	0.969849	+	0.243706i	$0.0783631\pi$
$199$	96.9948i	0.487411i	−0.969849	+	0.243706i	$0.921637\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	166.000	0.786730	0.393365	−	0.919382i	$-0.371311\pi$
$211$	166.000	0.786730	0.393365	+	0.919382i	$0.371311\pi$
$212$	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
$221$	0	0
$222$	0	0
$223$	− 290.985i	− 1.30486i	−0.757848	−	0.652432i	$-0.773749\pi$
$223$	− 290.985i	− 1.30486i	0.757848	−	0.652432i	$-0.226251\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 251.147i	− 1.09671i	−0.836245	−	0.548357i	$-0.815254\pi$
$229$	− 251.147i	− 1.09671i	0.836245	−	0.548357i	$-0.184746\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	− 387.979i	− 1.60987i	−0.593361	−	0.804936i	$-0.702199\pi$
$241$	− 387.979i	− 1.60987i	0.593361	−	0.804936i	$-0.297801\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	225.000	0.910931
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	− 484.974i	− 1.78957i	−0.446494	−	0.894786i	$-0.647328\pi$
$271$	− 484.974i	− 1.78957i	0.446494	−	0.894786i	$-0.352672\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−529.000	−1.90975	−0.954874	−	0.297012i	$-0.904010\pi$
$277$	−529.000	−1.90975	−0.954874	+	0.297012i	$0.904010\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	562.917i	1.98910i	0.104240	+	0.994552i	$0.466759\pi$
$283$	562.917i	1.98910i	−0.104240	+	0.994552i	$0.533241\pi$
$284$	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
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	559.452i	1.82232i	0.412052	+	0.911160i	$0.364812\pi$
$307$	559.452i	1.82232i	−0.412052	+	0.911160i	$0.635188\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	427.817i	1.36683i	0.730032	+	0.683413i	$0.239505\pi$
$313$	427.817i	1.36683i	−0.730032	+	0.683413i	$0.760495\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 649.519i	− 1.99852i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−299.000	−0.903323	−0.451662	−	0.892189i	$-0.649169\pi$
$331$	−299.000	−0.903323	−0.451662	+	0.892189i	$0.649169\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−167.000	−0.495549	−0.247774	−	0.968818i	$-0.579699\pi$
$337$	−167.000	−0.495549	−0.247774	+	0.968818i	$0.579699\pi$
$338$	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	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	484.974i	1.38961i	0.719198	+	0.694805i	$0.244510\pi$
$349$	484.974i	1.38961i	−0.719198	+	0.694805i	$0.755490\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	286.000	0.792244
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	698.016i	1.90195i	0.309264	+	0.950976i	$0.399917\pi$
$367$	698.016i	1.90195i	−0.309264	+	0.950976i	$0.600083\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−121.000	−0.324397	−0.162198	−	0.986758i	$-0.551858\pi$
$373$	−121.000	−0.324397	−0.162198	+	0.986758i	$0.551858\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−83.0000	−0.218997	−0.109499	−	0.993987i	$-0.534925\pi$
$379$	−83.0000	−0.218997	−0.109499	+	0.993987i	$0.534925\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 39.8372i	− 0.100346i	−0.998741	−	0.0501728i	$-0.984023\pi$
$397$	− 39.8372i	− 0.100346i	0.998741	−	0.0501728i	$-0.0159772\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	495.000	1.22829
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	− 805.404i	− 1.96920i	−0.174817	−	0.984601i	$-0.555933\pi$
$409$	− 805.404i	− 1.96920i	0.174817	−	0.984601i	$-0.444067\pi$
$410$	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			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	839.000	1.99287	0.996437	−	0.0843398i	$-0.0268781\pi$
$421$	839.000	1.99287	0.996437	+	0.0843398i	$0.0268781\pi$
$422$	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		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 704.945i	− 1.62805i	−0.580831	−	0.814024i	$-0.697272\pi$
$433$	− 704.945i	− 1.62805i	0.580831	−	0.814024i	$-0.302728\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	872.954i	1.98850i	0.107062	+	0.994252i	$0.465856\pi$
$439$	872.954i	1.98850i	−0.107062	+	0.994252i	$0.534144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−47.0000	−0.102845	−0.0514223	−	0.998677i	$-0.516375\pi$
$457$	−47.0000	−0.102845	−0.0514223	+	0.998677i	$0.516375\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	923.000	1.99352	0.996760	−	0.0804300i	$-0.0256293\pi$
$463$	923.000	1.99352	0.996760	+	0.0804300i	$0.0256293\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	216.506i	0.455803i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	1221.10i	2.53866i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	613.000	1.25873	0.629363	−	0.777111i	$-0.283316\pi$
