LMFDB - Embedded newform 2736.3.o.k.721.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(721,2736)]

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([0, 0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2736.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.o (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:	yes
Analytic conductor:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 11x^{2} + 16 $ x^4 - 11*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			721.3
Root			$-1.31342$ of defining polynomial
Character	$\chi$	$=$	2736.721

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.61478 q^{5} -8.82475 q^{7} +O(q^{10})$	$q+5.61478 q^{5} -8.82475 q^{7} -13.4953 q^{11} -2.26577 q^{17} -19.0000 q^{19} +34.8712 q^{23} +6.52575 q^{25} -49.5490 q^{35} -31.1752 q^{43} +93.2847 q^{47} +28.8762 q^{49} -75.7733 q^{55} +108.124 q^{61} +137.072 q^{73} +119.093 q^{77} +139.485 q^{83} -12.7218 q^{85} -106.681 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{7}+O(q^{10}) $ 4 * q + 10 * q^7	$ 4 q + 10 q^{7} - 76 q^{19} + 162 q^{25} - 170 q^{43} + 342 q^{49} + 14 q^{55} + 206 q^{61} + 50 q^{73} + 538 q^{85}+O(q^{100}) $ 4 * q + 10 * q^7 - 76 * q^19 + 162 * q^25 - 170 * q^43 + 342 * q^49 + 14 * q^55 + 206 * q^61 + 50 * q^73 + 538 * q^85

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
$5$	5.61478	1.12296	0.561478	−	0.827492i	$-0.310233\pi$
$5$	5.61478	1.12296	0.561478	+	0.827492i	$0.310233\pi$
$6$	0	0
$7$	−8.82475	−1.26068	−0.630339	−	0.776320i	$-0.717084\pi$
$7$	−8.82475	−1.26068	−0.630339	+	0.776320i	$0.717084\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−13.4953	−1.22685	−0.613424	−	0.789754i	$-0.710208\pi$
$11$	−13.4953	−1.22685	−0.613424	+	0.789754i	$0.710208\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.26577	−0.133280	−0.0666402	−	0.997777i	$-0.521228\pi$
$17$	−2.26577	−0.133280	−0.0666402	+	0.997777i	$0.521228\pi$
$18$	0	0
$19$	−19.0000	−1.00000
$19$	−19.0000	−1.00000
$20$	0	0
$21$	0	0
$22$	0	0
$23$	34.8712	1.51614	0.758069	−	0.652174i	$-0.226143\pi$
$23$	34.8712	1.51614	0.758069	+	0.652174i	$0.226143\pi$
$24$	0	0
$25$	6.52575	0.261030
$26$	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
$35$	−49.5490	−1.41569
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	−31.1752	−0.725006	−0.362503	−	0.931983i	$-0.618078\pi$
$43$	−31.1752	−0.725006	−0.362503	+	0.931983i	$0.618078\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	93.2847	1.98478	0.992391	−	0.123127i	$-0.0392922\pi$
$47$	93.2847	1.98478	0.992391	+	0.123127i	$0.0392922\pi$
$48$	0	0
$49$	28.8762	0.589311
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−75.7733	−1.37770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	108.124	1.77252	0.886260	−	0.463187i	$-0.153294\pi$
$61$	108.124	1.77252	0.886260	+	0.463187i	$0.153294\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	137.072	1.87770	0.938851	−	0.344323i	$-0.111892\pi$
$73$	137.072	1.87770	0.938851	+	0.344323i	$0.111892\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	119.093	1.54666
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	139.485	1.68054	0.840270	−	0.542169i	$-0.182397\pi$
$83$	139.485	1.68054	0.840270	+	0.542169i	$0.182397\pi$
$84$	0	0
$85$	−12.7218	−0.149668
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−106.681	−1.12296
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	174.356	1.72630	0.863148	−	0.504950i	$-0.168489\pi$
$101$	174.356	1.72630	0.863148	+	0.504950i	$0.168489\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	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	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	195.794	1.70256
$116$	0	0
$117$	0	0
$118$	0	0
$119$	19.9948	0.168024
$120$	0	0
$121$	61.1238	0.505155
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−103.729	−0.829831
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−260.744	−1.99041	−0.995207	−	0.0977943i	$-0.968821\pi$
$131$	−260.744	−1.99041	−0.995207	+	0.0977943i	$0.968821\pi$
$132$	0	0
$133$	167.670	1.26068
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−170.709	−1.24605	−0.623026	−	0.782201i	$-0.714097\pi$
$137$	−170.709	−1.24605	−0.623026	+	0.782201i	$0.714097\pi$
$138$	0	0
$139$	268.371	1.93073	0.965364	−	0.260906i	$-0.0840212\pi$
$139$	268.371	1.93073	0.965364	+	0.260906i	$0.0840212\pi$
