LMFDB - Embedded newform 1856.2.g.a.289.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1856,2,Mod(289,1856)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1856.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1856, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{29})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 15x^{2} + 49 $ x^4 + 15*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{59}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			289.3
Root			$-3.19258i$ of defining polynomial
Character	$\chi$	$=$	1856.289
Dual form			1856.2.g.a.289.4

$q$-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 $\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$639$
$\chi(n)$	$-1$	$-1$	$1$

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		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	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
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.38516	1.00000
$29$	5.38516	1.00000
$30$	0	0
$31$	− 10.7703i	− 1.93441i	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	− 10.7703i	− 1.93441i	0.254000	−	0.967204i	$-0.418254\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.7703	1.77063	0.885316	−	0.464991i	$-0.153942\pi$
$37$	10.7703	1.77063	0.885316	+	0.464991i	$0.153942\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 10.7703i	− 1.57101i	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	− 10.7703i	− 1.57101i	0.618853	−	0.785507i	$-0.287598\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.00000i	− 0.260378i	−0.991489	−	0.130189i	$-0.958442\pi$
$59$	− 2.00000i	− 0.260378i	0.991489	−	0.130189i	$-0.0415584\pi$
$60$	0	0
$61$	10.7703	1.37900	0.689500	−	0.724286i	$-0.257830\pi$
$61$	10.7703	1.37900	0.689500	+	0.724286i	$0.257830\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	6.00000i	0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$	6.00000i	0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	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
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 10.7703i	− 1.21176i	−0.795557	−	0.605878i	$-0.792822\pi$
$79$	− 10.7703i	− 1.21176i	0.795557	−	0.605878i	$-0.207178\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 10.0000i	− 1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$	− 10.0000i	− 1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$	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
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.7703	1.07169	0.535844	−	0.844317i	$-0.319994\pi$
$101$	10.7703	1.07169	0.535844	+	0.844317i	$0.319994\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.0000i	1.35343i	0.736245	+	0.676716i	$0.236597\pi$
$107$	14.0000i	1.35343i	−0.736245	+	0.676716i	$0.763403\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	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
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 10.7703i	− 0.955712i	−0.878438	−	0.477856i	$-0.841414\pi$
$127$	− 10.7703i	− 0.955712i	0.878438	−	0.477856i	$-0.158586\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	− 18.0000i	− 1.52674i	−0.645961	−	0.763370i	$-0.723543\pi$
$139$	− 18.0000i	− 1.52674i	0.645961	−	0.763370i	$-0.276457\pi$
$140$	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	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.7703	0.859566	0.429783	−	0.902932i	$-0.358590\pi$
$157$	10.7703	0.859566	0.429783	+	0.902932i	$0.358590\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	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
$179$	22.0000i	1.64436i	0.569230	+	0.822179i	$0.307242\pi$
$179$	22.0000i	1.64436i	−0.569230	+	0.822179i	$0.692758\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	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
$191$	− 10.7703i	− 0.779314i	−0.920960	−	0.389657i	$-0.872594\pi$
$191$	− 10.7703i	− 0.779314i	0.920960	−	0.389657i	$-0.127406\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	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
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			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
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−15.0000	−1.00000
$226$	0	0
$227$	− 26.0000i	− 1.72568i	−0.505477	−	0.862840i	$-0.668683\pi$
$227$	− 26.0000i	− 1.72568i	0.505477	−	0.862840i	$-0.331317\pi$
$228$	0	0
$229$	10.7703	0.711723	0.355862	−	0.934539i	$-0.384187\pi$
