LMFDB - Embedded newform 1728.3.b.h.1567.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(1567,1728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.1567");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1567.3
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1728.1567
Dual form			1728.3.b.h.1567.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16693i q^{5} -3.48528i q^{7} +O(q^{10})$	$q-3.16693i q^{5} -3.48528i q^{7} +1.13770 q^{11} +14.9706 q^{25} -29.3939i q^{29} -23.4264i q^{31} -11.0376 q^{35} +36.8528 q^{49} -56.9115i q^{53} -3.60303i q^{55} -117.576 q^{59} -93.7939 q^{73} -3.96522i q^{77} +58.0000i q^{79} -67.0576 q^{83} +61.0589 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{25} - 384 q^{49} + 200 q^{73} + 760 q^{97}+O(q^{100}) $ 8 * q - 16 * q^25 - 384 * q^49 + 200 * q^73 + 760 * q^97

Display 100 coefficients 10 coefficients

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
$5$	− 3.16693i	− 0.633386i	−0.948528	−	0.316693i	$-0.897428\pi$
$5$	− 3.16693i	− 0.633386i	0.948528	−	0.316693i	$-0.102572\pi$
$6$	0	0
$7$	− 3.48528i	− 0.497897i	−0.968517	−	0.248949i	$-0.919915\pi$
$7$	− 3.48528i	− 0.497897i	0.968517	−	0.248949i	$-0.0800850\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.13770	0.103428	0.0517139	−	0.998662i	$-0.483532\pi$
$11$	1.13770	0.103428	0.0517139	+	0.998662i	$0.483532\pi$
$12$	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		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	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	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	14.9706	0.598823
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 29.3939i	− 1.01358i	−0.862069	−	0.506791i	$-0.830832\pi$
$29$	− 29.3939i	− 1.01358i	0.862069	−	0.506791i	$-0.169168\pi$
$30$	0	0
$31$	− 23.4264i	− 0.755691i	−0.925869	−	0.377845i	$-0.876665\pi$
$31$	− 23.4264i	− 0.755691i	0.925869	−	0.377845i	$-0.123335\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−11.0376	−0.315361
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	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
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	36.8528	0.752098
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 56.9115i	− 1.07380i	−0.843645	−	0.536901i	$-0.819595\pi$
$53$	− 56.9115i	− 1.07380i	0.843645	−	0.536901i	$-0.180405\pi$
$54$	0	0
$55$	− 3.60303i	− 0.0655096i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−117.576	−1.99281	−0.996403	−	0.0847458i	$-0.972992\pi$
$59$	−117.576	−1.99281	−0.996403	+	0.0847458i	$0.972992\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−93.7939	−1.28485	−0.642424	−	0.766349i	$-0.722071\pi$
$73$	−93.7939	−1.28485	−0.642424	+	0.766349i	$0.722071\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.96522i	− 0.0514964i
$78$	0	0
$79$	58.0000i	0.734177i	0.930186	+	0.367089i	$0.119645\pi$
$79$	58.0000i	0.734177i	−0.930186	+	0.367089i	$0.880355\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−67.0576	−0.807923	−0.403962	−	0.914776i	$-0.632367\pi$
$83$	−67.0576	−0.807923	−0.403962	+	0.914776i	$0.632367\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	61.0589	0.629473	0.314736	−	0.949179i	$-0.398084\pi$
$97$	61.0589	0.629473	0.314736	+	0.949179i	$0.398084\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 198.838i	− 1.96869i	−0.176253	−	0.984345i	$-0.556398\pi$
$101$	− 198.838i	− 1.96869i	0.176253	−	0.984345i	$-0.443602\pi$
$102$	0	0
$103$	10.0000i	0.0970874i	0.998821	+	0.0485437i	$0.0154580\pi$
$103$	10.0000i	0.0970874i	−0.998821	+	0.0485437i	$0.984542\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−172.458	−1.61175	−0.805877	−	0.592082i	$-0.798306\pi$
$107$	−172.458	−1.61175	−0.805877	+	0.592082i	$0.798306\pi$
$108$	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		−	1.00000i	$-0.5\pi$
$113$	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
$121$	−119.706	−0.989303
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 126.584i	− 1.01267i
$126$	0	0
$127$	− 21.6619i	− 0.170566i	−0.996357	−	0.0852831i	$-0.972821\pi$
$127$	− 21.6619i	− 0.170566i	0.996357	−	0.0852831i	$-0.0271795\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−177.315	−1.35355	−0.676773	−	0.736192i	$-0.736622\pi$
