LMFDB - Embedded newform 1728.3.h.c.161.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,3,Mod(161,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.161"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-100,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-88] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)

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

Self dual:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.161
Dual form			1728.3.h.c.161.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−5.19615	−0.742307	−0.371154	−	0.928571i	$-0.621038\pi$
$7$	−5.19615	−0.742307	−0.371154	+	0.928571i	$0.621038\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 25.9808i	− 1.99852i	−0.0384615	−	0.999260i	$-0.512246\pi$
$13$	− 25.9808i	− 1.99852i	0.0384615	−	0.999260i	$-0.487754\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	11.0000i	0.578947i	0.957186	+	0.289474i	$0.0934803\pi$
$19$	11.0000i	0.578947i	−0.957186	+	0.289474i	$0.906520\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−25.0000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	41.5692	1.34094	0.670471	−	0.741935i	$-0.266092\pi$
$31$	41.5692	1.34094	0.670471	+	0.741935i	$0.266092\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	57.1577i	1.54480i	0.635135	+	0.772401i	$0.280944\pi$
$37$	57.1577i	1.54480i	−0.635135	+	0.772401i	$0.719056\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	22.0000i	0.511628i	0.966726	+	0.255814i	$0.0823435\pi$
$43$	22.0000i	0.511628i	−0.966726	+	0.255814i	$0.917657\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−22.0000	−0.448980
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	− 15.5885i	− 0.255548i	−0.991803	−	0.127774i	$-0.959217\pi$
$61$	− 15.5885i	− 0.255548i	0.991803	−	0.127774i	$-0.0407833\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	109.000i	1.62687i	0.581659	+	0.813433i	$0.302404\pi$
$67$	109.000i	1.62687i	−0.581659	+	0.813433i	$0.697596\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−97.0000	−1.32877	−0.664384	−	0.747392i	$-0.731306\pi$
$73$	−97.0000	−1.32877	−0.664384	+	0.747392i	$0.731306\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−88.3346	−1.11816	−0.559080	−	0.829114i	$-0.688845\pi$
$79$	−88.3346	−1.11816	−0.559080	+	0.829114i	$0.688845\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	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
$91$	135.000i	1.48352i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−167.000	−1.72165	−0.860825	−	0.508902i	$-0.830052\pi$
$97$	−167.000	−1.72165	−0.860825	+	0.508902i	$0.830052\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−202.650	−1.96748	−0.983738	−	0.179612i	$-0.942516\pi$
$103$	−202.650	−1.96748	−0.983738	+	0.179612i	$0.942516\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	− 41.5692i	− 0.381369i	−0.981651	−	0.190684i	$-0.938929\pi$
$109$	− 41.5692i	− 0.381369i	0.981651	−	0.190684i	$-0.0610707\pi$
$110$	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
$121$	−121.000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	207.846	1.63658	0.818292	−	0.574803i	$-0.194921\pi$
$127$	207.846	1.63658	0.818292	+	0.574803i	$0.194921\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	− 57.1577i	− 0.429757i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	251.000i	1.80576i	0.429898	+	0.902878i	$0.358550\pi$
$139$	251.000i	1.80576i	−0.429898	+	0.902878i	$0.641450\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	296.181	1.96146	0.980731	−	0.195364i	$-0.0625889\pi$
$151$	296.181	1.96146	0.980731	+	0.195364i	$0.0625889\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	290.985i	1.85340i	0.375796	+	0.926702i	$0.377369\pi$
$157$	290.985i	1.85340i	−0.375796	+	0.926702i	$0.622631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	299.000i	1.83436i	0.398478	+	0.917178i	$0.369539\pi$
$163$	299.000i	1.83436i	−0.398478	+	0.917178i	$0.630461\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−506.000	−2.99408
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	129.904	0.742307
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	− 181.865i	− 1.00478i	−0.864641	−	0.502390i	$-0.832454\pi$
$181$	− 181.865i	− 1.00478i	0.864641	−	0.502390i	$-0.167546\pi$
$182$	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
$193$	143.000	0.740933	0.370466	−	0.928846i	$-0.379198\pi$
$193$	143.000	0.740933	0.370466	+	0.928846i	$0.379198\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−285.788	−1.43612	−0.718061	−	0.695980i	$-0.754970\pi$
$199$	−285.788	−1.43612	−0.718061	+	0.695980i	$0.754970\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	253.000i	1.19905i	0.800355	+	0.599526i	$0.204644\pi$