$229$	10.7703	0.711723	0.355862	+	0.934539i	$0.384187\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	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
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−16.1555	−1.00000
$262$	0	0
$263$	32.3110i	1.99238i	0.0872041	+	0.996190i	$0.472207\pi$
$263$	32.3110i	1.99238i	−0.0872041	+	0.996190i	$0.527793\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−32.3110	−1.97004	−0.985018	−	0.172452i	$-0.944831\pi$
$269$	−32.3110	−1.97004	−0.985018	+	0.172452i	$0.944831\pi$
$270$	0	0
$271$	− 10.7703i	− 0.654251i	−0.944981	−	0.327125i	$-0.893920\pi$
$271$	− 10.7703i	− 0.654251i	0.944981	−	0.327125i	$-0.106080\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	32.3110i	1.93441i
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	30.0000i	1.78331i	0.452711	+	0.891657i	$0.350457\pi$
$283$	30.0000i	1.78331i	−0.452711	+	0.891657i	$0.649543\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.3110	−1.88763	−0.943814	−	0.330477i	$-0.892790\pi$
$293$	−32.3110	−1.88763	−0.943814	+	0.330477i	$0.892790\pi$
$294$	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
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.3110i	1.83219i	0.400963	+	0.916094i	$0.368675\pi$
$311$	32.3110i	1.83219i	−0.400963	+	0.916094i	$0.631325\pi$
$312$	0	0
$313$	18.0000	1.01742	0.508710	−	0.860938i	$-0.330123\pi$
$313$	18.0000	1.01742	0.508710	+	0.860938i	$0.330123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.7703	0.604922	0.302461	−	0.953162i	$-0.402192\pi$
$317$	10.7703	0.604922	0.302461	+	0.953162i	$0.402192\pi$
$318$	0	0
$319$	0	0
$320$	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		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	−32.3110	−1.77063
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	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
$347$	− 34.0000i	− 1.82522i	−0.408836	−	0.912608i	$-0.634065\pi$
$347$	− 34.0000i	− 1.82522i	0.408836	−	0.912608i	$-0.365935\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.0000	−1.17094	−0.585471	−	0.810693i	$-0.699090\pi$
$353$	−22.0000	−1.17094	−0.585471	+	0.810693i	$0.699090\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.3110i	1.70531i	0.522475	+	0.852654i	$0.325009\pi$
$359$	32.3110i	1.70531i	−0.522475	+	0.852654i	$0.674991\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 10.7703i	− 0.562207i	−0.959678	−	0.281103i	$-0.909300\pi$
$367$	− 10.7703i	− 0.562207i	0.959678	−	0.281103i	$-0.0907004\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.3110	−1.63823	−0.819116	−	0.573628i	$-0.805536\pi$
$389$	−32.3110	−1.63823	−0.819116	+	0.573628i	$0.805536\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	38.0000i	1.85642i	0.372055	+	0.928211i	$0.378653\pi$
$419$	38.0000i	1.85642i	−0.372055	+	0.928211i	$0.621347\pi$
$420$	0	0
$421$	10.7703	0.524914	0.262457	−	0.964944i	$-0.415467\pi$
$421$	10.7703	0.524914	0.262457	+	0.964944i	$0.415467\pi$
$422$	0	0
$423$	32.3110i	1.57101i
$424$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	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	1.00000			$0$
$449$	0	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
$457$	−30.0000	−1.40334	−0.701670	−	0.712502i	$-0.747562\pi$
$457$	−30.0000	−1.40334	−0.701670	+	0.712502i	$0.747562\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−32.3110	−1.50487	−0.752436	−	0.658665i	$-0.771121\pi$
$461$	−32.3110	−1.50487	−0.752436	+	0.658665i	$0.771121\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			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
$479$	− 10.7703i	− 0.492109i	−0.969256	−	0.246054i	$-0.920866\pi$
$479$	− 10.7703i	− 0.492109i	0.969256	−	0.246054i	$-0.0791342\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	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
$499$	− 42.0000i	− 1.88018i	−0.340929	−	0.940089i	$-0.610742\pi$
$499$	− 42.0000i	− 1.88018i	0.340929	−	0.940089i	$-0.389258\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	32.3110i	1.44068i	0.693623	+	0.720338i	$0.256013\pi$
$503$	32.3110i	1.44068i	−0.693623	+	0.720338i	$0.743987\pi$
$504$	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
$521$	34.0000	1.48957	0.744784	−	0.667306i	$-0.232553\pi$
$521$	34.0000	1.48957	0.744784	+	0.667306i	$0.232553\pi$
$522$	0	0
$523$	− 2.00000i	− 0.0874539i	−0.999044	−	0.0437269i	$-0.986077\pi$
$523$	− 2.00000i	− 0.0874539i	0.999044	−	0.0437269i	$-0.0139232\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	6.00000i	0.260378i
$532$	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
