LMFDB - Embedded newform 1728.3.b.h.1567.2

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			1567.2
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1728.1567
Dual form			1728.3.b.h.1567.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.63103i q^{5} +13.4853i q^{7} +O(q^{10})$	$q-6.63103i q^{5} +13.4853i q^{7} +18.4582 q^{11} -18.9706 q^{25} -29.3939i q^{29} +61.4264i q^{31} +89.4213 q^{35} -132.853 q^{49} +105.901i q^{53} -122.397i q^{55} -117.576 q^{59} +143.794 q^{73} +248.914i q^{77} +58.0000i q^{79} +165.037 q^{83} +128.941 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 6.63103i	− 1.32621i	−0.748528	−	0.663103i	$-0.769239\pi$
$5$	− 6.63103i	− 1.32621i	0.748528	−	0.663103i	$-0.230761\pi$
$6$	0	0
$7$	13.4853i	1.92647i	0.268662	+	0.963234i	$0.413418\pi$
$7$	13.4853i	1.92647i	−0.268662	+	0.963234i	$0.586582\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18.4582	1.67802	0.839010	−	0.544116i	$-0.183135\pi$
$11$	18.4582	1.67802	0.839010	+	0.544116i	$0.183135\pi$
$12$	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
$25$	−18.9706	−0.758823
$26$	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
$31$	61.4264i	1.98150i	0.135711	+	0.990748i	$0.456668\pi$
$31$	61.4264i	1.98150i	−0.135711	+	0.990748i	$0.543332\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	89.4213	2.55489
$36$	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
$49$	−132.853	−2.71128
$50$	0	0
$51$	0	0
$52$	0	0
$53$	105.901i	1.99814i	0.0431471	+	0.999069i	$0.486262\pi$
$53$	105.901i	1.99814i	−0.0431471	+	0.999069i	$0.513738\pi$
$54$	0	0
$55$	− 122.397i	− 2.22540i
$56$	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
$73$	143.794	1.96978	0.984890	−	0.173181i	$-0.0554046\pi$
$73$	143.794	1.96978	0.984890	+	0.173181i	$0.0554046\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	248.914i	3.23265i
$78$	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
$83$	165.037	1.98840	0.994200	−	0.107547i	$-0.0342996\pi$
$83$	165.037	1.98840	0.994200	+	0.107547i	$0.0342996\pi$
$84$	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
$97$	128.941	1.32929	0.664645	−	0.747159i	$-0.268583\pi$
$97$	128.941	1.32929	0.664645	+	0.747159i	$0.268583\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	130.252i	1.28962i	0.764341	+	0.644812i	$0.223064\pi$
$101$	130.252i	1.28962i	−0.764341	+	0.644812i	$0.776936\pi$
$102$	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
$107$	−23.5014	−0.219639	−0.109820	−	0.993952i	$-0.535027\pi$
$107$	−23.5014	−0.219639	−0.109820	+	0.993952i	$0.535027\pi$
$108$	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
$121$	219.706	1.81575
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 39.9814i	− 0.319851i
$126$	0	0
$127$	− 208.338i	− 1.64046i	−0.572036	−	0.820229i	$-0.693846\pi$
$127$	− 208.338i	− 1.64046i	0.572036	−	0.820229i	$-0.306154\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	255.698	1.95189	0.975947	−	0.218007i	$-0.0699555\pi$
$131$	255.698	1.95189	0.975947	+	0.218007i	$0.0699555\pi$
$132$	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
$145$	−194.912	−1.34422
$146$	0	0
$147$	0	0
$148$	0	0
$149$	285.440i	1.91571i	0.287258	+	0.957853i	$0.407256\pi$
$149$	285.440i	1.91571i	−0.287258	+	0.957853i	$0.592744\pi$
$150$	0	0
$151$	106.574i	0.705785i	0.935664	+	0.352893i	$0.114802\pi$
$151$	106.574i	0.705785i	−0.935664	+	0.352893i	$0.885198\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	407.320	2.62787
$156$	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
$173$	− 211.301i	− 1.22140i	−0.791864	−	0.610698i	$-0.790889\pi$
$173$	− 211.301i	− 1.22140i	0.791864	−	0.610698i	$-0.209111\pi$
$174$	0	0
$175$	− 255.823i	− 1.46185i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	336.357	1.87909	0.939545	−	0.342424i	$-0.111248\pi$
$179$	336.357	1.87909	0.939545	+	0.342424i	$0.111248\pi$
