LMFDB - Embedded newform 2736.2.k.l.2431.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,2,Mod(2431,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("2736.2431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2736.k (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:	$21.8470699930$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-6}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 26x^{2} + 625 $ x^4 - 26*x^2 + 625
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2431.3
Root			$-4.35890 - 2.44949i$ of defining polynomial
Character	$\chi$	$=$	2736.2431
Dual form			2736.2.k.l.2431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q+2.44949i q^{11} -4.35890 q^{19} -7.34847i q^{23} -5.00000 q^{25} +10.6771i q^{29} -8.71780 q^{31} -10.6771i q^{41} +12.2474i q^{47} +7.00000 q^{49} -10.6771i q^{53} +4.00000 q^{61} -8.71780 q^{67} -8.00000 q^{73} -17.4356 q^{79} -17.1464i q^{83} +10.6771i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 20 q^{25} + 28 q^{49} + 16 q^{61} - 32 q^{73}+O(q^{100}) $ 4 * q - 20 * q^25 + 28 * q^49 + 16 * q^61 - 32 * q^73

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.44949i	0.738549i	0.929320	+	0.369274i	$0.120394\pi$
$11$	2.44949i	0.738549i	−0.929320	+	0.369274i	$0.879606\pi$
$12$	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
$19$	−4.35890	−1.00000
$19$	−4.35890	−1.00000
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 7.34847i	− 1.53226i	−0.642685	−	0.766131i	$-0.722179\pi$
$23$	− 7.34847i	− 1.53226i	0.642685	−	0.766131i	$-0.277821\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	10.6771i	1.98268i	0.131306	+	0.991342i	$0.458083\pi$
$29$	10.6771i	1.98268i	−0.131306	+	0.991342i	$0.541917\pi$
$30$	0	0
$31$	−8.71780	−1.56576	−0.782881	−	0.622171i	$-0.786251\pi$
$31$	−8.71780	−1.56576	−0.782881	+	0.622171i	$0.786251\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.6771i	− 1.66748i	−0.552158	−	0.833740i	$-0.686195\pi$
$41$	− 10.6771i	− 1.66748i	0.552158	−	0.833740i	$-0.313805\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.2474i	1.78647i	0.449586	+	0.893237i	$0.351571\pi$
$47$	12.2474i	1.78647i	−0.449586	+	0.893237i	$0.648429\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.6771i	− 1.46661i	−0.679900	−	0.733305i	$-0.737977\pi$
$53$	− 10.6771i	− 1.46661i	0.679900	−	0.733305i	$-0.262023\pi$
$54$	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
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.71780	−1.06505	−0.532524	−	0.846415i	$-0.678756\pi$
$67$	−8.71780	−1.06505	−0.532524	+	0.846415i	$0.678756\pi$
$68$	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
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−17.4356	−1.96166	−0.980829	−	0.194871i	$-0.937571\pi$
$79$	−17.4356	−1.96166	−0.980829	+	0.194871i	$0.937571\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 17.1464i	− 1.88207i	−0.338316	−	0.941033i	$-0.609857\pi$
$83$	− 17.1464i	− 1.88207i	0.338316	−	0.941033i	$-0.390143\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.6771i	1.13177i	0.824485	+	0.565884i	$0.191465\pi$
$89$	10.6771i	1.13177i	−0.824485	+	0.565884i	$0.808535\pi$
$90$	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		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	−17.4356	−1.71798	−0.858990	−	0.511992i	$-0.828908\pi$
$103$	−17.4356	−1.71798	−0.858990	+	0.511992i	$0.828908\pi$
$104$	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
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.6771i	1.00442i	0.864747	+	0.502208i	$0.167479\pi$
$113$	10.6771i	1.00442i	−0.864747	+	0.502208i	$0.832521\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.71780	−0.773579	−0.386790	−	0.922168i	$-0.626416\pi$
$127$	−8.71780	−0.773579	−0.386790	+	0.922168i	$0.626416\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	22.0454i	1.92612i	0.269294	+	0.963058i	$0.413210\pi$
$131$	22.0454i	1.92612i	−0.269294	+	0.963058i	$0.586790\pi$
