LMFDB - Embedded newform 1728.4.f.f.863.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,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, 4, names="a")

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

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

Self dual:	no
Analytic conductor:	$101.955300490$
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^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			863.3
Root			$0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.4.f.f.863.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.9503 q^{5} -29.7279i q^{7} +O(q^{10})$	$q-11.9503 q^{5} -29.7279i q^{7} +41.2720i q^{11} +17.8091 q^{25} -218.238 q^{29} -257.742i q^{31} +355.257i q^{35} -540.749 q^{49} -247.534 q^{53} -493.212i q^{55} +717.069i q^{59} -1204.69 q^{73} +1226.93 q^{77} +1370.00i q^{79} -1504.75i q^{83} +1865.26 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 1568 q^{25} - 864 q^{49} - 1288 q^{73} + 2296 q^{97}+O(q^{100}) $ 8 * q + 1568 * q^25 - 864 * q^49 - 1288 * q^73 + 2296 * 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
$5$	−11.9503	−1.06887	−0.534433	−	0.845211i	$-0.679475\pi$
$5$	−11.9503	−1.06887	−0.534433	+	0.845211i	$0.679475\pi$
$6$	0	0
$7$	− 29.7279i	− 1.60516i	−0.596547	−	0.802578i	$-0.703461\pi$
$7$	− 29.7279i	− 1.60516i	0.596547	−	0.802578i	$-0.296539\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	41.2720i	1.13127i	0.824655	+	0.565636i	$0.191369\pi$
$11$	41.2720i	1.13127i	−0.824655	+	0.565636i	$0.808631\pi$
$12$	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
$25$	17.8091	0.142473
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−218.238	−1.39744	−0.698722	−	0.715394i	$-0.746247\pi$
$29$	−218.238	−1.39744	−0.698722	+	0.715394i	$0.746247\pi$
$30$	0	0
$31$	− 257.742i	− 1.49329i	−0.665225	−	0.746643i	$-0.731664\pi$
$31$	− 257.742i	− 1.49329i	0.665225	−	0.746643i	$-0.268336\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	355.257i	1.71570i
$36$	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
$49$	−540.749	−1.57653
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−247.534	−0.641536	−0.320768	−	0.947158i	$-0.603941\pi$
$53$	−247.534	−0.641536	−0.320768	+	0.947158i	$0.603941\pi$
$54$	0	0
$55$	− 493.212i	− 1.20918i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	717.069i	1.58228i	0.611636	+	0.791139i	$0.290512\pi$
$59$	717.069i	1.58228i	−0.611636	+	0.791139i	$0.709488\pi$
$60$	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
$73$	−1204.69	−1.93148	−0.965741	−	0.259506i	$-0.916440\pi$
$73$	−1204.69	−1.93148	−0.965741	+	0.259506i	$0.916440\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1226.93	1.81587
$78$	0	0
$79$	1370.00i	1.95110i	0.219774	+	0.975551i	$0.429468\pi$
$79$	1370.00i	1.95110i	−0.219774	+	0.975551i	$0.570532\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1504.75i	− 1.98998i	−0.0999988	−	0.994988i	$-0.531884\pi$
$83$	− 1504.75i	− 1.98998i	0.0999988	−	0.994988i	$-0.468116\pi$
$84$	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
$97$	1865.26	1.95246	0.976230	−	0.216735i	$-0.0695409\pi$
$97$	1865.26	1.95246	0.976230	+	0.216735i	$0.0695409\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2030.06	1.99998	0.999992	−	0.00396786i	$-0.00126301\pi$
$101$	2030.06	1.99998	0.999992	+	0.00396786i	$0.00126301\pi$
$102$	0	0
$103$	− 1582.00i	− 1.51339i	−0.653768	−	0.756695i	$-0.726813\pi$
$103$	− 1582.00i	− 1.51339i	0.653768	−	0.756695i	$-0.273187\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2072.87i	1.87282i	0.350904	+	0.936411i	$0.385874\pi$
$107$	2072.87i	1.87282i	−0.350904	+	0.936411i	$0.614126\pi$
$108$	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
$121$	−372.382	−0.279776
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1280.96	0.916581
$126$	0	0
$127$	381.623i	0.266642i	0.991073	+	0.133321i	$0.0425641\pi$
