LMFDB - Embedded newform 1728.4.f.b.863.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,4,Mod(863,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.863"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-500,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,216] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)

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

Self dual:	no
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			863.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.4.f.b.863.3

$q$-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 $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$-1$	$-1$

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		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	− 17.0000i	− 0.917914i	−0.888459	−	0.458957i	$-0.848223\pi$
$7$	− 17.0000i	− 0.917914i	0.888459	−	0.458957i	$-0.151777\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	29.4449i	0.628195i	0.949391	+	0.314098i	$0.101702\pi$
$13$	29.4449i	0.628195i	−0.949391	+	0.314098i	$0.898298\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	126.440	1.52670	0.763349	−	0.645986i	$-0.223554\pi$
$19$	126.440	1.52670	0.763349	+	0.645986i	$0.223554\pi$
$20$	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
$25$	−125.000	−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
$31$	308.000i	1.78447i	0.451576	+	0.892233i	$0.350862\pi$
$31$	308.000i	1.78447i	−0.451576	+	0.892233i	$0.649138\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 122.976i	− 0.546407i	−0.961956	−	0.273204i	$-0.911917\pi$
$37$	− 122.976i	− 0.546407i	0.961956	−	0.273204i	$-0.0880833\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−218.238	−0.773978	−0.386989	−	0.922084i	$-0.626485\pi$
$43$	−218.238	−0.773978	−0.386989	+	0.922084i	$0.626485\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	54.0000	0.157434
$50$	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
$61$	310.037i	0.650757i	0.945584	+	0.325379i	$0.105492\pi$
$61$	310.037i	0.650757i	−0.945584	+	0.325379i	$0.894508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	434.745	0.792724	0.396362	−	0.918094i	$-0.370272\pi$
$67$	434.745	0.792724	0.396362	+	0.918094i	$0.370272\pi$
$68$	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
$73$	271.000	0.434495	0.217248	−	0.976117i	$-0.430292\pi$
$73$	271.000	0.434495	0.217248	+	0.976117i	$0.430292\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	503.000i	0.716353i	0.933654	+	0.358177i	$0.116601\pi$
$79$	503.000i	0.716353i	−0.933654	+	0.358177i	$0.883399\pi$
$80$	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
$91$	500.563	0.576629
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1853.00	1.93963	0.969813	−	0.243851i	$-0.0784109\pi$
$97$	1853.00	1.93963	0.969813	+	0.243851i	$0.0784109\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 19.0000i	− 0.0181760i	−0.999959	−	0.00908799i	$-0.997107\pi$
$103$	− 19.0000i	− 0.0181760i	0.999959	−	0.00908799i	$-0.00289284\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 2182.38i	− 1.91775i	−0.283833	−	0.958874i	$-0.591606\pi$
$109$	− 2182.38i	− 1.91775i	0.283833	−	0.958874i	$-0.408394\pi$
$110$	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
$121$	1331.00	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 380.000i	− 0.265508i	−0.991149	−	0.132754i	$-0.957618\pi$
$127$	− 380.000i	− 0.265508i	0.991149	−	0.132754i	$-0.0423821\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 2149.48i	− 1.40138i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−1217.63	−0.743008	−0.371504	−	0.928431i	$-0.621158\pi$
$139$	−1217.63	−0.743008	−0.371504	+	0.928431i	$0.621158\pi$
$140$	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
$151$	3709.00i	1.99890i	0.0331378	+	0.999451i	$0.489450\pi$
$151$	3709.00i	1.99890i	−0.0331378	+	0.999451i	$0.510550\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	810.600i	0.412057i	0.978546	+	0.206028i	$0.0660539\pi$
$157$	810.600i	0.412057i	−0.978546	+	0.206028i	$0.933946\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	4144.80	1.99169	0.995845	−	0.0910600i	$-0.0290255\pi$
$163$	4144.80	1.99169	0.995845	+	0.0910600i	$0.0290255\pi$
$164$	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
$169$	1330.00	0.605371
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	2125.00i	0.917914i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	1279.99i	0.525639i	0.964845	+	0.262819i	$0.0846523\pi$
$181$	1279.99i	0.525639i	−0.964845	+	0.262819i	$0.915348\pi$
$182$	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
$193$	5111.00	1.90621	0.953103	−	0.302646i	$-0.0978698\pi$
$193$	5111.00	1.90621	0.953103	+	0.302646i	$0.0978698\pi$
$194$	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
$199$	4373.00i	1.55776i	0.627175	+	0.778879i	$0.284211\pi$
