LMFDB - Embedded newform 3744.2.m.f.1585.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(1585,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3744.1585");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.m (of order $2$, degree $1$, not 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:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.151613669376.21
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 5x^{4} + 16 $ x^8 + 5*x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1585.6
Root			$-1.19709 + 0.752986i$ of defining polynomial
Character	$\chi$	$=$	3744.1585
Dual form			3744.2.m.f.1585.5

$q$-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 $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	1.75265	0.783811	0.391905	−	0.920006i	$-0.371816\pi$
$5$	1.75265	0.783811	0.391905	+	0.920006i	$0.371816\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.54099	−1.97218	−0.986092	−	0.166200i	$-0.946850\pi$
$11$	−6.54099	−1.97218	−0.986092	+	0.166200i	$0.946850\pi$
$12$	0	0
$13$	3.60555i	1.00000i
$13$	3.60555i	1.00000i
$14$	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		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.92820	−0.385641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.1383i	− 1.58333i	−0.610954	−	0.791666i	$-0.709214\pi$
$41$	− 10.1383i	− 1.58333i	0.610954	−	0.791666i	$-0.290786\pi$
$42$	0	0
$43$	− 12.4900i	− 1.90471i	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	− 12.4900i	− 1.90471i	0.304997	−	0.952353i	$-0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.33123i	− 1.36110i	−0.732702	−	0.680550i	$-0.761741\pi$
$47$	− 9.33123i	− 1.36110i	0.732702	−	0.680550i	$-0.238259\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−11.4641	−1.54582
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.469622	0.0611396	0.0305698	−	0.999533i	$-0.490268\pi$
$59$	0.469622	0.0611396	0.0305698	+	0.999533i	$0.490268\pi$
$60$	0	0
$61$	− 7.21110i	− 0.923287i	−0.887066	−	0.461644i	$-0.847260\pi$
$61$	− 7.21110i	− 0.923287i	0.887066	−	0.461644i	$-0.152740\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.31928i	0.783811i
$66$	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
$71$	− 4.92144i	− 0.584067i	−0.956408	−	0.292034i	$-0.905668\pi$
$71$	− 4.92144i	− 0.584067i	0.956408	−	0.292034i	$-0.0943319\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−10.3923	−1.16923	−0.584613	−	0.811312i	$-0.698754\pi$
$79$	−10.3923	−1.16923	−0.584613	+	0.811312i	$0.698754\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.6124	−1.38439	−0.692194	−	0.721712i	$-0.743356\pi$
$83$	−12.6124	−1.38439	−0.692194	+	0.721712i	$0.743356\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 18.3671i	− 1.94690i	−0.228892	−	0.973452i	$-0.573510\pi$
$89$	− 18.3671i	− 1.94690i	0.228892	−	0.973452i	$-0.426490\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	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
$121$	31.7846	2.88951
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.1427	−1.08608
$126$	0	0
$127$	17.3205	1.53695	0.768473	−	0.639882i	$-0.221017\pi$
$127$	17.3205	1.53695	0.768473	+	0.639882i	$0.221017\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.5481i	− 1.24293i	−0.783444	−	0.621463i	$-0.786539\pi$
$137$	− 14.5481i	− 1.24293i	0.783444	−	0.621463i	$-0.213461\pi$
$138$	0	0
$139$	12.4900i	1.05939i	0.848189	+	0.529694i	$0.177693\pi$
$139$	12.4900i	1.05939i	−0.848189	+	0.529694i	$0.822307\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 23.5839i	− 1.97218i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−24.4113	−1.99985	−0.999927	−	0.0121001i	$-0.996148\pi$
$149$	−24.4113	−1.99985	−0.999927	+	0.0121001i	$0.996148\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	24.9800i	1.99362i	0.0798087	+	0.996810i	$0.474569\pi$
