LMFDB - Embedded newform 1728.2.f.g.863.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(863,1728)]

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, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1728.863");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.f (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:	$13.7981494693$
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^{10}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			863.2
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.2.f.g.863.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.18154 q^{5} +5.24264i q^{7} +O(q^{10})$	$q-4.18154 q^{5} +5.24264i q^{7} +3.16693i q^{11} +12.4853 q^{25} -10.3923 q^{29} +0.757359i q^{31} -21.9223i q^{35} -20.4853 q^{49} +10.5154 q^{53} -13.2426i q^{55} -10.3923i q^{59} -1.48528 q^{73} -16.6031 q^{77} -10.0000i q^{79} -13.5592i q^{83} -17.9706 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 32 q^{25} - 96 q^{49} + 56 q^{73} - 8 q^{97}+O(q^{100}) $ 8 * q + 32 * q^25 - 96 * q^49 + 56 * q^73 - 8 * 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
$5$	−4.18154	−1.87004	−0.935021	−	0.354593i	$-0.884620\pi$
$5$	−4.18154	−1.87004	−0.935021	+	0.354593i	$0.884620\pi$
$6$	0	0
$7$	5.24264i	1.98153i	0.135583	+	0.990766i	$0.456709\pi$
$7$	5.24264i	1.98153i	−0.135583	+	0.990766i	$0.543291\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.16693i	0.954865i	0.878668	+	0.477432i	$0.158432\pi$
$11$	3.16693i	0.954865i	−0.878668	+	0.477432i	$0.841568\pi$
$12$	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
$25$	12.4853	2.49706
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.3923	−1.92980	−0.964901	−	0.262613i	$-0.915416\pi$
$29$	−10.3923	−1.92980	−0.964901	+	0.262613i	$0.915416\pi$
$30$	0	0
$31$	0.757359i	0.136026i	0.997684	+	0.0680129i	$0.0216659\pi$
$31$	0.757359i	0.136026i	−0.997684	+	0.0680129i	$0.978334\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 21.9223i	− 3.70555i
$36$	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	1.00000			$0$
$41$	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
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−20.4853	−2.92647
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.5154	1.44440	0.722200	−	0.691684i	$-0.243131\pi$
$53$	10.5154	1.44440	0.722200	+	0.691684i	$0.243131\pi$
$54$	0	0
$55$	− 13.2426i	− 1.78564i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.3923i	− 1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	− 10.3923i	− 1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$	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		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−1.48528	−0.173839	−0.0869195	−	0.996215i	$-0.527702\pi$
$73$	−1.48528	−0.173839	−0.0869195	+	0.996215i	$0.527702\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−16.6031	−1.89210
$78$	0	0
$79$	− 10.0000i	− 1.12509i	−0.826767	−	0.562544i	$-0.809823\pi$
$79$	− 10.0000i	− 1.12509i	0.826767	−	0.562544i	$-0.190177\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 13.5592i	− 1.48832i	−0.668002	−	0.744160i	$-0.732850\pi$
$83$	− 13.5592i	− 1.48832i	0.668002	−	0.744160i	$-0.267150\pi$
$84$	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
$97$	−17.9706	−1.82463	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−17.9706	−1.82463	−0.912317	+	0.409484i	$0.865709\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.8785	−1.87848	−0.939239	−	0.343263i	$-0.888468\pi$
$101$	−18.8785	−1.87848	−0.939239	+	0.343263i	$0.888468\pi$
$102$	0	0
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.13770i	0.109986i	0.998487	+	0.0549930i	$0.0175137\pi$
$107$	1.13770i	0.109986i	−0.998487	+	0.0549930i	$0.982486\pi$
$108$	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
$121$	0.970563	0.0882330
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−31.3000	−2.79956
$126$	0	0
$127$	− 6.75736i	− 0.599619i	−0.953999	−	0.299809i	$-0.903077\pi$
$127$	− 6.75736i	− 0.599619i	0.953999	−	0.299809i	$-0.0969231\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	17.8639i	1.56077i	0.625297	+	0.780387i	$0.284978\pi$