$211$	253.000i	1.19905i	−0.800355	+	0.599526i	$0.795356\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−216.000	−0.995392
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−290.985	−1.30486	−0.652432	−	0.757848i	$-0.726251\pi$
$223$	−290.985	−1.30486	−0.652432	+	0.757848i	$0.726251\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	− 457.261i	− 1.99677i	−0.0567686	−	0.998387i	$-0.518080\pi$
$229$	− 457.261i	− 1.99677i	0.0567686	−	0.998387i	$-0.481920\pi$
$230$	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	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−193.000	−0.800830	−0.400415	−	0.916334i	$-0.631134\pi$
$241$	−193.000	−0.800830	−0.400415	+	0.916334i	$0.631134\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	285.788	1.15704
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	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
$259$	− 297.000i	− 1.14672i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−452.065	−1.66814	−0.834069	−	0.551661i	$-0.813994\pi$
$271$	−452.065	−1.66814	−0.834069	+	0.551661i	$0.813994\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	540.400i	1.95090i	0.220217	+	0.975451i	$0.429324\pi$
$277$	540.400i	1.95090i	−0.220217	+	0.975451i	$0.570676\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	458.000i	1.61837i	0.587551	+	0.809187i	$0.300092\pi$
$283$	458.000i	1.61837i	−0.587551	+	0.809187i	$0.699908\pi$
$284$	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
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 114.315i	− 0.379785i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	358.000i	1.16612i	0.812428	+	0.583062i	$0.198145\pi$
$307$	358.000i	1.16612i	−0.812428	+	0.583062i	$0.801855\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−599.000	−1.91374	−0.956869	−	0.290520i	$-0.906172\pi$
$313$	−599.000	−1.91374	−0.956869	+	0.290520i	$0.906172\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	649.519i	1.99852i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 661.000i	− 1.99698i	−0.0549442	−	0.998489i	$-0.517498\pi$
$331$	− 661.000i	− 1.99698i	0.0549442	−	0.998489i	$-0.482502\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	649.000	1.92582	0.962908	−	0.269830i	$-0.0869675\pi$
$337$	649.000	1.92582	0.962908	+	0.269830i	$0.0869675\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	368.927	1.07559
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	− 192.258i	− 0.550881i	−0.961318	−	0.275441i	$-0.911176\pi$
$349$	− 192.258i	− 0.550881i	0.961318	−	0.275441i	$-0.0888238\pi$
$350$	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	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	240.000	0.664820
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	545.596	1.48664	0.743319	−	0.668937i	$-0.233251\pi$
$367$	545.596	1.48664	0.743319	+	0.668937i	$0.233251\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	472.850i	1.26769i	0.773458	+	0.633847i	$0.218525\pi$
$373$	472.850i	1.26769i	−0.773458	+	0.633847i	$0.781475\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 83.0000i	− 0.218997i	−0.993987	−	0.109499i	$-0.965075\pi$
$379$	− 83.0000i	− 0.218997i	0.993987	−	0.109499i	$-0.0349245\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 706.677i	− 1.78004i	−0.455919	−	0.890021i	$-0.650689\pi$
$397$	− 706.677i	− 1.78004i	0.455919	−	0.890021i	$-0.349311\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 1080.00i	− 2.67990i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	143.000	0.349633	0.174817	−	0.984601i	$-0.444067\pi$
$409$	143.000	0.349633	0.174817	+	0.984601i	$0.444067\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	− 691.088i	− 1.64154i	−0.571259	−	0.820770i	$-0.693545\pi$
$421$	− 691.088i	− 1.64154i	0.571259	−	0.820770i	$-0.306455\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	81.0000i	0.189696i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	862.000	1.99076	0.995381	−	0.0960028i	$-0.0306058\pi$
$433$	862.000	1.99076	0.995381	+	0.0960028i	$0.0306058\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−872.954	−1.98850	−0.994252	−	0.107062i	$-0.965856\pi$
$439$	−872.954	−1.98850	−0.994252	+	0.107062i	$0.965856\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	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
$457$	−814.000	−1.78118	−0.890591	−	0.454805i	$-0.849709\pi$
$457$	−814.000	−1.78118	−0.890591	+	0.454805i	$0.849709\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	−836.581	−1.80687	−0.903435	−	0.428726i	$-0.858963\pi$
$463$	−836.581	−1.80687	−0.903435	+	0.428726i	$0.858963\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	− 566.381i	− 1.20763i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 275.000i	− 0.578947i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	1485.00	3.08732