$180$	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
$193$	−365.617	−1.89439	−0.947195	−	0.320658i	$-0.896096\pi$
$193$	−365.617	−1.89439	−0.947195	+	0.320658i	$0.896096\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3.45535i	− 0.0175399i	−0.999962	−	0.00876993i	$-0.997208\pi$
$197$	− 3.45535i	− 0.0175399i	0.999962	−	0.00876993i	$-0.00279159\pi$
$198$	0	0
$199$	195.985i	0.984848i	0.870355	+	0.492424i	$0.163889\pi$
$199$	195.985i	0.984848i	−0.870355	+	0.492424i	$0.836111\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	396.385	1.95263
$204$	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
$217$	−828.352	−3.81729
$218$	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
$245$	880.951i	3.59572i
$246$	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
$265$	702.235	2.64994
$266$	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
$271$	495.690i	1.82912i	0.404455	+	0.914558i	$0.367461\pi$
$271$	495.690i	1.82912i	−0.404455	+	0.914558i	$0.632539\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−350.163	−1.27332
$276$	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
$295$	779.647i	2.64287i
$296$	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
$313$	−23.4996	−0.0750785	−0.0375392	−	0.999295i	$-0.511952\pi$
$313$	−23.4996	−0.0750785	−0.0375392	+	0.999295i	$0.511952\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 19.8086i	− 0.0624877i	−0.999512	−	0.0312439i	$-0.990053\pi$
$317$	− 19.8086i	− 0.0624877i	0.999512	−	0.0312439i	$-0.00994685\pi$
$318$	0	0
$319$	− 542.558i	− 1.70081i
$320$	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
$341$	1133.82i	3.32499i
$342$	0	0
$343$	− 1130.78i	− 3.29673i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	251.130	0.723717	0.361859	−	0.932233i	$-0.382142\pi$
$347$	251.130	0.723717	0.361859	+	0.932233i	$0.382142\pi$
$348$	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
$365$	− 953.502i	− 2.61233i
$366$	0	0
$367$	193.780i	0.528010i	0.964521	+	0.264005i	$0.0850435\pi$
$367$	193.780i	0.528010i	−0.964521	+	0.264005i	$0.914956\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1428.11	−3.84935
$372$	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
$385$	1650.56	4.28716
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 541.766i	− 1.39272i	−0.717695	−	0.696358i	$-0.754803\pi$
$389$	− 541.766i	− 1.39272i	0.717695	−	0.696358i	$-0.245197\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	384.600	0.973670
$396$	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
$409$	−698.411	−1.70761	−0.853803	−	0.520595i	$-0.825710\pi$
$409$	−698.411	−1.70761	−0.853803	+	0.520595i	$0.825710\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 1585.54i	− 3.83908i
$414$	0	0
$415$	− 1094.37i	− 2.63703i
$416$	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
$433$	140.176	0.323732	0.161866	−	0.986813i	$-0.448249\pi$
$433$	140.176	0.323732	0.161866	+	0.986813i	$0.448249\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 582.249i	− 1.32631i	−0.748483	−	0.663154i	$-0.769218\pi$
$439$	− 582.249i	− 1.32631i	0.748483	−	0.663154i	$-0.230782\pi$
$440$	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
$457$	−379.881	−0.831250	−0.415625	−	0.909536i	$-0.636437\pi$
$457$	−379.881	−0.831250	−0.415625	+	0.909536i	$0.636437\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 81.7720i	− 0.177380i	−0.996059	−	0.0886899i	$-0.971732\pi$
$461$	− 81.7720i	− 0.177380i	0.996059	−	0.0886899i	$-0.0282680\pi$
$462$	0	0
$463$	897.161i	1.93771i	0.247625	+	0.968856i	$0.420350\pi$
$463$	897.161i	1.93771i	−0.247625	+	0.968856i	$0.579650\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−206.132	−0.441395	−0.220698	−	0.975342i	$-0.570833\pi$
$467$	−206.132	−0.441395	−0.220698	+	0.975342i	$0.570833\pi$
$468$	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
$485$	− 855.013i	− 1.76291i