$131$	17.8639i	1.56077i	−0.625297	+	0.780387i	$0.715022\pi$
$132$	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
$139$	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
$145$	43.4558	3.60881
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.5738	1.19394	0.596968	−	0.802265i	$-0.296372\pi$
$149$	14.5738	1.19394	0.596968	+	0.802265i	$0.296372\pi$
$150$	0	0
$151$	− 20.2132i	− 1.64493i	−0.568818	−	0.822464i	$-0.692599\pi$
$151$	− 20.2132i	− 1.64493i	0.568818	−	0.822464i	$-0.307401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.16693i	− 0.254374i
$156$	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		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.48617	0.645192	0.322596	−	0.946537i	$-0.395445\pi$
$173$	8.48617	0.645192	0.322596	+	0.946537i	$0.395445\pi$
$174$	0	0
$175$	65.4558i	4.94800i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	21.9223i	1.63855i	0.573400	+	0.819275i	$0.305624\pi$
$179$	21.9223i	1.63855i	−0.573400	+	0.819275i	$0.694376\pi$
$180$	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		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−4.51472	−0.324977	−0.162488	−	0.986710i	$-0.551952\pi$
$193$	−4.51472	−0.324977	−0.162488	+	0.986710i	$0.551952\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.123093	−0.00877002	−0.00438501	−	0.999990i	$-0.501396\pi$
$197$	−0.123093	−0.00877002	−0.00438501	+	0.999990i	$0.501396\pi$
$198$	0	0
$199$	14.2132i	1.00755i	0.863836	+	0.503774i	$0.168055\pi$
$199$	14.2132i	1.00755i	−0.863836	+	0.503774i	$0.831945\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 54.4831i	− 3.82397i
$204$	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
$217$	−3.97056	−0.269539
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 26.0000i	− 1.74109i	−0.492090	−	0.870544i	$-0.663767\pi$
$223$	− 26.0000i	− 1.74109i	0.492090	−	0.870544i	$-0.336233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 10.3923i	− 0.689761i	−0.938647	−	0.344881i	$-0.887919\pi$
$227$	− 10.3923i	− 0.689761i	0.938647	−	0.344881i	$-0.112081\pi$
$228$	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	1.00000			$0$
$233$	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
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	85.6600	5.47262
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	31.1769i	1.96787i	0.178529	+	0.983935i	$0.442866\pi$
$251$	31.1769i	1.96787i	−0.178529	+	0.983935i	$0.557134\pi$
$252$	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
$265$	−43.9706	−2.70109
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.3923	−0.633630	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	−10.3923	−0.633630	−0.316815	+	0.948487i	$0.602613\pi$
$270$	0	0
$271$	− 32.2132i	− 1.95681i	−0.206691	−	0.978406i	$-0.566269\pi$
$271$	− 32.2132i	− 1.95681i	0.206691	−	0.978406i	$-0.433731\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	39.5400i	2.38435i
$276$	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	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	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
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	31.1769	1.82137	0.910687	−	0.413096i	$-0.135553\pi$
$293$	31.1769	1.82137	0.910687	+	0.413096i	$0.135553\pi$
$294$	0	0
$295$	43.4558i	2.53010i
$296$	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		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−25.4853	−1.44051	−0.720257	−	0.693708i	$-0.755976\pi$
$313$	−25.4853	−1.44051	−0.720257	+	0.693708i	$0.755976\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−35.6046	−1.99976	−0.999878	−	0.0156238i	$-0.995027\pi$
$317$	−35.6046	−1.99976	−0.999878	+	0.0156238i	$0.995027\pi$
$318$	0	0
$319$	− 32.9117i	− 1.84270i
$320$	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
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2.39850	−0.129886
$342$	0	0
$343$	− 70.6985i	− 3.81736i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	36.6193i	1.96582i	0.184075	+	0.982912i	$0.441071\pi$
$347$	36.6193i	1.96582i	−0.184075	+	0.982912i	$0.558929\pi$
$348$	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	1.00000			$0$