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	909.327	1.86720	0.933600	−	0.358316i	$-0.116649\pi$
$487$	909.327	1.86720	0.933600	+	0.358316i	$0.116649\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 26.0000i	− 0.0521042i	−0.999661	−	0.0260521i	$-0.991706\pi$
$499$	− 26.0000i	− 0.0521042i	0.999661	−	0.0260521i	$-0.00829358\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	504.027	0.986354
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	− 803.000i	− 1.53537i	−0.640826	−	0.767686i	$-0.721408\pi$
$523$	− 803.000i	− 1.53537i	0.640826	−	0.767686i	$-0.278592\pi$
$524$	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
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1054.82i	1.94976i	0.222736	+	0.974879i	$0.428501\pi$
$541$	1054.82i	1.94976i	−0.222736	+	0.974879i	$0.571499\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1093.00i	− 1.99817i	−0.0427471	−	0.999086i	$-0.513611\pi$
$547$	− 1093.00i	− 1.99817i	0.0427471	−	0.999086i	$-0.486389\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	459.000	0.830018
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	571.577	1.02250
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	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
$571$	− 181.000i	− 0.316988i	−0.987360	−	0.158494i	$-0.949336\pi$
$571$	− 181.000i	− 0.316988i	0.987360	−	0.158494i	$-0.0506638\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1033.00	1.79029	0.895147	−	0.445770i	$-0.147070\pi$
$577$	1033.00	1.79029	0.895147	+	0.445770i	$0.147070\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	457.261i	0.776335i
$590$	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	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−526.000	−0.875208	−0.437604	−	0.899168i	$-0.644173\pi$
$601$	−526.000	−0.875208	−0.437604	+	0.899168i	$0.644173\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−254.611	−0.419459	−0.209729	−	0.977759i	$-0.567258\pi$
$607$	−254.611	−0.419459	−0.209729	+	0.977759i	$0.567258\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	732.657i	1.19520i	0.801794	+	0.597600i	$0.203879\pi$
$613$	732.657i	1.19520i	−0.801794	+	0.597600i	$0.796121\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	− 949.000i	− 1.53312i	−0.642174	−	0.766559i	$-0.721967\pi$
$619$	− 949.000i	− 1.53312i	0.642174	−	0.766559i	$-0.278033\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−1117.17	−1.77048	−0.885240	−	0.465135i	$-0.846006\pi$
$631$	−1117.17	−1.77048	−0.885240	+	0.465135i	$0.846006\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	571.577i	0.897295i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	314.000i	0.488336i	0.969733	+	0.244168i	$0.0785148\pi$
$643$	314.000i	0.488336i	−0.969733	+	0.244168i	$0.921485\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	− 763.834i	− 1.15557i	−0.816188	−	0.577787i	$-0.803916\pi$
$661$	− 763.834i	− 1.15557i	0.816188	−	0.577787i	$-0.196084\pi$
$662$	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
$673$	−23.0000	−0.0341753	−0.0170877	−	0.999854i	$-0.505439\pi$
$673$	−23.0000	−0.0341753	−0.0170877	+	0.999854i	$0.505439\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	867.757	1.27799
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 1318.00i	− 1.90738i	−0.300790	−	0.953690i	$-0.597250\pi$
$691$	− 1318.00i	− 1.90738i	0.300790	−	0.953690i	$-0.402750\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−628.734	−0.894359
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 275.396i	− 0.388429i	−0.980959	−	0.194214i	$-0.937784\pi$
$709$	− 275.396i	− 0.388429i	0.980959	−	0.194214i	$-0.0622158\pi$
$710$	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
$721$	1053.00	1.46047
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1371.78	−1.88691	−0.943455	−	0.331499i	$-0.892446\pi$
$727$	−1371.78	−1.88691	−0.943455	+	0.331499i	$0.892446\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 1039.23i	− 1.41778i	−0.705321	−	0.708888i	$-0.749197\pi$
$733$	− 1039.23i	− 1.41778i	0.705321	−	0.708888i	$-0.250803\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1222.00i	1.65359i	0.562506	+	0.826793i	$0.309837\pi$
$739$	1222.00i	1.65359i	−0.562506	+	0.826793i	$0.690163\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1491.30	1.98575	0.992873	−	0.119174i	$-0.0380248\pi$
$751$	1491.30	1.98575	0.992873	+	0.119174i	$0.0380248\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1356.20i	− 1.79154i	−0.444518	−	0.895770i	$-0.646625\pi$
$757$	− 1356.20i	− 1.79154i	0.444518	−	0.895770i	$-0.353375\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	216.000i	0.283093i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	863.000	1.12224	0.561118	−	0.827736i	$-0.310371\pi$