$131$	−177.315	−1.35355	−0.676773	+	0.736192i	$0.736622\pi$
$132$	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
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−93.0883	−0.641988
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 216.855i	− 1.45540i	−0.685896	−	0.727700i	$-0.740589\pi$
$149$	− 216.855i	− 1.45540i	0.685896	−	0.727700i	$-0.259411\pi$
$150$	0	0
$151$	191.426i	1.26772i	0.773446	+	0.633862i	$0.218531\pi$
$151$	191.426i	1.26772i	−0.773446	+	0.633862i	$0.781469\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−74.1898	−0.478644
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	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	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 131.627i	− 0.760850i	−0.924812	−	0.380425i	$-0.875778\pi$
$173$	− 131.627i	− 0.760850i	0.924812	−	0.380425i	$-0.124222\pi$
$174$	0	0
$175$	− 52.1766i	− 0.298152i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−62.0144	−0.346449	−0.173225	−	0.984882i	$-0.555419\pi$
$179$	−62.0144	−0.346449	−0.173225	+	0.984882i	$0.555419\pi$
$180$	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
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	75.6173	0.391800	0.195900	−	0.980624i	$-0.437237\pi$
$193$	75.6173	0.391800	0.195900	+	0.980624i	$0.437237\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 339.473i	− 1.72321i	−0.507576	−	0.861607i	$-0.669458\pi$
$197$	− 339.473i	− 1.72321i	0.507576	−	0.861607i	$-0.330542\pi$
$198$	0	0
$199$	− 397.985i	− 1.99992i	−0.00872575	−	0.999962i	$-0.502778\pi$
$199$	− 397.985i	− 1.99992i	0.00872575	−	0.999962i	$-0.497222\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−102.446	−0.504660
$204$	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
$217$	−81.6476	−0.376256
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	230.000i	1.03139i	0.856772	+	0.515695i	$0.172466\pi$
$223$	230.000i	1.03139i	−0.856772	+	0.515695i	$0.827534\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−293.939	−1.29488	−0.647442	−	0.762115i	$-0.724161\pi$
$227$	−293.939	−1.29488	−0.647442	+	0.762115i	$0.724161\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	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	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	382.000	1.58506	0.792531	−	0.609831i	$-0.208763\pi$
$241$	382.000	1.58506	0.792531	+	0.609831i	$0.208763\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 116.710i	− 0.476368i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−176.363	−0.702642	−0.351321	−	0.936255i	$-0.614267\pi$
$251$	−176.363	−0.702642	−0.351321	+	0.936255i	$0.614267\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−180.235	−0.680131
$266$	0	0
$267$	0	0
$268$	0	0
$269$	323.333i	1.20198i	0.799257	+	0.600990i	$0.205227\pi$
$269$	323.333i	1.20198i	−0.799257	+	0.600990i	$0.794773\pi$
$270$	0	0
$271$	− 437.690i	− 1.61509i	−0.589803	−	0.807547i	$-0.700795\pi$
$271$	− 437.690i	− 1.61509i	0.589803	−	0.807547i	$-0.299205\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	17.0321	0.0619348
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	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
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	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
$293$	− 440.908i	− 1.50481i	−0.658703	−	0.752403i	$-0.728895\pi$
$293$	− 440.908i	− 1.50481i	0.658703	−	0.752403i	$-0.271105\pi$
$294$	0	0
$295$	372.353i	1.26221i
$296$	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
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	553.500	1.76837	0.884185	−	0.467138i	$-0.154715\pi$
$313$	553.500	1.76837	0.884185	+	0.467138i	$0.154715\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	558.696i	1.76245i	0.472698	+	0.881225i	$0.343280\pi$
$317$	558.696i	1.76245i	−0.472698	+	0.881225i	$0.656720\pi$
$318$	0	0
$319$	− 33.4416i	− 0.104832i
$320$	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
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−190.000	−0.563798	−0.281899	−	0.959444i	$-0.590964\pi$
$337$	−190.000	−0.563798	−0.281899	+	0.959444i	$0.590964\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 26.6523i	− 0.0781594i