$353$	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
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.21076	0.325086
$366$	0	0
$367$	− 23.2426i	− 1.21326i	−0.794986	−	0.606628i	$-0.792522\pi$
$367$	− 23.2426i	− 1.21326i	0.794986	−	0.606628i	$-0.207478\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	55.1285i	2.86213i
$372$	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
$385$	69.4264	3.53830
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.8200	−0.751405	−0.375703	−	0.926740i	$-0.622599\pi$
$389$	−14.8200	−0.751405	−0.375703	+	0.926740i	$0.622599\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	41.8154i	2.10396i
$396$	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	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−38.9411	−1.92551	−0.962757	−	0.270367i	$-0.912855\pi$
$409$	−38.9411	−1.92551	−0.962757	+	0.270367i	$0.912855\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	54.4831	2.68094
$414$	0	0
$415$	56.6985i	2.78322i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 10.3923i	− 0.507697i	−0.967244	−	0.253849i	$-0.918303\pi$
$419$	− 10.3923i	− 0.507697i	0.967244	−	0.253849i	$-0.0816965\pi$
$420$	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		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−26.9411	−1.29471	−0.647354	−	0.762190i	$-0.724124\pi$
$433$	−26.9411	−1.29471	−0.647354	+	0.762190i	$0.724124\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	38.2132i	1.82382i	0.410394	+	0.911908i	$0.365391\pi$
$439$	38.2132i	1.82382i	−0.410394	+	0.911908i	$0.634609\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.1769i	1.48126i	0.671913	+	0.740630i	$0.265473\pi$
$443$	31.1769i	1.48126i	−0.671913	+	0.740630i	$0.734527\pi$
$444$	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
$457$	35.9706	1.68263	0.841316	−	0.540544i	$-0.181781\pi$
$457$	35.9706	1.68263	0.841316	+	0.540544i	$0.181781\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.90613	0.0887774	0.0443887	−	0.999014i	$-0.485866\pi$
$461$	1.90613	0.0887774	0.0443887	+	0.999014i	$0.485866\pi$
$462$	0	0
$463$	42.6985i	1.98437i	0.124788	+	0.992183i	$0.460175\pi$
$463$	42.6985i	1.98437i	−0.124788	+	0.992183i	$0.539825\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 42.9531i	− 1.98763i	−0.111035	−	0.993816i	$-0.535417\pi$
$467$	− 42.9531i	− 1.98763i	0.111035	−	0.993816i	$-0.464583\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	75.1446	3.41214
$486$	0	0
$487$	− 2.00000i	− 0.0906287i	−0.998973	−	0.0453143i	$-0.985571\pi$
$487$	− 2.00000i	− 0.0906287i	0.998973	−	0.0453143i	$-0.0144289\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	38.6485i	1.74418i	0.489344	+	0.872091i	$0.337236\pi$
$491$	38.6485i	1.74418i	−0.489344	+	0.872091i	$0.662764\pi$
$492$	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		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	78.9411	3.51283
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.9662	−1.10661	−0.553303	−	0.832980i	$-0.686633\pi$
$509$	−24.9662	−1.10661	−0.553303	+	0.832980i	$0.686633\pi$
$510$	0	0
$511$	− 7.78680i	− 0.344468i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 58.5416i	− 2.57965i
$516$	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
$523$	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
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 4.75736i	− 0.205679i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 64.8754i	− 2.79438i
$540$	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
$553$	52.4264	2.22940
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.2692	−0.435120	−0.217560	−	0.976047i	$-0.569810\pi$
$557$	−10.2692	−0.435120	−0.217560	+	0.976047i	$0.569810\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 5.44234i	− 0.229367i	−0.993402	−	0.114684i	$-0.963415\pi$
$563$	− 5.44234i	− 0.229367i	0.993402	−	0.114684i	$-0.0365854\pi$
$564$	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
$571$	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
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	71.0862	2.94915
$582$	0	0