$157$	24.9800i	1.99362i	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	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
$167$	11.5361i	0.892692i	0.894860	+	0.446346i	$0.147275\pi$
$167$	11.5361i	0.892692i	−0.894860	+	0.446346i	$0.852725\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	24.9800i	1.85675i	0.371647	+	0.928374i	$0.378793\pi$
$181$	24.9800i	1.85675i	−0.371647	+	0.928374i	$0.621207\pi$
$182$	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
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.8954	−0.990006	−0.495003	−	0.868891i	$-0.664833\pi$
$197$	−13.8954	−0.990006	−0.495003	+	0.868891i	$0.664833\pi$
$198$	0	0
$199$	24.2487	1.71895	0.859473	−	0.511182i	$-0.170792\pi$
$199$	24.2487	1.71895	0.859473	+	0.511182i	$0.170792\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 17.7689i	− 1.24103i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	28.8444i	1.98573i	0.119239	+	0.992866i	$0.461954\pi$
$211$	28.8444i	1.98573i	−0.119239	+	0.992866i	$0.538046\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 21.8906i	− 1.49293i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.60175	−0.371801	−0.185901	−	0.982569i	$-0.559520\pi$
$227$	−5.60175	−0.371801	−0.185901	+	0.982569i	$0.559520\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	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
$235$	− 16.3544i	− 1.06684i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 25.7888i	− 1.66814i	−0.551660	−	0.834069i	$-0.686005\pi$
$239$	− 25.7888i	− 1.66814i	0.551660	−	0.834069i	$-0.313995\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.2686	0.783811
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	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		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.6124	0.760554
$276$	0	0
$277$	− 24.9800i	− 1.50090i	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	− 24.9800i	− 1.50090i	0.660926	−	0.750451i	$-0.270164\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 13.9573i	− 0.832621i	−0.909223	−	0.416310i	$-0.863323\pi$
$281$	− 13.9573i	− 0.832621i	0.909223	−	0.416310i	$-0.136677\pi$
$282$	0	0
$283$	− 28.8444i	− 1.71462i	−0.514799	−	0.857311i	$-0.672133\pi$
$283$	− 28.8444i	− 1.71462i	0.514799	−	0.857311i	$-0.327867\pi$
$284$	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
$293$	10.3901	0.606996	0.303498	−	0.952832i	$-0.401846\pi$
$293$	10.3901	0.606996	0.303498	+	0.952832i	$0.401846\pi$
$294$	0	0
$295$	0.823085	0.0479219
$296$	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
$305$	− 12.6386i	− 0.723682i
$306$	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
$313$	−34.6410	−1.95803	−0.979013	−	0.203798i	$-0.934671\pi$
$313$	−34.6410	−1.95803	−0.979013	+	0.203798i	$0.934671\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−31.4219	−1.76483	−0.882416	−	0.470470i	$-0.844084\pi$
$317$	−31.4219	−1.76483	−0.882416	+	0.470470i	$0.844084\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 6.95224i	− 0.385641i
$326$	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
$337$	−6.92820	−0.377403	−0.188702	−	0.982034i	$-0.560428\pi$
$337$	−6.92820	−0.377403	−0.188702	+	0.982034i	$0.560428\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.50029i	0.133077i	0.997784	+	0.0665386i	$0.0211956\pi$
$353$	2.50029i	0.133077i	−0.997784	+	0.0665386i	$0.978804\pi$
$354$	0	0
$355$	− 8.62558i	− 0.457798i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	37.8366i	1.99694i	0.0553230	+	0.998469i	$0.482381\pi$
$359$	37.8366i	1.99694i	−0.0553230	+	0.998469i	$0.517619\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 36.0555i	− 1.86688i	−0.358729	−	0.933442i	$-0.616790\pi$
$373$	− 36.0555i	− 1.86688i	0.358729	−	0.933442i	$-0.383210\pi$
$374$	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
$383$	16.9692i	0.867086i	0.901133	+	0.433543i	$0.142737\pi$