$199$	4373.00i	1.55776i	−0.627175	+	0.778879i	$0.715789\pi$
$200$	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
$211$	5769.46	1.88240	0.941199	−	0.337852i	$-0.109700\pi$
$211$	5769.46	1.88240	0.941199	+	0.337852i	$0.109700\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5236.00	1.63799
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	3220.00i	0.966938i	0.875362	+	0.483469i	$0.160623\pi$
$223$	3220.00i	0.966938i	−0.875362	+	0.483469i	$0.839377\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 5300.08i	− 1.52943i	−0.644370	−	0.764714i	$-0.722880\pi$
$229$	− 5300.08i	− 1.52943i	0.644370	−	0.764714i	$-0.277120\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	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
$241$	−4769.00	−1.27468	−0.637341	−	0.770582i	$-0.719966\pi$
$241$	−4769.00	−1.27468	−0.637341	+	0.770582i	$0.719966\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3723.00i	0.959064i
$248$	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
$259$	−2090.59	−0.501555
$260$	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		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	8101.00i	1.81587i	0.419111	+	0.907935i	$0.362342\pi$
$271$	8101.00i	1.81587i	−0.419111	+	0.907935i	$0.637658\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8293.06i	− 1.79885i	−0.437074	−	0.899425i	$-0.643985\pi$
$277$	− 8293.06i	− 1.79885i	0.437074	−	0.899425i	$-0.356015\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	7700.70	1.61752	0.808761	−	0.588137i	$-0.200138\pi$
$283$	7700.70	1.61752	0.808761	+	0.588137i	$0.200138\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4913.00	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	3710.05i	0.710445i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1590.02	−0.295594	−0.147797	−	0.989018i	$-0.547218\pi$
$307$	−1590.02	−0.295594	−0.147797	+	0.989018i	$0.547218\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	9109.00	1.64496	0.822478	−	0.568797i	$-0.192591\pi$
$313$	9109.00	1.64496	0.822478	+	0.568797i	$0.192591\pi$
$314$	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
$325$	− 3680.61i	− 0.628195i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	5142.46	0.853943	0.426971	−	0.904265i	$-0.359580\pi$
$331$	5142.46	0.853943	0.426971	+	0.904265i	$0.359580\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	12293.0	1.98707	0.993535	−	0.113529i	$-0.0362155\pi$
$337$	12293.0	1.98707	0.993535	+	0.113529i	$0.0362155\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 6749.00i	− 1.06242i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	− 12967.9i	− 1.98898i	−0.104834	−	0.994490i	$-0.533431\pi$
$349$	− 12967.9i	− 1.98898i	0.104834	−	0.994490i	$-0.466569\pi$
$350$	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
$361$	9128.00	1.33081
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	9413.00i	1.33884i	0.742884	+	0.669420i	$0.233457\pi$
$367$	9413.00i	1.33884i	−0.742884	+	0.669420i	$0.766543\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6985.36i	0.969674i	0.874605	+	0.484837i	$0.161121\pi$
$373$	6985.36i	0.969674i	−0.874605	+	0.484837i	$0.838879\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	1432.41	0.194137	0.0970683	−	0.995278i	$-0.469053\pi$
$379$	1432.41	0.194137	0.0970683	+	0.995278i	$0.469053\pi$
$380$	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
$389$	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
$397$	15775.5i	1.99433i	0.0752196	+	0.997167i	$0.476034\pi$
$397$	15775.5i	1.99433i	−0.0752196	+	0.997167i	$0.523966\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−9069.02	−1.12099
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−8297.00	−1.00308	−0.501541	−	0.865134i	$-0.667233\pi$
$409$	−8297.00	−1.00308	−0.501541	+	0.865134i	$0.667233\pi$
$410$	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
$421$	− 15933.1i	− 1.84450i	−0.386597	−	0.922249i	$-0.626350\pi$
$421$	− 15933.1i	− 1.84450i	0.386597	−	0.922249i	$-0.373650\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5270.63	0.597339
$428$	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
$433$	2590.00	0.287454	0.143727	−	0.989617i	$-0.454091\pi$
$433$	2590.00	0.287454	0.143727	+	0.989617i	$0.454091\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	14924.0i	1.62251i	0.584690	+	0.811257i	$0.301216\pi$
$439$	14924.0i	1.62251i	−0.584690	+	0.811257i	$0.698784\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	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
$457$	−12710.0	−1.30098	−0.650491	−	0.759514i	$-0.725437\pi$
$457$	−12710.0	−1.30098	−0.650491	+	0.759514i	$0.725437\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	− 11969.0i	− 1.20140i	−0.799476	−	0.600698i	$-0.794889\pi$
$463$	− 11969.0i	− 1.20140i	0.799476	−	0.600698i	$-0.205111\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	− 7390.66i	− 0.727652i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−15805.0	−1.52670