$140$	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
$149$	−273.156	−1.83326	−0.916632	−	0.399733i	$-0.869103\pi$
$149$	−273.156	−1.83326	−0.916632	+	0.399733i	$0.869103\pi$
$150$	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
$157$	−10.0000	−0.0636943	−0.0318471	−	0.999493i	$-0.510139\pi$
$157$	−10.0000	−0.0636943	−0.0318471	+	0.999493i	$0.510139\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−307.730	−1.91136
$162$	0	0
$163$	250.000	1.53374	0.766871	−	0.641801i	$-0.221813\pi$
$163$	250.000	1.53374	0.766871	+	0.641801i	$0.221813\pi$
$164$	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
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	−57.5881	−0.329075
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	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
$187$	30.5772	0.163515
$188$	0	0
$189$	0	0
$190$	0	0
$191$	104.713	0.548235	0.274117	−	0.961696i	$-0.411614\pi$
$191$	104.713	0.548235	0.274117	+	0.961696i	$0.411614\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	383.583	1.94712	0.973561	−	0.228426i	$-0.0733580\pi$
$197$	383.583	1.94712	0.973561	+	0.228426i	$0.0733580\pi$
$198$	0	0
$199$	396.619	1.99306	0.996530	−	0.0832388i	$-0.0265264\pi$
$199$	396.619	1.99306	0.996530	+	0.0832388i	$0.0265264\pi$
$200$	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
$209$	256.411	1.22685
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−175.042	−0.814149
$216$	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
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	−404.866	−1.76798	−0.883988	−	0.467510i	$-0.845151\pi$
$229$	−404.866	−1.76798	−0.883988	+	0.467510i	$0.845151\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−417.958	−1.79381	−0.896905	−	0.442222i	$-0.854190\pi$
$233$	−417.958	−1.79381	−0.896905	+	0.442222i	$0.854190\pi$
$234$	0	0
$235$	523.773	2.22882
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−468.590	−1.96063	−0.980314	−	0.197443i	$-0.936736\pi$
$239$	−468.590	−1.96063	−0.980314	+	0.197443i	$0.936736\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	162.134	0.661770
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	227.254	0.905394	0.452697	−	0.891664i	$-0.350462\pi$
$251$	227.254	0.905394	0.452697	+	0.891664i	$0.350462\pi$
$252$	0	0
$253$	−470.598	−1.86007
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	182.923	0.695523	0.347762	−	0.937583i	$-0.386942\pi$
$263$	182.923	0.695523	0.347762	+	0.937583i	$0.386942\pi$
$264$	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
$271$	142.000	0.523985	0.261993	−	0.965070i	$-0.415620\pi$
$271$	142.000	0.523985	0.261993	+	0.965070i	$0.415620\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−88.0670	−0.320244
$276$	0	0
$277$	142.928	0.515985	0.257992	−	0.966147i	$-0.416939\pi$
$277$	142.928	0.515985	0.257992	+	0.966147i	$0.416939\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−153.567	−0.542641	−0.271320	−	0.962489i	$-0.587460\pi$
$283$	−153.567	−0.542641	−0.271320	+	0.962489i	$0.587460\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−283.866	−0.982236
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	275.114	0.913999
$302$	0	0
$303$	0	0
$304$	0	0
$305$	607.091	1.99046
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	445.933	1.43387	0.716933	−	0.697142i	$-0.245545\pi$
$311$	445.933	1.43387	0.716933	+	0.697142i	$0.245545\pi$
$312$	0	0
$313$	590.000	1.88498	0.942492	−	0.334229i	$-0.108476\pi$
$313$	590.000	1.88498	0.942492	+	0.334229i	$0.108476\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	43.0495	0.133280
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−823.215	−2.50217
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	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
$341$	0	0
$342$	0	0
$343$	177.587	0.517747
$344$	0	0
$345$	0	0
$346$	0	0
$347$	665.207	1.91702	0.958511	−	0.285055i	$-0.0920118\pi$
$347$	665.207	1.91702	0.958511	+	0.285055i	$0.0920118\pi$
$348$	0	0
$349$	132.866	0.380706	0.190353	−	0.981716i	$-0.439037\pi$
$349$	132.866	0.380706	0.190353	+	0.981716i	$0.439037\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	488.197	1.38299	0.691497	−	0.722380i	$-0.256952\pi$
$353$	488.197	1.38299	0.691497	+	0.722380i	$0.256952\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−127.370	−0.354792	−0.177396	−	0.984140i	$-0.556767\pi$