$127$	381.623i	0.266642i	−0.991073	+	0.133321i	$0.957436\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1672.00i	− 1.11514i	−0.830129	−	0.557571i	$-0.811733\pi$
$131$	− 1672.00i	− 1.11514i	0.830129	−	0.557571i	$-0.188267\pi$
$132$	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
$145$	2608.01	1.49368
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2784.86	1.53117	0.765586	−	0.643334i	$-0.222449\pi$
$149$	2784.86	1.53117	0.765586	+	0.643334i	$0.222449\pi$
$150$	0	0
$151$	2669.34i	1.43860i	0.694702	+	0.719298i	$0.255536\pi$
$151$	2669.34i	1.43860i	−0.694702	+	0.719298i	$0.744464\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3080.09i	1.59612i
$156$	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
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4074.19	1.79049	0.895246	−	0.445572i	$-0.147000\pi$
$173$	4074.19	1.79049	0.895246	+	0.445572i	$0.147000\pi$
$174$	0	0
$175$	− 529.427i	− 0.228691i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4625.71i	1.93152i	0.259442	+	0.965759i	$0.416461\pi$
$179$	4625.71i	1.93152i	−0.259442	+	0.965759i	$0.583539\pi$
$180$	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
$193$	2837.39	1.05824	0.529119	−	0.848548i	$-0.322523\pi$
$193$	2837.39	1.05824	0.529119	+	0.848548i	$0.322523\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4825.13	−1.74506	−0.872528	−	0.488564i	$-0.837521\pi$
$197$	−4825.13	−1.74506	−0.872528	+	0.488564i	$0.837521\pi$
$198$	0	0
$199$	2743.36i	0.977245i	0.872495	+	0.488622i	$0.162500\pi$
$199$	2743.36i	0.977245i	−0.872495	+	0.488622i	$0.837500\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6487.77i	2.24312i
$204$	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
$217$	−7662.14	−2.39696
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 182.000i	− 0.0546530i	−0.999627	−	0.0273265i	$-0.991301\pi$
$223$	− 182.000i	− 0.0546530i	0.999627	−	0.0273265i	$-0.00869938\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5954.79i	1.74112i	0.492066	+	0.870558i	$0.336242\pi$
$227$	5954.79i	1.74112i	−0.492066	+	0.870558i	$0.663758\pi$
$228$	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
$241$	−6230.00	−1.66518	−0.832592	−	0.553886i	$-0.813144\pi$
$241$	−6230.00	−1.66518	−0.832592	+	0.553886i	$0.813144\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6462.10	1.68510
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6827.74i	1.71699i	0.512826	+	0.858493i	$0.328599\pi$
$251$	6827.74i	1.71699i	−0.512826	+	0.858493i	$0.671401\pi$
$252$	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
$265$	2958.10	0.685715
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7264.22	1.64650	0.823248	−	0.567682i	$-0.192160\pi$
$269$	7264.22	1.64650	0.823248	+	0.567682i	$0.192160\pi$
$270$	0	0
$271$	− 899.412i	− 0.201607i	−0.994906	−	0.100803i	$-0.967859\pi$
$271$	− 899.412i	− 0.201607i	0.994906	−	0.100803i	$-0.0321413\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	735.018i	0.161175i
$276$	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
$289$	4913.00	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2899.45	0.578116	0.289058	−	0.957312i	$-0.406658\pi$
$293$	2899.45	0.578116	0.289058	+	0.957312i	$0.406658\pi$
$294$	0	0
$295$	− 8569.17i	− 1.69124i
$296$	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
$313$	−10842.1	−1.95793	−0.978964	−	0.204034i	$-0.934595\pi$
$313$	−10842.1	−1.95793	−0.978964	+	0.204034i	$0.934595\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6095.88	1.08006	0.540030	−	0.841646i	$-0.318413\pi$
$317$	6095.88	1.08006	0.540030	+	0.841646i	$0.318413\pi$
$318$	0	0
$319$	− 9007.15i	− 1.58089i
$320$	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
$337$	11594.0	1.87408	0.937041	−	0.349220i	$-0.113553\pi$
$337$	11594.0	1.87408	0.937041	+	0.349220i	$0.113553\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10637.5	1.68931
$342$	0	0