$383$	16.9692i	0.867086i	−0.901133	+	0.433543i	$0.857263\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−18.2141	−0.916452
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 31.0056i	− 1.54835i	−0.632973	−	0.774173i	$-0.718166\pi$
$401$	− 31.0056i	− 1.54835i	0.632973	−	0.774173i	$-0.281834\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−22.1051	−1.08510
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	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
$431$	− 40.0415i	− 1.92873i	−0.264578	−	0.964364i	$-0.585233\pi$
$431$	− 40.0415i	− 1.92873i	0.264578	−	0.964364i	$-0.414767\pi$
$432$	0	0
$433$	20.7846	0.998845	0.499422	−	0.866359i	$-0.333546\pi$
$433$	20.7846	0.998845	0.499422	+	0.866359i	$0.333546\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	40.0000	1.90910	0.954548	−	0.298057i	$-0.0963387\pi$
$439$	40.0000	1.90910	0.954548	+	0.298057i	$0.0963387\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	− 32.1911i	− 1.52600i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10.7291i	0.506336i	0.967422	+	0.253168i	$0.0814726\pi$
$449$	10.7291i	0.506336i	−0.967422	+	0.253168i	$0.918527\pi$
$450$	0	0
$451$	66.3144i	3.12262i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	34.9272	1.62672	0.813362	−	0.581758i	$-0.197635\pi$
$461$	34.9272	1.62672	0.813362	+	0.581758i	$0.197635\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	81.6970i	3.75643i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	33.4268i	1.52731i	0.645626	+	0.763654i	$0.276597\pi$
$479$	33.4268i	1.52731i	−0.645626	+	0.763654i	$0.723403\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	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
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	43.5647	1.93097	0.965485	−	0.260457i	$-0.0838733\pi$
$509$	43.5647	1.93097	0.965485	+	0.260457i	$0.0838733\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−28.0425	−1.23570
$516$	0	0
$517$	61.0355i	2.68434i
$518$	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
$523$	− 12.4900i	− 0.546149i	−0.961993	−	0.273075i	$-0.911959\pi$
$523$	− 12.4900i	− 0.546149i	0.961993	−	0.273075i	$-0.0880406\pi$
$524$	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
$533$	36.5541	1.58333
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−45.7870	−1.97218
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	37.4700i	1.60210i	0.598597	+	0.801050i	$0.295725\pi$
$547$	37.4700i	1.60210i	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−47.0700	−1.99442	−0.997210	−	0.0746496i	$-0.976216\pi$
$557$	−47.0700	−1.99442	−0.997210	+	0.0746496i	$0.976216\pi$
$558$	0	0
$559$	45.0333	1.90471
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	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
$571$	− 28.8444i	− 1.20710i	−0.797325	−	0.603550i	$-0.793752\pi$
$571$	− 28.8444i	− 1.20710i	0.797325	−	0.603550i	$-0.206248\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	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
$587$	24.7551	1.02175	0.510876	−	0.859654i	$-0.329321\pi$
$587$	24.7551	1.02175	0.510876	+	0.859654i	$0.329321\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 39.2344i	− 1.61116i	−0.592484	−	0.805582i	$-0.701853\pi$
$593$	− 39.2344i	− 1.61116i	0.592484	−	0.805582i	$-0.298147\pi$
$594$	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
$601$	48.4974	1.97825	0.989126	−	0.147074i	$-0.0469854\pi$
$601$	48.4974	1.97825	0.989126	+	0.147074i	$0.0469854\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	55.7074	2.26483
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	33.6442	1.36110
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	46.8724i	1.88701i	0.331357	+	0.943506i	$0.392494\pi$
$617$	46.8724i	1.88701i	−0.331357	+	0.943506i	$0.607506\pi$
$618$	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
$625$	−11.6410	−0.465641