$132$	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	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	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
$151$	−17.4356	−1.41889	−0.709444	−	0.704761i	$-0.751054\pi$
$151$	−17.4356	−1.41889	−0.709444	+	0.704761i	$0.751054\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 10.6771i	− 0.811763i	−0.913926	−	0.405882i	$-0.866965\pi$
$173$	− 10.6771i	− 0.811763i	0.913926	−	0.405882i	$-0.133035\pi$
$174$	0	0
$175$	0	0
$176$	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
$181$	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
$191$	− 26.9444i	− 1.94963i	−0.223021	−	0.974814i	$-0.571592\pi$
$191$	− 26.9444i	− 1.94963i	0.223021	−	0.974814i	$-0.428408\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	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
$209$	− 10.6771i	− 0.738549i
$210$	0	0
$211$	−8.71780	−0.600158	−0.300079	−	0.953914i	$-0.597013\pi$
$211$	−8.71780	−0.600158	−0.300079	+	0.953914i	$0.597013\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−17.4356	−1.16757	−0.583787	−	0.811907i	$-0.698430\pi$
$223$	−17.4356	−1.16757	−0.583787	+	0.811907i	$0.698430\pi$
$224$	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
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	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
$239$	− 2.44949i	− 0.158444i	−0.996857	−	0.0792222i	$-0.974756\pi$
$239$	− 2.44949i	− 0.158444i	0.996857	−	0.0792222i	$-0.0252436\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.34847i	0.463831i	0.972736	+	0.231916i	$0.0744993\pi$
$251$	7.34847i	0.463831i	−0.972736	+	0.231916i	$0.925501\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 32.0312i	− 1.99805i	−0.0441081	−	0.999027i	$-0.514045\pi$
$257$	− 32.0312i	− 1.99805i	0.0441081	−	0.999027i	$-0.485955\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	31.8434i	1.96355i	0.190057	+	0.981773i	$0.439133\pi$
$263$	31.8434i	1.96355i	−0.190057	+	0.981773i	$0.560867\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	32.0312i	1.95298i	0.215565	+	0.976489i	$0.430841\pi$
$269$	32.0312i	1.95298i	−0.215565	+	0.976489i	$0.569159\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 12.2474i	− 0.738549i
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	32.0312i	1.91082i	0.295277	+	0.955412i	$0.404588\pi$
$281$	32.0312i	1.91082i	−0.295277	+	0.955412i	$0.595412\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	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
$293$	10.6771i	0.623762i	0.950121	+	0.311881i	$0.100959\pi$
$293$	10.6771i	0.623762i	−0.950121	+	0.311881i	$0.899041\pi$
$294$	0	0
$295$	0	0
$296$	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
$307$	34.8712	1.99020	0.995102	−	0.0988534i	$-0.0315175\pi$
$307$	34.8712	1.99020	0.995102	+	0.0988534i	$0.0315175\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	17.1464i	0.972285i	0.873880	+	0.486142i	$0.161596\pi$
$311$	17.1464i	0.972285i	−0.873880	+	0.486142i	$0.838404\pi$
$312$	0	0
$313$	−32.0000	−1.80875	−0.904373	−	0.426742i	$-0.859661\pi$
$313$	−32.0000	−1.80875	−0.904373	+	0.426742i	$0.859661\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 32.0312i	− 1.79905i	−0.436866	−	0.899527i	$-0.643912\pi$
$317$	− 32.0312i	− 1.79905i	0.436866	−	0.899527i	$-0.356088\pi$
$318$	0	0
$319$	−26.1534	−1.46431
$320$	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
$331$	34.8712	1.91669	0.958347	−	0.285606i	$-0.0921950\pi$
$331$	34.8712	1.91669	0.958347	+	0.285606i	$0.0921950\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 21.3542i	− 1.15639i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 36.7423i	− 1.97243i	−0.165462	−	0.986216i	$-0.552911\pi$
$347$	− 36.7423i	− 1.97243i	0.165462	−	0.986216i	$-0.447089\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	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