$343$	5878.68i	0.925419i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	410.937i	0.0635742i	0.999495	+	0.0317871i	$0.0101199\pi$
$347$	410.937i	0.0635742i	−0.999495	+	0.0317871i	$0.989880\pi$
$348$	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
$361$	−6859.00	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14396.4	2.06449
$366$	0	0
$367$	− 11086.3i	− 1.57684i	−0.615135	−	0.788422i	$-0.710899\pi$
$367$	− 11086.3i	− 1.57684i	0.615135	−	0.788422i	$-0.289101\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	7358.67i	1.02977i
$372$	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
$385$	−14662.2	−1.94092
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12381.9	−1.61385	−0.806927	−	0.590652i	$-0.798871\pi$
$389$	−12381.9	−1.61385	−0.806927	+	0.590652i	$0.798871\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 16371.9i	− 2.08546i
$396$	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
$409$	16122.8	1.94920	0.974598	−	0.223959i	$-0.0718983\pi$
$409$	16122.8	1.94920	0.974598	+	0.223959i	$0.0718983\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	21317.0	2.53981
$414$	0	0
$415$	17982.2i	2.12702i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11940.8i	1.39223i	0.717930	+	0.696115i	$0.245090\pi$
$419$	11940.8i	1.39223i	−0.717930	+	0.696115i	$0.754910\pi$
$420$	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
$433$	323.047	0.0358537	0.0179268	−	0.999839i	$-0.494293\pi$
$433$	323.047	0.0358537	0.0179268	+	0.999839i	$0.494293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	12472.9i	1.35603i	0.735048	+	0.678015i	$0.237160\pi$
$439$	12472.9i	1.35603i	−0.735048	+	0.678015i	$0.762840\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 11130.2i	− 1.19370i	−0.802352	−	0.596851i	$-0.796418\pi$
$443$	− 11130.2i	− 1.19370i	0.802352	−	0.596851i	$-0.203582\pi$
$444$	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
$457$	18136.9	1.85648	0.928239	−	0.371984i	$-0.121322\pi$
$457$	18136.9	1.85648	0.928239	+	0.371984i	$0.121322\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15677.5	−1.58389	−0.791945	−	0.610593i	$-0.790931\pi$
$461$	−15677.5	−1.58389	−0.791945	+	0.610593i	$0.790931\pi$
$462$	0	0
$463$	− 2943.22i	− 0.295428i	−0.989030	−	0.147714i	$-0.952808\pi$
$463$	− 2943.22i	− 0.295428i	0.989030	−	0.147714i	$-0.0471915\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15261.8i	1.51228i	0.654411	+	0.756139i	$0.272917\pi$
$467$	15261.8i	1.51228i	−0.654411	+	0.756139i	$0.727083\pi$
$468$	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
$485$	−22290.4	−2.08692
$486$	0	0
$487$	2914.00i	0.271142i	0.990768	+	0.135571i	$0.0432868\pi$
$487$	2914.00i	0.271142i	−0.990768	+	0.135571i	$0.956713\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18431.6i	1.69410i	0.531510	+	0.847052i	$0.321625\pi$
$491$	18431.6i	1.69410i	−0.531510	+	0.847052i	$0.678375\pi$
$492$	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
$505$	−24259.8	−2.13771
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15001.6	−1.30636	−0.653178	−	0.757204i	$-0.726565\pi$
$509$	−15001.6	−1.30636	−0.653178	+	0.757204i	$0.726565\pi$
$510$	0	0
$511$	35812.9i	3.10033i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	18905.3i	1.61761i
$516$	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
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 24771.4i	− 2.00180i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 22317.8i	− 1.78348i
$540$	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
$553$	40727.3	3.13182
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9977.55	0.758999	0.379499	−	0.925192i	$-0.376096\pi$
$557$	9977.55	0.758999	0.379499	+	0.925192i	$0.376096\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 26291.4i	− 1.96812i	−0.177838	−	0.984060i	$-0.556910\pi$
$563$	− 26291.4i	− 1.96812i	0.177838	−	0.984060i	$-0.443090\pi$
$564$	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