$342$	0	0
$343$	− 299.221i	− 0.872365i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	434.727	1.25282	0.626408	−	0.779495i	$-0.284524\pi$
$347$	434.727	1.25282	0.626408	+	0.779495i	$0.284524\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−361.000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	297.039i	0.813805i
$366$	0	0
$367$	516.220i	1.40659i	0.710896	+	0.703297i	$0.248290\pi$
$367$	516.220i	1.40659i	−0.710896	+	0.703297i	$0.751710\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−198.353	−0.534643
$372$	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	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	−12.5576	−0.0326171
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 212.677i	− 0.546726i	−0.961911	−	0.273363i	$-0.911864\pi$
$389$	− 212.677i	− 0.546726i	0.961911	−	0.273363i	$-0.0881361\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	183.682	0.465017
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−19.5887	−0.0478942	−0.0239471	−	0.999713i	$-0.507623\pi$
$409$	−19.5887	−0.0478942	−0.0239471	+	0.999713i	$0.507623\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	409.784i	0.992212i
$414$	0	0
$415$	212.367i	0.511727i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	411.514	0.982134	0.491067	−	0.871122i	$-0.336607\pi$
$419$	411.514	0.982134	0.491067	+	0.871122i	$0.336607\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	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	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−810.176	−1.87108	−0.935538	−	0.353227i	$-0.885084\pi$
$433$	−810.176	−1.87108	−0.935538	+	0.353227i	$0.885084\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	860.249i	1.95956i	0.200066	+	0.979782i	$0.435884\pi$
$439$	860.249i	1.95956i	−0.200066	+	0.979782i	$0.564116\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	881.816	1.99056	0.995278	−	0.0970655i	$-0.0309456\pi$
$443$	881.816	1.99056	0.995278	+	0.0970655i	$0.0309456\pi$
$444$	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
$457$	909.881	1.99099	0.995494	−	0.0948261i	$-0.0302295\pi$
$457$	909.881	1.99099	0.995494	+	0.0948261i	$0.0302295\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	836.215i	1.81392i	0.421222	+	0.906958i	$0.361601\pi$
$461$	836.215i	1.81392i	−0.421222	+	0.906958i	$0.638399\pi$
$462$	0	0
$463$	− 647.161i	− 1.39776i	−0.715241	−	0.698878i	$-0.753683\pi$
$463$	− 647.161i	− 1.39776i	0.715241	−	0.698878i	$-0.246317\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	891.989	1.91004	0.955020	−	0.296541i	$-0.0958333\pi$
$467$	891.989	1.91004	0.955020	+	0.296541i	$0.0958333\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 193.369i	− 0.398699i
$486$	0	0
$487$	− 970.000i	− 1.99179i	−0.0905356	−	0.995893i	$-0.528858\pi$
$487$	− 970.000i	− 1.99179i	0.0905356	−	0.995893i	$-0.471142\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−24.0783	−0.0490392	−0.0245196	−	0.999699i	$-0.507806\pi$
$491$	−24.0783	−0.0490392	−0.0245196	+	0.999699i	$0.507806\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−629.705	−1.24694
$506$	0	0
$507$	0	0
$508$	0	0
$509$	810.999i	1.59332i	0.604429	+	0.796659i	$0.293401\pi$
$509$	810.999i	1.59332i	−0.604429	+	0.796659i	$0.706599\pi$
$510$	0	0
$511$	326.898i	0.639723i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	31.6693	0.0614938
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	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
$535$	546.161i	1.02086i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	41.9276	0.0777878
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	202.146	0.365545
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1109.98i	1.99278i	0.0848683	+	0.996392i	$0.472953\pi$
$557$	1109.98i	1.99278i	−0.0848683	+	0.996392i	$0.527047\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	821.212	1.45864	0.729318	−	0.684175i	$-0.239838\pi$
$563$	821.212	1.45864	0.729318	+	0.684175i	$0.239838\pi$
$564$	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
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−290.000	−0.502600	−0.251300	−	0.967909i	$-0.580858\pi$