$359$	− 22.0454i	− 1.16351i	−0.813363	−	0.581756i	$-0.802366\pi$
$359$	− 22.0454i	− 1.16351i	0.813363	−	0.581756i	$-0.197634\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	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
$379$	34.8712	1.79121	0.895606	−	0.444847i	$-0.146742\pi$
$379$	34.8712	1.79121	0.895606	+	0.444847i	$0.146742\pi$
$380$	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
$385$	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
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.0312i	1.59956i	0.600291	+	0.799782i	$0.295051\pi$
$401$	32.0312i	1.59956i	−0.600291	+	0.799782i	$0.704949\pi$
$402$	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
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	26.9444i	1.31632i	0.752878	+	0.658160i	$0.228665\pi$
$419$	26.9444i	1.31632i	−0.752878	+	0.658160i	$0.771335\pi$
$420$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	32.0312i	1.53226i
$438$	0	0
$439$	−17.4356	−0.832156	−0.416078	−	0.909329i	$-0.636596\pi$
$439$	−17.4356	−0.832156	−0.416078	+	0.909329i	$0.636596\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	41.6413i	1.97844i	0.146440	+	0.989220i	$0.453218\pi$
$443$	41.6413i	1.97844i	−0.146440	+	0.989220i	$0.546782\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 10.6771i	− 0.503882i	−0.967743	−	0.251941i	$-0.918931\pi$
$449$	− 10.6771i	− 0.503882i	0.967743	−	0.251941i	$-0.0810690\pi$
$450$	0	0
$451$	26.1534	1.23152
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	40.0000	1.87112	0.935561	−	0.353166i	$-0.114895\pi$
$457$	40.0000	1.87112	0.935561	+	0.353166i	$0.114895\pi$
$458$	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
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.44949i	− 0.113349i	−0.998393	−	0.0566744i	$-0.981950\pi$
$467$	− 2.44949i	− 0.113349i	0.998393	−	0.0566744i	$-0.0180497\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	21.7945	1.00000
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7.34847i	0.335760i	0.985807	+	0.167880i	$0.0536921\pi$
$479$	7.34847i	0.335760i	−0.985807	+	0.167880i	$0.946308\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	43.5890	1.97521	0.987603	−	0.156973i	$-0.0501737\pi$
$487$	43.5890	1.97521	0.987603	+	0.156973i	$0.0501737\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 31.8434i	− 1.43707i	−0.695490	−	0.718536i	$-0.744813\pi$
$491$	− 31.8434i	− 1.43707i	0.695490	−	0.718536i	$-0.255187\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 12.2474i	− 0.546087i	−0.962002	−	0.273043i	$-0.911970\pi$
$503$	− 12.2474i	− 0.546087i	0.962002	−	0.273043i	$-0.0880303\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 10.6771i	− 0.473253i	−0.971601	−	0.236627i	$-0.923958\pi$
$509$	− 10.6771i	− 0.473253i	0.971601	−	0.236627i	$-0.0760418\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−30.0000	−1.31940
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 32.0312i	− 1.40331i	−0.712515	−	0.701657i	$-0.752444\pi$
$521$	− 32.0312i	− 1.40331i	0.712515	−	0.701657i	$-0.247556\pi$
$522$	0	0
$523$	43.5890	1.90601	0.953007	−	0.302949i	$-0.0979712\pi$
$523$	43.5890	1.90601	0.953007	+	0.302949i	$0.0979712\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−31.0000	−1.34783
$530$	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
$539$	17.1464i	0.738549i
$540$	0	0
$541$	−44.0000	−1.89171	−0.945854	−	0.324593i	$-0.894773\pi$
$541$	−44.0000	−1.89171	−0.945854	+	0.324593i	$0.894773\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	34.8712	1.49098	0.745492	−	0.666514i	$-0.232214\pi$
$547$	34.8712	1.49098	0.745492	+	0.666514i	$0.232214\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 46.5403i	− 1.98268i
$552$	0	0
$553$	0	0
$554$	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
$559$	0	0
$560$	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
$569$	− 32.0312i	− 1.34282i	−0.741086	−	0.671410i	$-0.765689\pi$