$769$	863.000	1.12224	0.561118	+	0.827736i	$0.310371\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	−1039.23	−1.34094
$776$	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
$787$	− 613.000i	− 0.778907i	−0.921046	−	0.389454i	$-0.872664\pi$
$787$	− 613.000i	− 0.778907i	0.921046	−	0.389454i	$-0.127336\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−405.000	−0.510719
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	− 1514.00i	− 1.86683i	−0.358797	−	0.933416i	$-0.616813\pi$
$811$	− 1514.00i	− 1.86683i	0.358797	−	0.933416i	$-0.383187\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−242.000	−0.296206
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	−285.788	−0.347252	−0.173626	−	0.984812i	$-0.555548\pi$
$823$	−285.788	−0.347252	−0.173626	+	0.984812i	$0.555548\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	400.104i	0.482634i	0.970446	+	0.241317i	$0.0775794\pi$
$829$	400.104i	0.482634i	−0.970446	+	0.241317i	$0.922421\pi$
$830$	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
$841$	−841.000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	628.734	0.742307
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1636.79i	1.91886i	0.281946	+	0.959430i	$0.409020\pi$
$853$	1636.79i	1.91886i	−0.281946	+	0.959430i	$0.590980\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	1549.00i	1.80326i	0.432509	+	0.901630i	$0.357629\pi$
$859$	1549.00i	1.80326i	−0.432509	+	0.901630i	$0.642371\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2831.90	3.25132
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	306.573i	0.349570i	0.984607	+	0.174785i	$0.0559231\pi$
$877$	306.573i	0.349570i	−0.984607	+	0.174785i	$0.944077\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	443.000i	0.501699i	0.968026	+	0.250849i	$0.0807099\pi$
$883$	443.000i	0.501699i	−0.968026	+	0.250849i	$0.919290\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−1080.00	−1.21485
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1453.00i	1.60198i	0.598675	+	0.800992i	$0.295694\pi$
$907$	1453.00i	1.60198i	−0.598675	+	0.800992i	$0.704306\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1621.20	1.76409	0.882045	−	0.471164i	$-0.156166\pi$
$919$	1621.20	1.76409	0.882045	+	0.471164i	$0.156166\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 1428.94i	− 1.54480i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	− 242.000i	− 0.259936i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1847.00	−1.97118	−0.985592	−	0.169138i	$-0.945902\pi$
$937$	−1847.00	−1.97118	−0.985592	+	0.169138i	$0.945902\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	2520.13i	2.65557i
$950$	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
$961$	767.000	0.798127
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1917.38	−1.98281	−0.991407	−	0.130817i	$-0.958240\pi$
$967$	−1917.38	−1.98281	−0.991407	+	0.130817i	$0.958240\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	− 1304.23i	− 1.34043i
$974$	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	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−950.896	−0.959532	−0.479766	−	0.877397i	$-0.659278\pi$
$991$	−950.896	−0.959532	−0.479766	+	0.877397i	$0.659278\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 623.538i	− 0.625415i	−0.949850	−	0.312707i	$-0.898764\pi$
$997$	− 623.538i	− 0.625415i	0.949850	−	0.312707i	$-0.101236\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.h.c.161.1	✓	4
3.2	odd	2	CM	1728.3.h.c.161.1	✓	4
4.3	odd	2	inner	1728.3.h.c.161.3	yes	4
8.3	odd	2	inner	1728.3.h.c.161.4	yes	4
8.5	even	2	inner	1728.3.h.c.161.2	yes	4
12.11	even	2	inner	1728.3.h.c.161.3	yes	4
24.5	odd	2	inner	1728.3.h.c.161.2	yes	4
24.11	even	2	inner	1728.3.h.c.161.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.h.c.161.1	✓	4	1.1	even	1	trivial
1728.3.h.c.161.1	✓	4	3.2	odd	2	CM
1728.3.h.c.161.2	yes	4	8.5	even	2	inner
1728.3.h.c.161.2	yes	4	24.5	odd	2	inner
1728.3.h.c.161.3	yes	4	4.3	odd	2	inner
1728.3.h.c.161.3	yes	4	12.11	even	2	inner
1728.3.h.c.161.4	yes	4	8.3	odd	2	inner
1728.3.h.c.161.4	yes	4	24.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-5.19615 q^{7} -25.9808i q^{13} +11.0000i q^{19} -25.0000 q^{25} +41.5692 q^{31} +57.1577i q^{37} +22.0000i q^{43} -22.0000 q^{49} -15.5885i q^{61} +109.000i q^{67} -97.0000 q^{73} -88.3346 q^{79} +135.000i q^{91} -167.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 100 q^{25} - 88 q^{49} - 388 q^{73} - 668 q^{97}+O(q^{100}) \) 4 * q - 100 * q^25 - 88 * q^49 - 388 * q^73 - 668 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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