$577$	−290.000	−0.502600	−0.251300	+	0.967909i	$0.580858\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	233.715i	0.402263i
$582$	0	0
$583$	− 64.7485i	− 0.111061i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	364.988	0.621785	0.310893	−	0.950445i	$-0.399372\pi$
$587$	364.988	0.621785	0.310893	+	0.950445i	$0.399372\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	514.147	0.855486	0.427743	−	0.903900i	$-0.359309\pi$
$601$	514.147	0.855486	0.427743	+	0.903900i	$0.359309\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	379.099i	0.626610i
$606$	0	0
$607$	− 730.000i	− 1.20264i	−0.799010	−	0.601318i	$-0.794643\pi$
$607$	− 730.000i	− 1.20264i	0.799010	−	0.601318i	$-0.205357\pi$
$608$	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		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	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
$625$	−26.6182	−0.0425891
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 831.132i	− 1.31717i	−0.752508	−	0.658583i	$-0.771156\pi$
$631$	− 831.132i	− 1.31717i	0.752508	−	0.658583i	$-0.228844\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−68.6017	−0.108034
$636$	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
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−133.766	−0.206111
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1287.63i	1.97188i	0.167112	+	0.985938i	$0.446556\pi$
$653$	1287.63i	1.97188i	−0.167112	+	0.985938i	$0.553444\pi$
$654$	0	0
$655$	561.542i	0.857317i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1180.88	1.79193	0.895967	−	0.444121i	$-0.146484\pi$
$659$	1180.88	1.79193	0.895967	+	0.444121i	$0.146484\pi$
$660$	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
$673$	1059.00	1.57355	0.786774	−	0.617241i	$-0.211750\pi$
$673$	1059.00	1.57355	0.786774	+	0.617241i	$0.211750\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 146.969i	− 0.217089i	−0.994092	−	0.108545i	$-0.965381\pi$
$677$	− 146.969i	− 0.217089i	0.994092	−	0.108545i	$-0.0346190\pi$
$678$	0	0
$679$	− 212.807i	− 0.313413i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	293.939	0.430364	0.215182	−	0.976574i	$-0.430965\pi$
$683$	293.939	0.430364	0.215182	+	0.976574i	$0.430965\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	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
$701$	743.855i	1.06113i	0.847643	+	0.530567i	$0.178021\pi$
$701$	743.855i	1.06113i	−0.847643	+	0.530567i	$0.821979\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−693.005	−0.980205
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	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
$721$	34.8528	0.0483395
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 440.043i	− 0.606956i
$726$	0	0
$727$	818.338i	1.12564i	0.826581	+	0.562818i	$0.190283\pi$
$727$	818.338i	1.12564i	−0.826581	+	0.562818i	$0.809717\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	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
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−686.763	−0.921829
$746$	0	0
$747$	0	0
$748$	0	0
$749$	601.064i	0.802488i
$750$	0	0
$751$	− 234.310i	− 0.311997i	−0.987757	−	0.155998i	$-0.950141\pi$
$751$	− 234.310i	− 0.311997i	0.987757	−	0.155998i	$-0.0498595\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	606.234	0.802959
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−672.087	−0.873975	−0.436987	−	0.899468i	$-0.643955\pi$
$769$	−672.087	−0.873975	−0.436987	+	0.899468i	$0.643955\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1028.79i	− 1.33090i	−0.746442	−	0.665450i	$-0.768240\pi$
$773$	− 1028.79i	− 1.33090i	0.746442	−	0.665450i	$-0.231760\pi$
$774$	0	0
$775$	− 350.706i	− 0.452525i
$776$	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
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	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
$797$	655.234i	0.822125i	0.911607	+	0.411063i	$0.134842\pi$
$797$	655.234i	0.822125i	−0.911607	+	0.411063i	$0.865158\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−106.710	−0.132889
$804$	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
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			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
$821$	− 1499.09i	− 1.82593i	−0.408039	−	0.912965i	$-0.633787\pi$