$486$	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
$491$	−838.142	−1.70701	−0.853505	−	0.521084i	$-0.825528\pi$
$491$	−838.142	−1.70701	−0.853505	+	0.521084i	$0.825528\pi$
$492$	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
$505$	863.705	1.71031
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 938.372i	− 1.84356i	−0.387713	−	0.921780i	$-0.626735\pi$
$509$	− 938.372i	− 1.84356i	0.387713	−	0.921780i	$-0.373265\pi$
$510$	0	0
$511$	1939.10i	3.79472i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	66.3103	0.128758
$516$	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
$535$	155.839i	0.291287i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2452.23	−4.54958
$540$	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
$553$	−782.146	−1.41437
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 473.114i	− 0.849396i	−0.905335	−	0.424698i	$-0.860380\pi$
$557$	− 473.114i	− 0.849396i	0.905335	−	0.424698i	$-0.139620\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	256.563	0.455708	0.227854	−	0.973695i	$-0.426829\pi$
$563$	256.563	0.455708	0.227854	+	0.973695i	$0.426829\pi$
$564$	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
$581$	2225.57i	3.83059i
$582$	0	0
$583$	1954.75i	3.35291i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1148.82	−1.95711	−0.978556	−	0.205982i	$-0.933961\pi$
$587$	−1148.82	−1.95711	−0.978556	+	0.205982i	$0.933961\pi$
$588$	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
$601$	683.853	1.13786	0.568929	−	0.822387i	$-0.307358\pi$
$601$	683.853	1.13786	0.568929	+	0.822387i	$0.307358\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1456.87i	− 2.40806i
$606$	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
$625$	−739.382	−1.18301
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 406.868i	− 0.644799i	−0.946604	−	0.322399i	$-0.895511\pi$
$631$	− 406.868i	− 0.644799i	0.946604	−	0.322399i	$-0.104489\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1381.50	−2.17558
$636$	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
$649$	−2170.23	−3.34397
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 454.808i	− 0.696491i	−0.937403	−	0.348245i	$-0.886778\pi$
$653$	− 454.808i	− 0.696491i	0.937403	−	0.348245i	$-0.113222\pi$
$654$	0	0
$655$	− 1695.54i	− 2.58861i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−83.5128	−0.126727	−0.0633633	−	0.997991i	$-0.520183\pi$
$659$	−83.5128	−0.126727	−0.0633633	+	0.997991i	$0.520183\pi$
$660$	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
$673$	−1249.00	−1.85587	−0.927933	−	0.372746i	$-0.878416\pi$
$673$	−1249.00	−1.85587	−0.927933	+	0.372746i	$0.878416\pi$
$674$	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
$679$	1738.81i	2.56084i
$680$	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
$701$	657.253i	0.937593i	0.883306	+	0.468797i	$0.155312\pi$
$701$	657.253i	0.937593i	−0.883306	+	0.468797i	$0.844688\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1756.48	−2.48442
$708$	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
$721$	−134.853	−0.187036
$722$	0	0
$723$	0	0
$724$	0	0
$725$	557.618i	0.769129i
$726$	0	0
$727$	631.662i	0.868861i	0.900705	+	0.434430i	$0.143050\pi$
$727$	631.662i	0.868861i	−0.900705	+	0.434430i	$0.856950\pi$
$728$	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
$745$	1892.76	2.54062
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 316.923i	− 0.423128i
$750$	0	0
$751$	− 1167.69i	− 1.55485i	−0.628977	−	0.777424i	$-0.716526\pi$
$751$	− 1167.69i	− 1.55485i	0.628977	−	0.777424i	$-0.283474\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	706.693	0.936017
$756$	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
$769$	1534.09	1.99491	0.997456	−	0.0712917i	$-0.0227121\pi$
$769$	1534.09	1.99491	0.997456	+	0.0712917i	$0.0227121\pi$
$770$	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