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	3621.00	0.343250
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6103.00i	0.567871i	0.958843	+	0.283936i	$0.0916402\pi$
$487$	6103.00i	0.567871i	−0.958843	+	0.283936i	$0.908360\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16367.9	1.46839	0.734195	−	0.678938i	$-0.237560\pi$
$499$	16367.9	1.46839	0.734195	+	0.678938i	$0.237560\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	− 4607.00i	− 0.398829i
$512$	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
$523$	20762.1	1.73588	0.867938	−	0.496673i	$-0.165445\pi$
$523$	20762.1	1.73588	0.867938	+	0.496673i	$0.165445\pi$
$524$	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
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	25095.7i	1.99436i	0.0750596	+	0.997179i	$0.476085\pi$
$541$	25095.7i	1.99436i	−0.0750596	+	0.997179i	$0.523915\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−14187.2	−1.10896	−0.554481	−	0.832196i	$-0.687083\pi$
$547$	−14187.2	−1.10896	−0.554481	+	0.832196i	$0.687083\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	8551.00	0.657551
$554$	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
$559$	− 6426.00i	− 0.486209i
$560$	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	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	13096.0	0.959811	0.479905	−	0.877320i	$-0.340671\pi$
$571$	13096.0	0.959811	0.479905	+	0.877320i	$0.340671\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−27323.0	−1.97135	−0.985677	−	0.168644i	$-0.946061\pi$
$577$	−27323.0	−1.97135	−0.985677	+	0.168644i	$0.946061\pi$
$578$	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
$589$	38943.4i	2.72434i
$590$	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
$601$	−29302.0	−1.98877	−0.994387	−	0.105801i	$-0.966259\pi$
$601$	−29302.0	−1.98877	−0.994387	+	0.105801i	$0.966259\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 6137.00i	− 0.410368i	−0.978723	−	0.205184i	$-0.934221\pi$
$607$	− 6137.00i	− 0.410368i	0.978723	−	0.205184i	$-0.0657792\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 27499.8i	− 1.81192i	−0.423366	−	0.905959i	$-0.639152\pi$
$613$	− 27499.8i	− 1.81192i	0.423366	−	0.905959i	$-0.360848\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	−30801.1	−2.00000	−0.999999	−	0.00120126i	$-0.999618\pi$
$619$	−30801.1	−2.00000	−0.999999	+	0.00120126i	$0.999618\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 26459.0i	− 1.66928i	−0.550795	−	0.834640i	$-0.685675\pi$
$631$	− 26459.0i	− 1.66928i	0.550795	−	0.834640i	$-0.314325\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1590.02i	0.0988995i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−29836.3	−1.82991	−0.914953	−	0.403561i	$-0.867772\pi$
$643$	−29836.3	−1.82991	−0.914953	+	0.403561i	$0.867772\pi$
$644$	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
$653$	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	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	4175.97i	0.245729i	0.992423	+	0.122864i	$0.0392080\pi$
$661$	4175.97i	0.245729i	−0.992423	+	0.122864i	$0.960792\pi$
$662$	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
$673$	−9899.00	−0.566981	−0.283491	−	0.958975i	$-0.591492\pi$
$673$	−9899.00	−0.566981	−0.283491	+	0.958975i	$0.591492\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	− 31501.0i	− 1.78041i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−32579.9	−1.79363	−0.896814	−	0.442408i	$-0.854124\pi$
$691$	−32579.9	−1.79363	−0.896814	+	0.442408i	$0.854124\pi$
$692$	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
$703$	− 15549.0i	− 0.834199i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 36759.3i	− 1.94714i	−0.228381	−	0.973572i	$-0.573343\pi$
$709$	− 36759.3i	− 1.94714i	0.228381	−	0.973572i	$-0.426657\pi$
$710$	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	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−323.000	−0.0166840
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10780.0i	0.549942i	0.961452	+	0.274971i	$0.0886683\pi$
$727$	10780.0i	0.549942i	−0.961452	+	0.274971i	$0.911332\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	36726.4i	1.85064i	0.379184	+	0.925321i	$0.376205\pi$
$733$	36726.4i	1.85064i	−0.379184	+	0.925321i	$0.623795\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	25097.4	1.24929	0.624644	−	0.780910i	$-0.285244\pi$
$739$	25097.4	1.24929	0.624644	+	0.780910i	$0.285244\pi$
$740$	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
$751$	− 17569.0i	− 0.853664i	−0.904331	−	0.426832i	$-0.859630\pi$
$751$	− 17569.0i	− 0.853664i	0.904331	−	0.426832i	$-0.140370\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	33949.9i	1.63003i	0.579441	+	0.815014i	$0.303271\pi$