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	30.3569	1.20468
$636$	0	0
$637$	25.2389i	1.00000i
$638$	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		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−3.07180	−0.120579
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	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
$671$	47.1678i	1.82089i
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	51.8583	1.98430	0.992152	−	0.125038i	$-0.0399051\pi$
$683$	51.8583	1.98430	0.992152	+	0.125038i	$0.0399051\pi$
$684$	0	0
$685$	− 25.4977i	− 0.974218i
$686$	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
$695$	21.8906i	0.830359i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 41.3344i	− 1.54582i
$716$	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
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	51.9615	1.92715	0.963573	−	0.267445i	$-0.0861794\pi$
$727$	51.9615	1.92715	0.963573	+	0.267445i	$0.0861794\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	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
$743$	54.2941i	1.99186i	0.0901418	+	0.995929i	$0.471268\pi$
$743$	54.2941i	1.99186i	−0.0901418	+	0.995929i	$0.528732\pi$
$744$	0	0
$745$	−42.7846	−1.56751
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−40.0000	−1.45962	−0.729810	−	0.683650i	$-0.760392\pi$
$751$	−40.0000	−1.45962	−0.729810	+	0.683650i	$0.760392\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 36.0555i	− 1.31046i	−0.755429	−	0.655230i	$-0.772572\pi$
$757$	− 36.0555i	− 1.31046i	0.755429	−	0.655230i	$-0.227428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27.1866i	0.985515i	0.870167	+	0.492757i	$0.164011\pi$
$761$	27.1866i	0.985515i	−0.870167	+	0.492757i	$0.835989\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.69325i	0.0611396i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−54.0806	−1.94514	−0.972572	−	0.232601i	$-0.925276\pi$
$773$	−54.0806	−1.94514	−0.972572	+	0.232601i	$0.925276\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	32.1911i	1.15189i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	43.7813i	1.56262i
$786$	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
$793$	26.0000	0.923287
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	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		−	1.00000i	$-0.5\pi$
$809$	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
$821$	−0.125835	−0.00439167	−0.00219583	−	0.999998i	$-0.500699\pi$
$821$	−0.125835	−0.00439167	−0.00219583	+	0.999998i	$0.500699\pi$
$822$	0	0
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	27.5728	0.958802	0.479401	−	0.877596i	$-0.340854\pi$
$827$	27.5728	0.958802	0.479401	+	0.877596i	$0.340854\pi$
$828$	0	0
$829$	50.4777i	1.75316i	0.481253	+	0.876582i	$0.340182\pi$
$829$	50.4777i	1.75316i	−0.481253	+	0.876582i	$0.659818\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	20.2188i	0.699702i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 44.4512i	− 1.53463i	−0.641272	−	0.767314i	$-0.721593\pi$
$839$	− 44.4512i	− 1.53463i	0.641272	−	0.767314i	$-0.278407\pi$
$840$	0	0
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22.7845	−0.783811
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	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
$859$	− 57.6888i	− 1.96832i	−0.177290	−	0.984159i	$-0.556733\pi$
$859$	− 57.6888i	− 1.96832i	0.177290	−	0.984159i	$-0.443267\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.6794i	1.62303i	0.584334	+	0.811513i	$0.301356\pi$
$863$	47.6794i	1.62303i	−0.584334	+	0.811513i	$0.698644\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	67.9760	2.30593
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	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
$883$	28.8444i	0.970692i	0.874322	+	0.485346i	$0.161306\pi$
$883$	28.8444i	0.970692i	−0.874322	+	0.485346i	$0.838694\pi$