$577$	10906.0	0.786868	0.393434	−	0.919353i	$-0.371287\pi$
$577$	10906.0	0.786868	0.393434	+	0.919353i	$0.371287\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−44733.1	−3.19422
$582$	0	0
$583$	− 10216.2i	− 0.725751i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1404.57i	0.0987615i	0.998780	+	0.0493807i	$0.0157248\pi$
$587$	1404.57i	0.0987615i	−0.998780	+	0.0493807i	$0.984275\pi$
$588$	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
$601$	27127.6	1.84119	0.920596	−	0.390516i	$-0.127703\pi$
$601$	27127.6	1.84119	0.920596	+	0.390516i	$0.127703\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4450.07	0.299043
$606$	0	0
$607$	− 29414.0i	− 1.96685i	−0.181318	−	0.983425i	$-0.558036\pi$
$607$	− 29414.0i	− 1.96685i	0.181318	−	0.983425i	$-0.441964\pi$
$608$	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
$625$	−17534.0	−1.12217
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	23104.5i	1.45765i	0.684701	+	0.728824i	$0.259933\pi$
$631$	23104.5i	1.45765i	−0.684701	+	0.728824i	$0.740067\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 4560.50i	− 0.285004i
$636$	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
$649$	−29594.9	−1.78999
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29066.7	1.74191	0.870957	−	0.491359i	$-0.163500\pi$
$653$	29066.7	1.74191	0.870957	+	0.491359i	$0.163500\pi$
$654$	0	0
$655$	19980.9i	1.19194i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 19623.4i	− 1.15997i	−0.814629	−	0.579983i	$-0.803059\pi$
$659$	− 19623.4i	− 1.15997i	0.814629	−	0.579983i	$-0.196941\pi$
$660$	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
$673$	−1722.56	−0.0986626	−0.0493313	−	0.998782i	$-0.515709\pi$
$673$	−1722.56	−0.0986626	−0.0493313	+	0.998782i	$0.515709\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34762.3	−1.97344	−0.986722	−	0.162416i	$-0.948071\pi$
$677$	−34762.3	−1.97344	−0.986722	+	0.162416i	$0.948071\pi$
$678$	0	0
$679$	− 55450.4i	− 3.13401i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 33827.0i	− 1.89510i	−0.319610	−	0.947549i	$-0.603552\pi$
$683$	− 33827.0i	− 1.89510i	0.319610	−	0.947549i	$-0.396448\pi$
$684$	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
$701$	−36317.6	−1.95677	−0.978385	−	0.206790i	$-0.933698\pi$
$701$	−36317.6	−1.95677	−0.978385	+	0.206790i	$0.933698\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 60349.4i	− 3.21029i
$708$	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
$721$	−47029.6	−2.42923
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3886.63	−0.199098
$726$	0	0
$727$	31564.7i	1.61028i	0.593087	+	0.805138i	$0.297909\pi$
$727$	31564.7i	1.61028i	−0.593087	+	0.805138i	$0.702091\pi$
$728$	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
$745$	−33279.8	−1.63662
$746$	0	0
$747$	0	0
$748$	0	0
$749$	61622.2	3.00617
$750$	0	0
$751$	− 41146.5i	− 1.99928i	−0.0268390	−	0.999640i	$-0.508544\pi$
$751$	− 41146.5i	− 1.99928i	0.0268390	−	0.999640i	$-0.491456\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 31899.4i	− 1.53766i
$756$	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
$769$	25148.7	1.17930	0.589651	−	0.807658i	$-0.299265\pi$
$769$	25148.7	1.17930	0.589651	+	0.807658i	$0.299265\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19797.3	−0.921165	−0.460583	−	0.887617i	$-0.652360\pi$
$773$	−19797.3	−0.921165	−0.460583	+	0.887617i	$0.652360\pi$
$774$	0	0
$775$	− 4590.15i	− 0.212753i
$776$	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
$797$	−5021.94	−0.223195	−0.111597	−	0.993754i	$-0.535597\pi$
$797$	−5021.94	−0.223195	−0.111597	+	0.993754i	$0.535597\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 49720.0i	− 2.18503i
$804$	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
$821$	−46484.8	−1.97604	−0.988021	−	0.154321i	$-0.950681\pi$
$821$	−46484.8	−1.97604	−0.988021	+	0.154321i	$0.950681\pi$