$583$	33.3015i	1.37921i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	30.5316i	1.26017i	0.776525	+	0.630087i	$0.216981\pi$
$587$	30.5316i	1.26017i	−0.776525	+	0.630087i	$0.783019\pi$
$588$	0	0
$589$	0	0
$590$	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
$601$	−43.4264	−1.77140	−0.885700	−	0.464258i	$-0.846321\pi$
$601$	−43.4264	−1.77140	−0.885700	+	0.464258i	$0.846321\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.05845	−0.164999
$606$	0	0
$607$	22.0000i	0.892952i	0.894795	+	0.446476i	$0.147321\pi$
$607$	22.0000i	0.892952i	−0.894795	+	0.446476i	$0.852679\pi$
$608$	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
$625$	68.4558	2.73823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	29.2426i	1.16413i	0.813142	+	0.582066i	$0.197755\pi$
$631$	29.2426i	1.16413i	−0.813142	+	0.582066i	$0.802245\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	28.2562i	1.12131i
$636$	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
$643$	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		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	32.9117	1.29190
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−50.3016	−1.96845	−0.984226	−	0.176913i	$-0.943389\pi$
$653$	−50.3016	−1.96845	−0.984226	+	0.176913i	$0.943389\pi$
$654$	0	0
$655$	− 74.6985i	− 2.91871i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	51.3162i	1.99900i	0.0316976	+	0.999498i	$0.489909\pi$
$659$	51.3162i	1.99900i	−0.0316976	+	0.999498i	$0.510091\pi$
$660$	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
$673$	−50.9411	−1.96364	−0.981818	−	0.189824i	$-0.939208\pi$
$673$	−50.9411	−1.96364	−0.981818	+	0.189824i	$0.939208\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−51.9615	−1.99704	−0.998522	−	0.0543526i	$-0.982690\pi$
$677$	−51.9615	−1.99704	−0.998522	+	0.0543526i	$0.982690\pi$
$678$	0	0
$679$	− 94.2132i	− 3.61557i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 51.9615i	− 1.98825i	−0.108227	−	0.994126i	$-0.534517\pi$
$683$	− 51.9615i	− 1.98825i	0.108227	−	0.994126i	$-0.465483\pi$
$684$	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
$701$	−12.7908	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$701$	−12.7908	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 98.9731i	− 3.72227i
$708$	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		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−73.3970	−2.73345
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−129.751	−4.81883
$726$	0	0
$727$	− 45.6690i	− 1.69377i	−0.531775	−	0.846886i	$-0.678475\pi$
$727$	− 45.6690i	− 1.69377i	0.531775	−	0.846886i	$-0.321525\pi$
$728$	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		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	−60.9411	−2.23271
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−5.96458	−0.217941
$750$	0	0
$751$	51.6690i	1.88543i	0.333599	+	0.942715i	$0.391737\pi$
$751$	51.6690i	1.88543i	−0.333599	+	0.942715i	$0.608263\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	84.5223i	3.07608i
$756$	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	1.00000			$0$
$761$	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
$769$	−55.4264	−1.99873	−0.999364	−	0.0356685i	$-0.988644\pi$
$769$	−55.4264	−1.99873	−0.999364	+	0.0356685i	$0.988644\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−51.9615	−1.86893	−0.934463	−	0.356060i	$-0.884120\pi$
$773$	−51.9615	−1.86893	−0.934463	+	0.356060i	$0.884120\pi$
$774$	0	0
$775$	9.45584i	0.339664i
$776$	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
$797$	37.8801	1.34178	0.670890	−	0.741557i	$-0.265912\pi$
$797$	37.8801	1.34178	0.670890	+	0.741557i	$0.265912\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 4.70378i	− 0.165993i
$804$	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
$811$	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
$821$	31.1769	1.08808	0.544041	−	0.839059i	$-0.316894\pi$
$821$	31.1769	1.08808	0.544041	+	0.839059i	$0.316894\pi$
$822$	0	0
$823$	6.69848i	0.233495i	0.993162	+	0.116747i	$0.0372467\pi$