$884$	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
$889$	0	0
$890$	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
$905$	43.7813i	1.45534i
$906$	0	0
$907$	− 57.6888i	− 1.91553i	−0.287559	−	0.957763i	$-0.592844\pi$
$907$	− 57.6888i	− 1.91553i	0.287559	−	0.957763i	$-0.407156\pi$
$908$	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
$913$	82.4974	2.73027
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−58.8897	−1.94259	−0.971296	−	0.237872i	$-0.923550\pi$
$919$	−58.8897	−1.94259	−0.971296	+	0.237872i	$0.923550\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	17.7445	0.584067
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	59.5110i	1.95249i	0.216667	+	0.976245i	$0.430481\pi$
$929$	59.5110i	1.95249i	−0.216667	+	0.976245i	$0.569519\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.6410	1.13167	0.565836	−	0.824518i	$-0.308553\pi$
$937$	34.6410	1.13167	0.565836	+	0.824518i	$0.308553\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−57.5859	−1.87725	−0.938624	−	0.344943i	$-0.887898\pi$
$941$	−57.5859	−1.87725	−0.938624	+	0.344943i	$0.887898\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	57.9297	1.88246	0.941231	−	0.337763i	$-0.109670\pi$
$947$	57.9297	1.88246	0.941231	+	0.337763i	$0.109670\pi$
$948$	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
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35.4154i	− 1.13304i	−0.824048	−	0.566520i	$-0.808289\pi$
$977$	− 35.4154i	− 1.13304i	0.824048	−	0.566520i	$-0.191711\pi$
$978$	0	0
$979$	120.139i	3.83965i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.5594i	0.400583i	0.979736	+	0.200292i	$0.0641890\pi$
$983$	12.5594i	0.400583i	−0.979736	+	0.200292i	$0.935811\pi$
$984$	0	0
$985$	−24.3538	−0.775978
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	42.4996	1.34733
$996$	0	0
$997$	24.9800i	0.791124i	0.918439	+	0.395562i	$0.129450\pi$
$997$	24.9800i	0.791124i	−0.918439	+	0.395562i	$0.870550\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.m.f.1585.6		8
3.2	odd	2	inner	3744.2.m.f.1585.4		8
4.3	odd	2		936.2.m.g.181.1	✓	8
8.3	odd	2		936.2.m.g.181.7	yes	8
8.5	even	2	inner	3744.2.m.f.1585.3		8
12.11	even	2		936.2.m.g.181.8	yes	8
13.12	even	2	inner	3744.2.m.f.1585.4		8
24.5	odd	2	inner	3744.2.m.f.1585.5		8
24.11	even	2		936.2.m.g.181.2	yes	8
39.38	odd	2	CM	3744.2.m.f.1585.6		8
52.51	odd	2		936.2.m.g.181.8	yes	8
104.51	odd	2		936.2.m.g.181.2	yes	8
104.77	even	2	inner	3744.2.m.f.1585.5		8
156.155	even	2		936.2.m.g.181.1	✓	8
312.77	odd	2	inner	3744.2.m.f.1585.3		8
312.155	even	2		936.2.m.g.181.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.m.g.181.1	✓	8	4.3	odd	2
936.2.m.g.181.1	✓	8	156.155	even	2
936.2.m.g.181.2	yes	8	24.11	even	2
936.2.m.g.181.2	yes	8	104.51	odd	2
936.2.m.g.181.7	yes	8	8.3	odd	2
936.2.m.g.181.7	yes	8	312.155	even	2
936.2.m.g.181.8	yes	8	12.11	even	2
936.2.m.g.181.8	yes	8	52.51	odd	2
3744.2.m.f.1585.3		8	8.5	even	2	inner
3744.2.m.f.1585.3		8	312.77	odd	2	inner
3744.2.m.f.1585.4		8	3.2	odd	2	inner
3744.2.m.f.1585.4		8	13.12	even	2	inner
3744.2.m.f.1585.5		8	24.5	odd	2	inner
3744.2.m.f.1585.5		8	104.77	even	2	inner
3744.2.m.f.1585.6		8	1.1	even	1	trivial
3744.2.m.f.1585.6		8	39.38	odd	2	CM

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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.75265 q^{5} +O(q^{10})\)	\(q+1.75265 q^{5} -6.54099 q^{11} +3.60555i q^{13} -1.92820 q^{25} -10.1383i q^{41} -12.4900i q^{43} -9.33123i q^{47} +7.00000 q^{49} -11.4641 q^{55} +0.469622 q^{59} -7.21110i q^{61} +6.31928i q^{65} -4.92144i q^{71} -10.3923 q^{79} -12.6124 q^{83} -18.3671i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 40 q^{25} + 56 q^{49} - 64 q^{55}+O(q^{100}) \) 8 * q + 40 * q^25 + 56 * q^49 - 64 * q^55

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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