$822$	0	0
$823$	46519.1i	1.97030i	0.171701	+	0.985149i	$0.445074\pi$
$823$	46519.1i	1.97030i	−0.171701	+	0.985149i	$0.554926\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24660.9i	1.03693i	0.855097	+	0.518467i	$0.173497\pi$
$827$	24660.9i	1.03693i	−0.855097	+	0.518467i	$0.826503\pi$
$828$	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
$841$	23239.0	0.952848
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26254.8	−1.06887
$846$	0	0
$847$	11070.1i	0.449084i
$848$	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
$865$	−48687.7	−1.91379
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−56542.7	−2.20723
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 38080.3i	− 1.47126i
$876$	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
$889$	11344.9	0.428002
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 55278.5i	− 2.06453i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	56249.2i	2.08678i
$900$	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
$913$	62104.2	2.25120
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−49705.2	−1.78998
$918$	0	0
$919$	40528.3i	1.45474i	0.686246	+	0.727370i	$0.259257\pi$
$919$	40528.3i	1.45474i	−0.686246	+	0.727370i	$0.740743\pi$
$920$	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
$937$	25150.6	0.876876	0.438438	−	0.898761i	$-0.355532\pi$
$937$	25150.6	0.876876	0.438438	+	0.898761i	$0.355532\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4842.34	−0.167753	−0.0838766	−	0.996476i	$-0.526730\pi$
$941$	−4842.34	−0.167753	−0.0838766	+	0.996476i	$0.526730\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 9728.84i	− 0.333838i	−0.985971	−	0.166919i	$-0.946618\pi$
$947$	− 9728.84i	− 0.333838i	0.985971	−	0.166919i	$-0.0533819\pi$
$948$	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
$961$	−36640.0	−1.22990
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−33907.6	−1.13111
$966$	0	0
$967$	− 41439.1i	− 1.37807i	−0.724729	−	0.689034i	$-0.758035\pi$
$967$	− 41439.1i	− 1.37807i	0.724729	−	0.689034i	$-0.241965\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	56704.0i	1.87407i	0.349240	+	0.937033i	$0.386440\pi$
$971$	56704.0i	1.87407i	−0.349240	+	0.937033i	$0.613560\pi$
$972$	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
$985$	57661.6	1.86523
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	61677.9i	1.97706i	0.151033	+	0.988529i	$0.451740\pi$
$991$	61677.9i	1.97706i	−0.151033	+	0.988529i	$0.548260\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 32783.9i	− 1.04454i
$996$	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	1728.4.f.f.863.3	✓	8
3.2	odd	2	inner	1728.4.f.f.863.5	yes	8
4.3	odd	2	inner	1728.4.f.f.863.4	yes	8
8.3	odd	2	inner	1728.4.f.f.863.6	yes	8
8.5	even	2	inner	1728.4.f.f.863.5	yes	8
12.11	even	2	inner	1728.4.f.f.863.6	yes	8
24.5	odd	2	CM	1728.4.f.f.863.3	✓	8
24.11	even	2	inner	1728.4.f.f.863.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.f.f.863.3	✓	8	1.1	even	1	trivial
1728.4.f.f.863.3	✓	8	24.5	odd	2	CM
1728.4.f.f.863.4	yes	8	4.3	odd	2	inner
1728.4.f.f.863.4	yes	8	24.11	even	2	inner
1728.4.f.f.863.5	yes	8	3.2	odd	2	inner
1728.4.f.f.863.5	yes	8	8.5	even	2	inner
1728.4.f.f.863.6	yes	8	8.3	odd	2	inner
1728.4.f.f.863.6	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-11.9503 q^{5} -29.7279i q^{7} +O(q^{10})\)	\(q-11.9503 q^{5} -29.7279i q^{7} +41.2720i q^{11} +17.8091 q^{25} -218.238 q^{29} -257.742i q^{31} +355.257i q^{35} -540.749 q^{49} -247.534 q^{53} -493.212i q^{55} +717.069i q^{59} -1204.69 q^{73} +1226.93 q^{77} +1370.00i q^{79} -1504.75i q^{83} +1865.26 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 1568 q^{25} - 864 q^{49} - 1288 q^{73} + 2296 q^{97}+O(q^{100}) \) 8 * q + 1568 * q^25 - 864 * q^49 - 1288 * q^73 + 2296 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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