$569$	− 32.0312i	− 1.34282i	0.741086	−	0.671410i	$-0.234311\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	36.7423i	1.53226i
$576$	0	0
$577$	16.0000	0.666089	0.333044	−	0.942911i	$-0.391924\pi$
$577$	16.0000	0.666089	0.333044	+	0.942911i	$0.391924\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	26.1534	1.08316
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 22.0454i	− 0.909911i	−0.890514	−	0.454956i	$-0.849655\pi$
$587$	− 22.0454i	− 0.909911i	0.890514	−	0.454956i	$-0.150345\pi$
$588$	0	0
$589$	38.0000	1.56576
$590$	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		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.71780	−0.353845	−0.176922	−	0.984225i	$-0.556614\pi$
$607$	−8.71780	−0.353845	−0.176922	+	0.984225i	$0.556614\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	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	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	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
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	10.6771i	0.421719i	0.977516	+	0.210860i	$0.0676263\pi$
$641$	10.6771i	0.421719i	−0.977516	+	0.210860i	$0.932374\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.9444i	1.05929i	0.848218	+	0.529647i	$0.177675\pi$
$647$	26.9444i	1.05929i	−0.848218	+	0.529647i	$0.822325\pi$
$648$	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
$661$	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
$667$	78.4602	3.03799
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.79796i	0.378246i
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.0312i	1.23106i	0.788113	+	0.615530i	$0.211058\pi$
$677$	32.0312i	1.23106i	−0.788113	+	0.615530i	$0.788942\pi$
$678$	0	0
$679$	0	0
$680$	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
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	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
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	64.0625i	2.39916i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 31.8434i	− 1.18756i	−0.804628	−	0.593779i	$-0.797636\pi$
$719$	− 31.8434i	− 1.18756i	0.804628	−	0.593779i	$-0.202364\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 53.3854i	− 1.98268i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	52.0000	1.92066	0.960332	−	0.278859i	$-0.0899564\pi$
$733$	52.0000	1.92066	0.960332	+	0.278859i	$0.0899564\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 21.3542i	− 0.786590i
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	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
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17.4356	−0.636234	−0.318117	−	0.948051i	$-0.603050\pi$
$751$	−17.4356	−0.636234	−0.318117	+	0.948051i	$0.603050\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−46.0000	−1.67190	−0.835949	−	0.548807i	$-0.815082\pi$
$757$	−46.0000	−1.67190	−0.835949	+	0.548807i	$0.815082\pi$
$758$	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
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	53.3854i	1.92014i	0.279761	+	0.960070i	$0.409745\pi$
$773$	53.3854i	1.92014i	−0.279761	+	0.960070i	$0.590255\pi$
$774$	0	0
$775$	43.5890	1.56576
$776$	0	0
$777$	0	0
$778$	0	0
$779$	46.5403i	1.66748i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−8.71780	−0.310756	−0.155378	−	0.987855i	$-0.549660\pi$
$787$	−8.71780	−0.310756	−0.155378	+	0.987855i	$0.549660\pi$
$788$	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
$797$	53.3854i	1.89101i	0.325611	+	0.945504i	$0.394430\pi$
$797$	53.3854i	1.89101i	−0.325611	+	0.945504i	$0.605570\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 19.5959i	− 0.691525i
$804$	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
$811$	34.8712	1.22449	0.612247	−	0.790667i	$-0.290266\pi$
$811$	34.8712	1.22449	0.612247	+	0.790667i	$0.290266\pi$
$812$	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	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	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