$757$	33949.9i	1.63003i	−0.579441	+	0.815014i	$0.696729\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	−37100.5	−1.76033
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−34417.0	−1.61393	−0.806963	−	0.590602i	$-0.798890\pi$
$769$	−34417.0	−1.61393	−0.806963	+	0.590602i	$0.798890\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	− 38500.0i	− 1.78447i
$776$	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
$787$	41654.1	1.88667	0.943334	−	0.331844i	$-0.107671\pi$
$787$	41654.1	1.88667	0.943334	+	0.331844i	$0.107671\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9129.00	−0.408802
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	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
$811$	−24162.1	−1.04617	−0.523087	−	0.852280i	$-0.675220\pi$
$811$	−24162.1	−1.04617	−0.523087	+	0.852280i	$0.675220\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−27594.0	−1.18163
$818$	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
$823$	− 45611.0i	− 1.93183i	−0.258852	−	0.965917i	$-0.583344\pi$
$823$	− 45611.0i	− 1.93183i	0.258852	−	0.965917i	$-0.416656\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	37071.1i	1.55311i	0.630047	+	0.776557i	$0.283036\pi$
$829$	37071.1i	1.55311i	−0.630047	+	0.776557i	$0.716964\pi$
$830$	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		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 22627.0i	− 0.917914i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 49133.1i	− 1.97220i	−0.166159	−	0.986099i	$-0.553137\pi$
$853$	− 49133.1i	− 1.97220i	0.166159	−	0.986099i	$-0.446863\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	7390.66	0.293558	0.146779	−	0.989169i	$-0.453109\pi$
$859$	7390.66	0.293558	0.146779	+	0.989169i	$0.453109\pi$
$860$	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
$871$	12801.0i	0.497985i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	36665.8i	1.41176i	0.708330	+	0.705881i	$0.249449\pi$
$877$	36665.8i	1.41176i	−0.708330	+	0.705881i	$0.750551\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	−6205.94	−0.236519	−0.118260	−	0.992983i	$-0.537731\pi$
$883$	−6205.94	−0.236519	−0.118260	+	0.992983i	$0.537731\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−6460.00	−0.243714
$890$	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
$907$	−54436.6	−1.99288	−0.996438	−	0.0843291i	$-0.973125\pi$
$907$	−54436.6	−1.99288	−0.996438	+	0.0843291i	$0.973125\pi$
$908$	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
$919$	− 2756.00i	− 0.0989250i	−0.998776	−	0.0494625i	$-0.984249\pi$
$919$	− 2756.00i	− 0.0989250i	0.998776	−	0.0494625i	$-0.0157508\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	15372.0i	0.546407i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	6827.74	0.240355
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−15227.0	−0.530891	−0.265445	−	0.964126i	$-0.585519\pi$
$937$	−15227.0	−0.530891	−0.265445	+	0.964126i	$0.585519\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	7979.56i	0.272948i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−65073.0	−2.18432
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	53927.0i	1.79336i	0.442683	+	0.896678i	$0.354027\pi$
$967$	53927.0i	1.79336i	−0.442683	+	0.896678i	$0.645973\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	20699.7i	0.682018i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	59669.0i	1.91266i	0.292286	+	0.956331i	$0.405584\pi$
$991$	59669.0i	1.91266i	−0.292286	+	0.956331i	$0.594416\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 55931.4i	− 1.77669i	−0.459172	−	0.888347i	$-0.651854\pi$
$997$	− 55931.4i	− 1.77669i	0.459172	−	0.888347i	$-0.348146\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.f.b.863.2	yes	4
3.2	odd	2	CM	1728.4.f.b.863.2	yes	4
4.3	odd	2	inner	1728.4.f.b.863.4	yes	4
8.3	odd	2	inner	1728.4.f.b.863.3	yes	4
8.5	even	2	inner	1728.4.f.b.863.1	✓	4
12.11	even	2	inner	1728.4.f.b.863.4	yes	4
24.5	odd	2	inner	1728.4.f.b.863.1	✓	4
24.11	even	2	inner	1728.4.f.b.863.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.f.b.863.1	✓	4	8.5	even	2	inner
1728.4.f.b.863.1	✓	4	24.5	odd	2	inner
1728.4.f.b.863.2	yes	4	1.1	even	1	trivial
1728.4.f.b.863.2	yes	4	3.2	odd	2	CM
1728.4.f.b.863.3	yes	4	8.3	odd	2	inner
1728.4.f.b.863.3	yes	4	24.11	even	2	inner
1728.4.f.b.863.4	yes	4	4.3	odd	2	inner
1728.4.f.b.863.4	yes	4	12.11	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-17.0000i q^{7} +29.4449i q^{13} +126.440 q^{19} -125.000 q^{25} +308.000i q^{31} -122.976i q^{37} -218.238 q^{43} +54.0000 q^{49} +310.037i q^{61} +434.745 q^{67} +271.000 q^{73} +503.000i q^{79} +500.563 q^{91} +1853.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 500 q^{25} + 216 q^{49} + 1084 q^{73} + 7412 q^{97}+O(q^{100}) \) 4 * q - 500 * q^25 + 216 * q^49 + 1084 * q^73 + 7412 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more