$487$	613.000	1.25873	0.629363	+	0.777111i	$0.283316\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	877.000	1.75752	0.878758	−	0.477269i	$-0.158373\pi$
$499$	877.000	1.75752	0.878758	+	0.477269i	$0.158373\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	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
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 670.304i	− 1.28165i	−0.767686	−	0.640826i	$-0.778592\pi$
$523$	− 670.304i	− 1.28165i	0.767686	−	0.640826i	$-0.221408\pi$
$524$	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
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−793.000	−1.46580	−0.732902	−	0.680334i	$-0.761835\pi$
$541$	−793.000	−1.46580	−0.732902	+	0.680334i	$0.761835\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	506.000	0.925046	0.462523	−	0.886607i	$-0.346944\pi$
$547$	506.000	0.925046	0.462523	+	0.886607i	$0.346944\pi$
$548$	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
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	2156.40i	3.85761i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	0	0
$571$	1067.00	1.86865	0.934326	−	0.356420i	$-0.116003\pi$
$571$	1067.00	1.86865	0.934326	+	0.356420i	$0.116003\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 514.419i	− 0.891541i	−0.895147	−	0.445770i	$-0.852930\pi$
$577$	− 514.419i	− 0.891541i	0.895147	−	0.445770i	$-0.147070\pi$
$578$	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	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	−165.000	−0.280136
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	995.929i	1.65712i	0.559900	+	0.828560i	$0.310839\pi$
$601$	995.929i	1.65712i	−0.559900	+	0.828560i	$0.689161\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1155.28i	1.90326i	0.307249	+	0.951629i	$0.400592\pi$
$607$	1155.28i	1.90326i	−0.307249	+	0.951629i	$0.599408\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1126.00	−1.83687	−0.918434	−	0.395574i	$-0.870546\pi$
$613$	−1126.00	−1.83687	−0.918434	+	0.395574i	$0.870546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	− 795.011i	− 1.28435i	−0.766559	−	0.642174i	$-0.778033\pi$
$619$	− 795.011i	− 1.28435i	0.766559	−	0.642174i	$-0.221967\pi$
$620$	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
$631$	674.000	1.06815	0.534073	−	0.845438i	$-0.320661\pi$
$631$	674.000	1.06815	0.534073	+	0.845438i	$0.320661\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	895.470i	1.39264i	0.717729	+	0.696322i	$0.245181\pi$
$643$	895.470i	1.39264i	−0.717729	+	0.696322i	$0.754819\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	552.524i	0.835891i	0.908472	+	0.417946i	$0.137250\pi$
$661$	552.524i	0.835891i	−0.908472	+	0.417946i	$0.862750\pi$
$662$	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
$673$	1177.00	1.74889	0.874443	−	0.485129i	$-0.161227\pi$
$673$	1177.00	1.74889	0.874443	+	0.485129i	$0.161227\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 1349.27i	− 1.95263i	−0.216353	−	0.976315i	$-0.569416\pi$
$691$	− 1349.27i	− 1.95263i	0.216353	−	0.976315i	$-0.430584\pi$
$692$	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
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	− 407.032i	− 0.578993i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	934.000	1.31735	0.658674	−	0.752428i	$-0.271118\pi$
$709$	934.000	1.31735	0.658674	+	0.752428i	$0.271118\pi$
$710$	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			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 268.468i	− 0.369282i	−0.982806	−	0.184641i	$-0.940888\pi$
$727$	− 268.468i	− 0.369282i	0.982806	−	0.184641i	$-0.0591122\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1415.09i	1.93054i	0.261255	+	0.965270i	$0.415864\pi$
$733$	1415.09i	1.93054i	−0.261255	+	0.965270i	$0.584136\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−109.000	−0.147497	−0.0737483	−	0.997277i	$-0.523496\pi$
$739$	−109.000	−0.147497	−0.0737483	+	0.997277i	$0.523496\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1381.00	1.83888	0.919441	−	0.393229i	$-0.128642\pi$
$751$	1381.00	1.83888	0.919441	+	0.393229i	$0.128642\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	838.000	1.10700	0.553501	−	0.832849i	$-0.313292\pi$
$757$	838.000	1.10700	0.553501	+	0.832849i	$0.313292\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	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
$769$	− 1383.91i	− 1.79962i	−0.436281	−	0.899811i	$-0.643705\pi$
$769$	− 1383.91i	− 1.79962i	0.436281	−	0.899811i	$-0.356295\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	476.314i	0.614599i
$776$	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
$787$	− 193.990i	− 0.246493i	−0.992376	−	0.123246i	$-0.960669\pi$
$787$	− 193.990i	− 0.246493i	0.992376	−	0.123246i	$-0.0393305\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2520.00	−3.17781