$829$	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
$841$	−85.0000	−2.93103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	50.0000	1.71197	0.855984	−	0.517003i	$-0.172952\pi$
$853$	50.0000	1.71197	0.855984	+	0.517003i	$0.172952\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 53.3854i	− 1.82361i	−0.410623	−	0.911805i	$-0.634689\pi$
$857$	− 53.3854i	− 1.82361i	0.410623	−	0.911805i	$-0.365311\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	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
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 42.7083i	− 1.44878i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	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	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	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
$893$	− 53.3854i	− 1.78647i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 93.0806i	− 3.10441i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	43.5890	1.44735	0.723674	−	0.690142i	$-0.242452\pi$
$907$	43.5890	1.44735	0.723674	+	0.690142i	$0.242452\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	42.0000	1.39000
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	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
$931$	−30.5123	−1.00000
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 32.0312i	− 1.04419i	−0.852888	−	0.522094i	$-0.825151\pi$
$941$	− 32.0312i	− 1.04419i	0.852888	−	0.522094i	$-0.174849\pi$
$942$	0	0
$943$	−78.4602	−2.55502
$944$	0	0
$945$	0	0
$946$	0	0
$947$	61.2372i	1.98994i	0.100158	+	0.994972i	$0.468065\pi$
$947$	61.2372i	1.98994i	−0.100158	+	0.994972i	$0.531935\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 10.6771i	− 0.345864i	−0.984934	−	0.172932i	$-0.944676\pi$
$953$	− 10.6771i	− 0.345864i	0.984934	−	0.172932i	$-0.0553242\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	45.0000	1.45161
$962$	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		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.3854i	1.70795i	0.520314	+	0.853975i	$0.325815\pi$
$977$	53.3854i	1.70795i	−0.520314	+	0.853975i	$0.674185\pi$
$978$	0	0
$979$	−26.1534	−0.835866
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−8.71780	−0.276930	−0.138465	−	0.990367i	$-0.544217\pi$
$991$	−8.71780	−0.276930	−0.138465	+	0.990367i	$0.544217\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−44.0000	−1.39349	−0.696747	−	0.717317i	$-0.745370\pi$
$997$	−44.0000	−1.39349	−0.696747	+	0.717317i	$0.745370\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.2.k.l.2431.3	yes	4
3.2	odd	2	inner	2736.2.k.l.2431.1	✓	4
4.3	odd	2	inner	2736.2.k.l.2431.2	yes	4
12.11	even	2	inner	2736.2.k.l.2431.4	yes	4
19.18	odd	2	inner	2736.2.k.l.2431.4	yes	4
57.56	even	2	inner	2736.2.k.l.2431.2	yes	4
76.75	even	2	inner	2736.2.k.l.2431.1	✓	4
228.227	odd	2	CM	2736.2.k.l.2431.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2736.2.k.l.2431.1	✓	4	3.2	odd	2	inner
2736.2.k.l.2431.1	✓	4	76.75	even	2	inner
2736.2.k.l.2431.2	yes	4	4.3	odd	2	inner
2736.2.k.l.2431.2	yes	4	57.56	even	2	inner
2736.2.k.l.2431.3	yes	4	1.1	even	1	trivial
2736.2.k.l.2431.3	yes	4	228.227	odd	2	CM
2736.2.k.l.2431.4	yes	4	12.11	even	2	inner
2736.2.k.l.2431.4	yes	4	19.18	odd	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2736.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q+2.44949i q^{11} -4.35890 q^{19} -7.34847i q^{23} -5.00000 q^{25} +10.6771i q^{29} -8.71780 q^{31} -10.6771i q^{41} +12.2474i q^{47} +7.00000 q^{49} -10.6771i q^{53} +4.00000 q^{61} -8.71780 q^{67} -8.00000 q^{73} -17.4356 q^{79} -17.1464i q^{83} +10.6771i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 20 q^{25} + 28 q^{49} + 16 q^{61} - 32 q^{73}+O(q^{100}) \) 4 * q - 20 * q^25 + 28 * q^49 + 16 * q^61 - 32 * q^73

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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