$823$	6.69848i	0.233495i	−0.993162	+	0.116747i	$0.962753\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 10.3923i	− 0.361376i	−0.983540	−	0.180688i	$-0.942168\pi$
$827$	− 10.3923i	− 0.361376i	0.983540	−	0.180688i	$-0.0578324\pi$
$828$	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		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	79.0000	2.72414
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−54.3600	−1.87004
$846$	0	0
$847$	5.08831i	0.174836i
$848$	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	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	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
$865$	−35.4853	−1.20654
$866$	0	0
$867$	0	0
$868$	0	0
$869$	31.6693	1.07431
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 164.095i	− 5.54741i
$876$	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
$883$	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		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	35.4264	1.18816
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 91.6690i	− 3.06416i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 7.87071i	− 0.262503i
$900$	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		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	42.9411	1.42114
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−93.6538	−3.09272
$918$	0	0
$919$	54.6985i	1.80434i	0.431384	+	0.902168i	$0.358025\pi$
$919$	54.6985i	1.80434i	−0.431384	+	0.902168i	$0.641975\pi$
$920$	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	1.00000			$0$
$929$	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
$937$	−12.0294	−0.392985	−0.196492	−	0.980505i	$-0.562955\pi$
$937$	−12.0294	−0.392985	−0.196492	+	0.980505i	$0.562955\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	60.6939	1.97856	0.989282	−	0.146017i	$-0.0466455\pi$
$941$	60.6939	1.97856	0.989282	+	0.146017i	$0.0466455\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 36.8654i	− 1.19797i	−0.800762	−	0.598983i	$-0.795572\pi$
$947$	− 36.8654i	− 1.19797i	0.800762	−	0.598983i	$-0.204428\pi$
$948$	0	0
$949$	0	0
$950$	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
$961$	30.4264	0.981497
$962$	0	0
$963$	0	0
$964$	0	0
$965$	18.8785	0.607720
$966$	0	0
$967$	− 35.2426i	− 1.13333i	−0.823949	−	0.566663i	$-0.808234\pi$
$967$	− 35.2426i	− 1.13333i	0.823949	−	0.566663i	$-0.191766\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 55.6208i	− 1.78496i	−0.451090	−	0.892479i	$-0.648965\pi$
$971$	− 55.6208i	− 1.78496i	0.451090	−	0.892479i	$-0.351035\pi$
$972$	0	0
$973$	0	0
$974$	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
$985$	0.514719	0.0164003
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	50.2132i	1.59507i	0.603269	+	0.797537i	$0.293864\pi$
$991$	50.2132i	1.59507i	−0.603269	+	0.797537i	$0.706136\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 59.4331i	− 1.88416i
$996$	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	1728.2.f.g.863.2	yes	8
3.2	odd	2	inner	1728.2.f.g.863.8	yes	8
4.3	odd	2	inner	1728.2.f.g.863.1	✓	8
8.3	odd	2	inner	1728.2.f.g.863.7	yes	8
8.5	even	2	inner	1728.2.f.g.863.8	yes	8
12.11	even	2	inner	1728.2.f.g.863.7	yes	8
24.5	odd	2	CM	1728.2.f.g.863.2	yes	8
24.11	even	2	inner	1728.2.f.g.863.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.2.f.g.863.1	✓	8	4.3	odd	2	inner
1728.2.f.g.863.1	✓	8	24.11	even	2	inner
1728.2.f.g.863.2	yes	8	1.1	even	1	trivial
1728.2.f.g.863.2	yes	8	24.5	odd	2	CM
1728.2.f.g.863.7	yes	8	8.3	odd	2	inner
1728.2.f.g.863.7	yes	8	12.11	even	2	inner
1728.2.f.g.863.8	yes	8	3.2	odd	2	inner
1728.2.f.g.863.8	yes	8	8.5	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.18154 q^{5} +5.24264i q^{7} +O(q^{10})\)	\(q-4.18154 q^{5} +5.24264i q^{7} +3.16693i q^{11} +12.4853 q^{25} -10.3923 q^{29} +0.757359i q^{31} -21.9223i q^{35} -20.4853 q^{49} +10.5154 q^{53} -13.2426i q^{55} -10.3923i q^{59} -1.48528 q^{73} -16.6031 q^{77} -10.0000i q^{79} -13.5592i q^{83} -17.9706 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 32 q^{25} - 96 q^{49} + 56 q^{73} - 8 q^{97}+O(q^{100}) \) 8 * q + 32 * q^25 - 96 * q^49 + 56 * q^73 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more