$359$	−127.370	−0.354792	−0.177396	+	0.984140i	$0.556767\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	769.631	2.10858
$366$	0	0
$367$	−50.0000	−0.136240	−0.0681199	−	0.997677i	$-0.521700\pi$
$367$	−50.0000	−0.136240	−0.0681199	+	0.997677i	$0.521700\pi$
$368$	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
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	668.680	1.73683
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−513.906	−1.32109	−0.660547	−	0.750785i	$-0.729676\pi$
$389$	−513.906	−1.32109	−0.660547	+	0.750785i	$0.729676\pi$
$390$	0	0
$391$	−79.0099	−0.202071
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	134.680	0.339245	0.169622	−	0.985509i	$-0.445745\pi$
$397$	134.680	0.339245	0.169622	+	0.985509i	$0.445745\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	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
$415$	783.176	1.88717
$416$	0	0
$417$	0	0
$418$	0	0
$419$	348.712	0.832248	0.416124	−	0.909308i	$-0.363388\pi$
$419$	348.712	0.832248	0.416124	+	0.909308i	$0.363388\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−14.7858	−0.0347901
$426$	0	0
$427$	−954.165	−2.23458
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−662.553	−1.51614
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−481.399	−1.08668	−0.543340	−	0.839513i	$-0.682841\pi$
$443$	−481.399	−1.08668	−0.543340	+	0.839513i	$0.682841\pi$
$444$	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
$457$	890.062	1.94762	0.973810	−	0.227363i	$-0.0730105\pi$
$457$	890.062	1.94762	0.973810	+	0.227363i	$0.0730105\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	790.312	1.71434	0.857171	−	0.515032i	$-0.172220\pi$
$461$	790.312	1.71434	0.857171	+	0.515032i	$0.172220\pi$
$462$	0	0
$463$	841.815	1.81817	0.909087	−	0.416606i	$-0.136780\pi$
$463$	841.815	1.81817	0.909087	+	0.416606i	$0.136780\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−698.697	−1.49614	−0.748070	−	0.663620i	$-0.769019\pi$
$467$	−698.697	−1.49614	−0.748070	+	0.663620i	$0.769019\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	420.720	0.889472
$474$	0	0
$475$	−123.989	−0.261030
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−174.356	−0.364000	−0.182000	−	0.983299i	$-0.558257\pi$
$479$	−174.356	−0.364000	−0.182000	+	0.983299i	$0.558257\pi$
$480$	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
$491$	348.712	0.710208	0.355104	−	0.934827i	$-0.384446\pi$
$491$	348.712	0.710208	0.355104	+	0.934827i	$0.384446\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−997.609	−1.99922	−0.999608	−	0.0279946i	$-0.991088\pi$
$499$	−997.609	−1.99922	−0.999608	+	0.0279946i	$0.991088\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−383.583	−0.762591	−0.381295	−	0.924453i	$-0.624522\pi$
$503$	−383.583	−0.762591	−0.381295	+	0.924453i	$0.624522\pi$
$504$	0	0
$505$	978.970	1.93855
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−1209.63	−2.36718
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1258.91	−2.43502
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	687.000	1.29868
$530$	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
$539$	−389.694	−0.722995
$540$	0	0
$541$	1077.86	1.99234	0.996170	−	0.0874330i	$-0.0278664\pi$
$541$	1077.86	1.99234	0.996170	+	0.0874330i	$0.0278664\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	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
$557$	−845.864	−1.51861	−0.759303	−	0.650737i	$-0.774460\pi$
$557$	−845.864	−1.51861	−0.759303	+	0.650737i	$0.774460\pi$
$558$	0	0
$559$	0	0
$560$	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	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−458.000	−0.802102	−0.401051	−	0.916056i	$-0.631355\pi$
$571$	−458.000	−0.802102	−0.401051	+	0.916056i	$0.631355\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	227.561	0.395757
$576$	0	0
$577$	−697.072	−1.20810	−0.604049	−	0.796947i	$-0.706447\pi$
$577$	−697.072	−1.20810	−0.604049	+	0.796947i	$0.706447\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1230.92	−2.11862
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	806.461	1.37387	0.686934	−	0.726719i	$-0.258956\pi$
$587$	806.461	1.37387	0.686934	+	0.726719i	$0.258956\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1185.62	−1.99936	−0.999680	−	0.0252951i	$-0.991947\pi$
$593$	−1185.62	−1.99936	−0.999680	+	0.0252951i	$0.991947\pi$
$594$	0	0
$595$	112.266	0.188683