$541$	10.7703	0.463053	0.231526	−	0.972829i	$-0.425628\pi$
$541$	10.7703	0.463053	0.231526	+	0.972829i	$0.425628\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 10.0000i	− 0.427569i	−0.976881	−	0.213785i	$-0.931421\pi$
$547$	− 10.0000i	− 0.427569i	0.976881	−	0.213785i	$-0.0685791\pi$
$548$	0	0
$549$	−32.3110	−1.37900
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	14.0000i	0.585882i	0.956131	+	0.292941i	$0.0946339\pi$
$571$	14.0000i	0.585882i	−0.956131	+	0.292941i	$0.905366\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	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
$587$	46.0000i	1.89862i	0.314337	+	0.949312i	$0.398218\pi$
$587$	46.0000i	1.89862i	−0.314337	+	0.949312i	$0.601782\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−38.0000	−1.56047	−0.780236	−	0.625485i	$-0.784901\pi$
$593$	−38.0000	−1.56047	−0.780236	+	0.625485i	$0.784901\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.3110i	1.32019i	0.751182	+	0.660096i	$0.229484\pi$
$599$	32.3110i	1.32019i	−0.751182	+	0.660096i	$0.770516\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	− 18.0000i	− 0.733017i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 10.7703i	− 0.437154i	−0.975820	−	0.218577i	$-0.929858\pi$
$607$	− 10.7703i	− 0.437154i	0.975820	−	0.218577i	$-0.0701415\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	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	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	22.0000i	0.867595i	0.901010	+	0.433798i	$0.142827\pi$
$643$	22.0000i	0.867595i	−0.901010	+	0.433798i	$0.857173\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.3110	−1.26443	−0.632213	−	0.774794i	$-0.717853\pi$
$653$	−32.3110	−1.26443	−0.632213	+	0.774794i	$0.717853\pi$
$654$	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
$661$	0	0	1.00000			$0$
$661$	0	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
$673$	42.0000	1.61898	0.809491	−	0.587133i	$-0.199743\pi$
$673$	42.0000	1.61898	0.809491	+	0.587133i	$0.199743\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10.7703	0.413937	0.206969	−	0.978348i	$-0.433640\pi$
$677$	10.7703	0.413937	0.206969	+	0.978348i	$0.433640\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 50.0000i	− 1.91320i	−0.291409	−	0.956598i	$-0.594124\pi$
$683$	− 50.0000i	− 1.91320i	0.291409	−	0.956598i	$-0.405876\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 26.0000i	− 0.989087i	−0.869153	−	0.494543i	$-0.835335\pi$
$691$	− 26.0000i	− 0.989087i	0.869153	−	0.494543i	$-0.164665\pi$
$692$	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	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	32.3110i	1.21176i
$712$	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	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	26.9258	1.00000
$726$	0	0
$727$	− 53.8516i	− 1.99725i	−0.0524503	−	0.998624i	$-0.516703\pi$
$727$	− 53.8516i	− 1.99725i	0.0524503	−	0.998624i	$-0.483297\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	53.8516	1.98906	0.994528	−	0.104470i	$-0.0333147\pi$
$733$	53.8516	1.98906	0.994528	+	0.104470i	$0.0333147\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 53.8516i	− 1.97563i	−0.155647	−	0.987813i	$-0.549746\pi$
$743$	− 53.8516i	− 1.97563i	0.155647	−	0.987813i	$-0.450254\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	30.0000i	1.09764i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 10.7703i	− 0.393015i	−0.980502	−	0.196507i	$-0.937040\pi$
$751$	− 10.7703i	− 0.393015i	0.980502	−	0.196507i	$-0.0629600\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	53.8516	1.95727	0.978636	−	0.205602i	$-0.0659152\pi$
$757$	53.8516	1.95727	0.978636	+	0.205602i	$0.0659152\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.0000	−1.66750	−0.833749	−	0.552143i	$-0.813810\pi$
$761$	−46.0000	−1.66750	−0.833749	+	0.552143i	$0.813810\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−32.3110	−1.16215	−0.581073	−	0.813852i	$-0.697367\pi$
$773$	−32.3110	−1.16215	−0.581073	+	0.813852i	$0.697367\pi$
$774$	0	0
$775$	− 53.8516i	− 1.93441i
$776$	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
$787$	54.0000i	1.92489i	0.271473	+	0.962446i	$0.412489\pi$
$787$	54.0000i	1.92489i	−0.271473	+	0.962446i	$0.587511\pi$
$788$	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
$797$	53.8516	1.90752	0.953762	−	0.300564i	$-0.0971749\pi$
$797$	53.8516	1.90752	0.953762	+	0.300564i	$0.0971749\pi$