$821$	− 1499.09i	− 1.82593i	0.408039	−	0.912965i	$-0.366213\pi$
$822$	0	0
$823$	1131.13i	1.37440i	0.726469	+	0.687199i	$0.241160\pi$
$823$	1131.13i	1.37440i	−0.726469	+	0.687199i	$0.758840\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	587.878	0.710856	0.355428	−	0.934704i	$-0.384335\pi$
$827$	587.878	0.710856	0.355428	+	0.934704i	$0.384335\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	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
$841$	−23.0000	−0.0273484
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 535.211i	− 0.633386i
$846$	0	0
$847$	417.208i	0.492571i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	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	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	−416.854	−0.481912
$866$	0	0
$867$	0	0
$868$	0	0
$869$	65.9869i	0.0759343i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−441.181	−0.504206
$876$	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		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−75.4978	−0.0849244
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	196.395i	0.219436i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−688.593	−0.765954
$900$	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
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−76.2918	−0.0835616
$914$	0	0
$915$	0	0
$916$	0	0
$917$	617.991i	0.673927i
$918$	0	0
$919$	− 1815.92i	− 1.97598i	−0.154523	−	0.987989i	$-0.549384\pi$
$919$	− 1815.92i	− 1.97598i	0.154523	−	0.987989i	$-0.450616\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−239.293	−0.255382	−0.127691	−	0.991814i	$-0.540757\pi$
$937$	−239.293	−0.255382	−0.127691	+	0.991814i	$0.540757\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1288.50i	− 1.36929i	−0.728878	−	0.684644i	$-0.759958\pi$
$941$	− 1288.50i	− 1.36929i	0.728878	−	0.684644i	$-0.240042\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	445.172	0.470087	0.235043	−	0.971985i	$-0.424477\pi$
$947$	445.172	0.470087	0.235043	+	0.971985i	$0.424477\pi$
$948$	0	0
$949$	0	0
$950$	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
$961$	412.203	0.428932
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 239.475i	− 0.248160i
$966$	0	0
$967$	1218.04i	1.25961i	0.776753	+	0.629805i	$0.216865\pi$
$967$	1218.04i	1.25961i	−0.776753	+	0.629805i	$0.783135\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1779.21	−1.83234	−0.916172	−	0.400785i	$-0.868738\pi$
$971$	−1779.21	−1.83234	−0.916172	+	0.400785i	$0.868738\pi$
$972$	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
$985$	−1075.09	−1.09146
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1921.37i	1.93882i	0.245457	+	0.969408i	$0.421062\pi$
$991$	1921.37i	1.93882i	−0.245457	+	0.969408i	$0.578938\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1260.39	−1.26672
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.b.h.1567.3	✓	8
3.2	odd	2	inner	1728.3.b.h.1567.5	yes	8
4.3	odd	2	inner	1728.3.b.h.1567.4	yes	8
8.3	odd	2	inner	1728.3.b.h.1567.6	yes	8
8.5	even	2	inner	1728.3.b.h.1567.5	yes	8
12.11	even	2	inner	1728.3.b.h.1567.6	yes	8
24.5	odd	2	CM	1728.3.b.h.1567.3	✓	8
24.11	even	2	inner	1728.3.b.h.1567.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.b.h.1567.3	✓	8	1.1	even	1	trivial
1728.3.b.h.1567.3	✓	8	24.5	odd	2	CM
1728.3.b.h.1567.4	yes	8	4.3	odd	2	inner
1728.3.b.h.1567.4	yes	8	24.11	even	2	inner
1728.3.b.h.1567.5	yes	8	3.2	odd	2	inner
1728.3.b.h.1567.5	yes	8	8.5	even	2	inner
1728.3.b.h.1567.6	yes	8	8.3	odd	2	inner
1728.3.b.h.1567.6	yes	8	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-3.16693i q^{5} -3.48528i q^{7} +O(q^{10})\)	\(q-3.16693i q^{5} -3.48528i q^{7} +1.13770 q^{11} +14.9706 q^{25} -29.3939i q^{29} -23.4264i q^{31} -11.0376 q^{35} +36.8528 q^{49} -56.9115i q^{53} -3.60303i q^{55} -117.576 q^{59} -93.7939 q^{73} -3.96522i q^{77} +58.0000i q^{79} -67.0576 q^{83} +61.0589 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{25} - 384 q^{49} + 200 q^{73} + 760 q^{97}+O(q^{100}) \) 8 * q - 16 * q^25 - 384 * q^49 + 200 * q^73 + 760 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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