$775$	− 1165.29i	− 1.50360i
$776$	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
$797$	− 1586.04i	− 1.99001i	−0.0998129	−	0.995006i	$-0.531824\pi$
$797$	− 1586.04i	− 1.99001i	0.0998129	−	0.995006i	$-0.468176\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2654.18	3.30533
$804$	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
$823$	− 1601.13i	− 1.94548i	−0.231898	−	0.972740i	$-0.574493\pi$
$823$	− 1601.13i	− 1.94548i	0.231898	−	0.972740i	$-0.425507\pi$
$824$	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
$845$	− 1120.64i	− 1.32621i
$846$	0	0
$847$	2962.79i	3.49798i
$848$	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
$865$	−1401.15	−1.61982
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1070.58i	1.23196i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	539.160	0.616183
$876$	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
$889$	2809.50	3.16029
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 2230.40i	− 2.49206i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1805.56	2.00841
$900$	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
$913$	3046.29	3.33657
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3448.16i	3.76026i
$918$	0	0
$919$	1153.92i	1.25563i	0.778362	+	0.627815i	$0.216051\pi$
$919$	1153.92i	1.25563i	−0.778362	+	0.627815i	$0.783949\pi$
$920$	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
$937$	1729.29	1.84556	0.922782	−	0.385323i	$-0.125910\pi$
$937$	1729.29	1.84556	0.922782	+	0.385323i	$0.125910\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 543.718i	− 0.577809i	−0.957358	−	0.288904i	$-0.906709\pi$
$941$	− 543.718i	− 0.577809i	0.957358	−	0.288904i	$-0.0932910\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1816.89	−1.91857	−0.959285	−	0.282439i	$-0.908857\pi$
$947$	−1816.89	−1.91857	−0.959285	+	0.282439i	$0.908857\pi$
$948$	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
$961$	−2812.20	−2.92633
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2424.42i	2.51235i
$966$	0	0
$967$	691.956i	0.715570i	0.933804	+	0.357785i	$0.116468\pi$
$967$	691.956i	0.715570i	−0.933804	+	0.357785i	$0.883532\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1563.65	1.61035	0.805176	−	0.593036i	$-0.202071\pi$
$971$	1563.65	1.61035	0.805176	+	0.593036i	$0.202071\pi$
$972$	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
$985$	−22.9126	−0.0232615
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 539.366i	− 0.544264i	−0.962260	−	0.272132i	$-0.912271\pi$
$991$	− 539.366i	− 0.544264i	0.962260	−	0.272132i	$-0.0877288\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1299.58	1.30611
$996$	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.1567.2	yes	8
3.2	odd	2	inner	1728.3.b.h.1567.8	yes	8
4.3	odd	2	inner	1728.3.b.h.1567.1	✓	8
8.3	odd	2	inner	1728.3.b.h.1567.7	yes	8
8.5	even	2	inner	1728.3.b.h.1567.8	yes	8
12.11	even	2	inner	1728.3.b.h.1567.7	yes	8
24.5	odd	2	CM	1728.3.b.h.1567.2	yes	8
24.11	even	2	inner	1728.3.b.h.1567.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.b.h.1567.1	✓	8	4.3	odd	2	inner
1728.3.b.h.1567.1	✓	8	24.11	even	2	inner
1728.3.b.h.1567.2	yes	8	1.1	even	1	trivial
1728.3.b.h.1567.2	yes	8	24.5	odd	2	CM
1728.3.b.h.1567.7	yes	8	8.3	odd	2	inner
1728.3.b.h.1567.7	yes	8	12.11	even	2	inner
1728.3.b.h.1567.8	yes	8	3.2	odd	2	inner
1728.3.b.h.1567.8	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.63103i q^{5} +13.4853i q^{7} +O(q^{10})\)	\(q-6.63103i q^{5} +13.4853i q^{7} +18.4582 q^{11} -18.9706 q^{25} -29.3939i q^{29} +61.4264i q^{31} +89.4213 q^{35} -132.853 q^{49} +105.901i q^{53} -122.397i q^{55} -117.576 q^{59} +143.794 q^{73} +248.914i q^{77} +58.0000i q^{79} +165.037 q^{83} +128.941 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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

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