$798$	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	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	− 34.0000i	− 1.19390i	−0.802278	−	0.596951i	$-0.796379\pi$
$811$	− 34.0000i	− 1.19390i	0.802278	−	0.596951i	$-0.203621\pi$
$812$	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
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 53.8516i	− 1.87715i	−0.345075	−	0.938575i	$-0.612146\pi$
$823$	− 53.8516i	− 1.87715i	0.345075	−	0.938575i	$-0.387854\pi$
$824$	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
$829$	10.7703	0.374069	0.187035	−	0.982353i	$-0.440112\pi$
$829$	10.7703	0.374069	0.187035	+	0.982353i	$0.440112\pi$
$830$	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
$839$	32.3110i	1.11550i	0.830009	+	0.557750i	$0.188335\pi$
$839$	32.3110i	1.11550i	−0.830009	+	0.557750i	$0.811665\pi$
$840$	0	0
$841$	29.0000	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
$853$	53.8516	1.84385	0.921923	−	0.387374i	$-0.126618\pi$
$853$	53.8516	1.84385	0.921923	+	0.387374i	$0.126618\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	50.0000	1.70797	0.853984	−	0.520300i	$-0.174180\pi$
$857$	50.0000	1.70797	0.853984	+	0.520300i	$0.174180\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	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
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	38.0000i	1.27880i	0.768874	+	0.639401i	$0.220818\pi$
$883$	38.0000i	1.27880i	−0.768874	+	0.639401i	$0.779182\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 53.8516i	− 1.80816i	−0.427362	−	0.904081i	$-0.640557\pi$
$887$	− 53.8516i	− 1.80816i	0.427362	−	0.904081i	$-0.359443\pi$
$888$	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
$899$	− 58.0000i	− 1.93441i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−32.3110	−1.07169
$910$	0	0
$911$	− 10.7703i	− 0.356837i	−0.983955	−	0.178418i	$-0.942902\pi$
$911$	− 10.7703i	− 0.356837i	0.983955	−	0.178418i	$-0.0570981\pi$
$912$	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	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	53.8516	1.77063
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2.00000	0.0656179	0.0328089	−	0.999462i	$-0.489555\pi$
$929$	2.00000	0.0656179	0.0328089	+	0.999462i	$0.489555\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−6.00000	−0.196011	−0.0980057	−	0.995186i	$-0.531246\pi$
$937$	−6.00000	−0.196011	−0.0980057	+	0.995186i	$0.531246\pi$
$938$	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
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−85.0000	−2.74194
$962$	0	0
$963$	− 42.0000i	− 1.35343i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 53.8516i	− 1.73175i	−0.500258	−	0.865876i	$-0.666762\pi$
$967$	− 53.8516i	− 1.73175i	0.500258	−	0.865876i	$-0.333238\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14.0000	−0.447900	−0.223950	−	0.974601i	$-0.571895\pi$
$977$	−14.0000	−0.447900	−0.223950	+	0.974601i	$0.571895\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32.3110i	1.03056i	0.857022	+	0.515280i	$0.172312\pi$
$983$	32.3110i	1.03056i	−0.857022	+	0.515280i	$0.827688\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	10.7703	0.341100	0.170550	−	0.985349i	$-0.445446\pi$
$997$	10.7703	0.341100	0.170550	+	0.985349i	$0.445446\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.g.a.289.3	yes	4
4.3	odd	2	inner	1856.2.g.a.289.4	yes	4
8.3	odd	2	inner	1856.2.g.a.289.2	yes	4
8.5	even	2	inner	1856.2.g.a.289.1	✓	4
29.28	even	2	inner	1856.2.g.a.289.2	yes	4
116.115	odd	2	inner	1856.2.g.a.289.1	✓	4
232.115	odd	2	CM	1856.2.g.a.289.3	yes	4
232.173	even	2	inner	1856.2.g.a.289.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1856.2.g.a.289.1	✓	4	8.5	even	2	inner
1856.2.g.a.289.1	✓	4	116.115	odd	2	inner
1856.2.g.a.289.2	yes	4	8.3	odd	2	inner
1856.2.g.a.289.2	yes	4	29.28	even	2	inner
1856.2.g.a.289.3	yes	4	1.1	even	1	trivial
1856.2.g.a.289.3	yes	4	232.115	odd	2	CM
1856.2.g.a.289.4	yes	4	4.3	odd	2	inner
1856.2.g.a.289.4	yes	4	232.173	even	2	inner

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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{9} +5.00000 q^{25} +5.38516 q^{29} -10.7703i q^{31} +10.7703 q^{37} -10.7703i q^{47} -7.00000 q^{49} -2.00000i q^{59} +10.7703 q^{61} +6.00000i q^{67} -10.7703i q^{79} +9.00000 q^{81} -10.0000i q^{83} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} + 20 q^{25} - 28 q^{49} + 36 q^{81}+O(q^{100}) \) 4 * q - 12 * q^9 + 20 * q^25 - 28 * q^49 + 36 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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