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	343.196	0.567267
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−883.052	−1.44054	−0.720271	−	0.693693i	$-0.755983\pi$
$613$	−883.052	−1.44054	−0.720271	+	0.693693i	$0.755983\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1233.98	1.99996	0.999982	−	0.00592639i	$-0.00188644\pi$
$617$	1233.98	1.99996	0.999982	+	0.00592639i	$0.00188644\pi$
$618$	0	0
$619$	−662.000	−1.06947	−0.534733	−	0.845021i	$-0.679588\pi$
$619$	−662.000	−1.06947	−0.534733	+	0.845021i	$0.679588\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−745.558	−1.19289
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1141.36	1.80881	0.904407	−	0.426671i	$-0.140314\pi$
$631$	1141.36	1.80881	0.904407	+	0.426671i	$0.140314\pi$
$632$	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
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	1112.41	1.73004	0.865018	−	0.501741i	$-0.167307\pi$
$643$	1112.41	1.73004	0.865018	+	0.501741i	$0.167307\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1277.91	1.97514	0.987568	−	0.157194i	$-0.0502449\pi$
$647$	1277.91	1.97514	0.987568	+	0.157194i	$0.0502449\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	300.742	0.460555	0.230278	−	0.973125i	$-0.426037\pi$
$653$	300.742	0.460555	0.230278	+	0.973125i	$0.426037\pi$
$654$	0	0
$655$	−1464.02	−2.23515
$656$	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	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	941.432	1.41569
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1459.17	−2.17461
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	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
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−958.494	−1.39926
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	1110.60	1.60723	0.803617	−	0.595147i	$-0.202906\pi$
$691$	1110.60	1.60723	0.803617	+	0.595147i	$0.202906\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1506.85	2.16812
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−871.780	−1.24362	−0.621812	−	0.783167i	$-0.713603\pi$
$701$	−871.780	−1.24362	−0.621812	+	0.783167i	$0.713603\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1538.65	−2.17631
$708$	0	0
$709$	−1318.00	−1.85896	−0.929478	−	0.368877i	$-0.879742\pi$
$709$	−1318.00	−1.85896	−0.929478	+	0.368877i	$0.879742\pi$
$710$	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
$719$	−1367.95	−1.90257	−0.951285	−	0.308313i	$-0.900235\pi$
$719$	−1367.95	−1.90257	−0.951285	+	0.308313i	$0.900235\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1299.55	−1.78755	−0.893774	−	0.448518i	$-0.851952\pi$
$727$	−1299.55	−1.78755	−0.893774	+	0.448518i	$0.851952\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	70.6358	0.0966290
$732$	0	0
$733$	−1270.00	−1.73261	−0.866303	−	0.499519i	$-0.833510\pi$
$733$	−1270.00	−1.73261	−0.866303	+	0.499519i	$0.833510\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−1462.60	−1.97916	−0.989580	−	0.143986i	$-0.954008\pi$
$739$	−1462.60	−1.97916	−0.989580	+	0.143986i	$0.954008\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−1533.71	−2.05867
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1513.65	−1.99954	−0.999769	−	0.0214884i	$-0.993160\pi$
$757$	−1513.65	−1.99954	−0.999769	+	0.0214884i	$0.993160\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1181.77	1.55291	0.776456	−	0.630171i	$-0.217015\pi$
$761$	1181.77	1.55291	0.776456	+	0.630171i	$0.217015\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	1494.10	1.94292	0.971459	−	0.237208i	$-0.0762322\pi$
$769$	1494.10	1.94292	0.971459	+	0.237208i	$0.0762322\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	−56.1478	−0.0715258
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	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
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	−211.361	−0.264532
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1849.83	−2.30365
$804$	0	0
$805$	−1727.83	−2.14638
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1237.91	−1.53018	−0.765089	−	0.643924i	$-0.777305\pi$
$809$	−1237.91	−1.53018	−0.765089	+	0.643924i	$0.777305\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1403.69	1.72232
$816$	0	0
$817$	592.330	0.725006
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1588.21	−1.93448	−0.967239	−	0.253869i	$-0.918297\pi$