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	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
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	581.969i	0.717594i	0.933416	+	0.358797i	$0.116813\pi$
$811$	581.969i	0.717594i	−0.933416	+	0.358797i	$0.883187\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 718.801i	− 0.879805i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	1058.00	1.28554	0.642770	−	0.766059i	$-0.277785\pi$
$823$	1058.00	1.28554	0.642770	+	0.766059i	$0.277785\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	− 1193.38i	− 1.43955i	−0.694210	−	0.719773i	$-0.744246\pi$
$829$	− 1193.38i	− 1.43955i	0.694210	−	0.719773i	$-0.255754\pi$
$830$	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			$0$
$839$	0	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
$853$	− 1234.95i	− 1.44778i	−0.689918	−	0.723888i	$-0.742353\pi$
$853$	− 1234.95i	− 1.44778i	0.689918	−	0.723888i	$-0.257647\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	969.948i	1.12916i	0.825378	+	0.564580i	$0.190962\pi$
$859$	969.948i	1.12916i	−0.825378	+	0.564580i	$0.809038\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2831.90i	3.25132i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−598.000	−0.681870	−0.340935	−	0.940087i	$-0.610744\pi$
$877$	−598.000	−0.681870	−0.340935	+	0.940087i	$0.610744\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	−1259.00	−1.42582	−0.712911	−	0.701255i	$-0.752623\pi$
$883$	−1259.00	−1.42582	−0.712911	+	0.701255i	$0.752623\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	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
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1453.00	1.60198	0.800992	−	0.598675i	$-0.204306\pi$
$907$	1453.00	1.60198	0.800992	+	0.598675i	$0.204306\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			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
$919$	−971.000	−1.05658	−0.528292	−	0.849063i	$-0.677167\pi$
$919$	−971.000	−1.05658	−0.528292	+	0.849063i	$0.677167\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1175.00	−1.27027
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	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
$937$	316.965i	0.338277i	0.985592	+	0.169138i	$0.0540985\pi$
$937$	316.965i	0.338277i	−0.985592	+	0.169138i	$0.945902\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	765.000	0.806112
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	598.000	0.622268
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−253.000	−0.261634	−0.130817	−	0.991407i	$-0.541760\pi$
$967$	−253.000	−0.261634	−0.130817	+	0.991407i	$0.541760\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1739.00	−1.75479	−0.877397	−	0.479766i	$-0.840722\pi$
$991$	−1739.00	−1.75479	−0.877397	+	0.479766i	$0.840722\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1328.48i	− 1.33248i	−0.745737	−	0.666240i	$-0.767903\pi$
$997$	− 1328.48i	− 1.33248i	0.745737	−	0.666240i	$-0.232097\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.d.b.685.1		2
3.2	odd	2	CM	1764.3.d.b.685.1		2
7.2	even	3		1764.3.z.c.325.1		2
7.3	odd	6		1764.3.z.c.901.1		2
7.4	even	3		252.3.z.c.145.1	yes	2
7.5	odd	6		252.3.z.c.73.1	✓	2
7.6	odd	2	inner	1764.3.d.b.685.2		2
21.2	odd	6		1764.3.z.c.325.1		2
21.5	even	6		252.3.z.c.73.1	✓	2
21.11	odd	6		252.3.z.c.145.1	yes	2
21.17	even	6		1764.3.z.c.901.1		2
21.20	even	2	inner	1764.3.d.b.685.2		2
28.11	odd	6		1008.3.cg.e.145.1		2
28.19	even	6		1008.3.cg.e.577.1		2
84.11	even	6		1008.3.cg.e.145.1		2
84.47	odd	6		1008.3.cg.e.577.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.z.c.73.1	✓	2	7.5	odd	6
252.3.z.c.73.1	✓	2	21.5	even	6
252.3.z.c.145.1	yes	2	7.4	even	3
252.3.z.c.145.1	yes	2	21.11	odd	6
1008.3.cg.e.145.1		2	28.11	odd	6
1008.3.cg.e.145.1		2	84.11	even	6
1008.3.cg.e.577.1		2	28.19	even	6
1008.3.cg.e.577.1		2	84.47	odd	6
1764.3.d.b.685.1		2	1.1	even	1	trivial
1764.3.d.b.685.1		2	3.2	odd	2	CM
1764.3.d.b.685.2		2	7.6	odd	2	inner
1764.3.d.b.685.2		2	21.20	even	2	inner
1764.3.z.c.325.1		2	7.2	even	3
1764.3.z.c.325.1		2	21.2	odd	6
1764.3.z.c.901.1		2	7.3	odd	6
1764.3.z.c.901.1		2	21.17	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-25.9808i q^{13} +8.66025i q^{19} +25.0000 q^{25} +19.0526i q^{31} -47.0000 q^{37} -83.0000 q^{43} -96.9948i q^{61} -109.000 q^{67} +29.4449i q^{73} +131.000 q^{79} -193.990i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 50 q^{25} - 94 q^{37} - 166 q^{43} - 218 q^{67} + 262 q^{79}+O(q^{100}) \) 2 * q + 50 * q^25 - 94 * q^37 - 166 * q^43 - 218 * q^67 + 262 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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