$821$	−1588.21	−1.93448	−0.967239	+	0.253869i	$0.918297\pi$
$822$	0	0
$823$	1224.17	1.48744	0.743721	−	0.668490i	$-0.233059\pi$
$823$	1224.17	1.48744	0.743721	+	0.668490i	$0.233059\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−65.4268	−0.0785436
$834$	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
$845$	948.898	1.12296
$846$	0	0
$847$	−539.402	−0.636838
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−1030.00	−1.20750	−0.603751	−	0.797173i	$-0.706328\pi$
$853$	−1030.00	−1.20750	−0.603751	+	0.797173i	$0.706328\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	10.3911	0.0120968	0.00604839	−	0.999982i	$-0.498075\pi$
$859$	10.3911	0.0120968	0.00604839	+	0.999982i	$0.498075\pi$
$860$	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
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	915.381	1.04615
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−374.032	−0.424554	−0.212277	−	0.977209i	$-0.568088\pi$
$881$	−374.032	−0.424554	−0.212277	+	0.977209i	$0.568088\pi$
$882$	0	0
$883$	930.145	1.05339	0.526696	−	0.850054i	$-0.323431\pi$
$883$	930.145	1.05339	0.526696	+	0.850054i	$0.323431\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1772.41	−1.98478
$894$	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
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−1882.39	−2.06177
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2301.00	2.50927
$918$	0	0
$919$	1762.00	1.91730	0.958651	−	0.284585i	$-0.0918559\pi$
$919$	1762.00	1.91730	0.958651	+	0.284585i	$0.0918559\pi$
$920$	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
$929$	1743.56	1.87681	0.938407	−	0.345533i	$-0.112302\pi$
$929$	1743.56	1.87681	0.938407	+	0.345533i	$0.112302\pi$
$930$	0	0
$931$	−548.649	−0.589311
$932$	0	0
$933$	0	0
$934$	0	0
$935$	171.684	0.183620
$936$	0	0
$937$	−1764.29	−1.88291	−0.941457	−	0.337134i	$-0.890543\pi$
$937$	−1764.29	−1.88291	−0.941457	+	0.337134i	$0.890543\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−488.197	−0.515519	−0.257760	−	0.966209i	$-0.582984\pi$
$947$	−488.197	−0.515519	−0.257760	+	0.966209i	$0.582984\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	587.939	0.615643
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1506.47	1.57087
$960$	0	0
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1790.00	−1.85109	−0.925543	−	0.378643i	$-0.876391\pi$
$967$	−1790.00	−1.85109	−0.925543	+	0.378643i	$0.876391\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	−2368.31	−2.43403
$974$	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
$985$	2153.73	2.18653
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1087.12	−1.09921
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2226.93	2.23812
$996$	0	0
$997$	1225.32	1.22901	0.614503	−	0.788914i	$-0.289356\pi$
$997$	1225.32	1.22901	0.614503	+	0.788914i	$0.289356\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.o.k.721.3		4
3.2	odd	2	inner	2736.3.o.k.721.2		4
4.3	odd	2		684.3.h.d.37.3	yes	4
12.11	even	2		684.3.h.d.37.2	✓	4
19.18	odd	2	CM	2736.3.o.k.721.3		4
57.56	even	2	inner	2736.3.o.k.721.2		4
76.75	even	2		684.3.h.d.37.3	yes	4
228.227	odd	2		684.3.h.d.37.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.h.d.37.2	✓	4	12.11	even	2
684.3.h.d.37.2	✓	4	228.227	odd	2
684.3.h.d.37.3	yes	4	4.3	odd	2
684.3.h.d.37.3	yes	4	76.75	even	2
2736.3.o.k.721.2		4	3.2	odd	2	inner
2736.3.o.k.721.2		4	57.56	even	2	inner
2736.3.o.k.721.3		4	1.1	even	1	trivial
2736.3.o.k.721.3		4	19.18	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:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.61478 q^{5} -8.82475 q^{7} +O(q^{10})\)	\(q+5.61478 q^{5} -8.82475 q^{7} -13.4953 q^{11} -2.26577 q^{17} -19.0000 q^{19} +34.8712 q^{23} +6.52575 q^{25} -49.5490 q^{35} -31.1752 q^{43} +93.2847 q^{47} +28.8762 q^{49} -75.7733 q^{55} +108.124 q^{61} +137.072 q^{73} +119.093 q^{77} +139.485 q^{83} -12.7218 q^{85} -106.681 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{7}+O(q^{10}) \) 4 * q + 10 * q^7	\( 4 q + 10 q^{7} - 76 q^{19} + 162 q^{25} - 170 q^{43} + 342 q^{49} + 14 q^{55} + 206 q^{61} + 50 q^{73} + 538 q^{85}+O(q^{100}) \) 4 * q + 10 * q^7 - 76 * q^19 + 162 * q^25 - 170 * q^43 + 342 * q^49 + 14 * q^55 + 206 * q^61 + 50 * q^73 + 538 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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