LMFDB - Embedded newform 2736.3.o.k.721.1

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			721.1
Root			$-3.04547$ 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)

$f(q)$	$=$	$q-9.97368 q^{5} +13.8248 q^{7} +O(q^{10})$	$q-9.97368 q^{5} +13.8248 q^{7} -8.29917 q^{11} -28.2465 q^{17} -19.0000 q^{19} +34.8712 q^{23} +74.4743 q^{25} -137.884 q^{35} -53.8248 q^{43} -36.6191 q^{47} +142.124 q^{49} +82.7733 q^{55} -5.12376 q^{61} -112.072 q^{73} -114.734 q^{77} +139.485 q^{83} +281.722 q^{85} +189.500 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−9.97368	−1.99474	−0.997368	−	0.0725083i	$-0.976900\pi$
$5$	−9.97368	−1.99474	−0.997368	+	0.0725083i	$0.976900\pi$
$6$	0	0
$7$	13.8248	1.97496	0.987482	−	0.157730i	$-0.0504176\pi$
$7$	13.8248	1.97496	0.987482	+	0.157730i	$0.0504176\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−8.29917	−0.754470	−0.377235	−	0.926118i	$-0.623125\pi$
$11$	−8.29917	−0.754470	−0.377235	+	0.926118i	$0.623125\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−28.2465	−1.66156	−0.830780	−	0.556601i	$-0.812105\pi$
$17$	−28.2465	−1.66156	−0.830780	+	0.556601i	$0.812105\pi$
$18$	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
$25$	74.4743	2.97897
$26$	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
$35$	−137.884	−3.93953
$36$	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
$43$	−53.8248	−1.25174	−0.625869	−	0.779928i	$-0.715256\pi$
$43$	−53.8248	−1.25174	−0.625869	+	0.779928i	$0.715256\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−36.6191	−0.779129	−0.389564	−	0.920999i	$-0.627374\pi$
$47$	−36.6191	−0.779129	−0.389564	+	0.920999i	$0.627374\pi$
$48$	0	0
$49$	142.124	2.90048
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	82.7733	1.50497
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	−5.12376	−0.0839960	−0.0419980	−	0.999118i	$-0.513372\pi$
$61$	−5.12376	−0.0839960	−0.0419980	+	0.999118i	$0.513372\pi$
$62$	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
$73$	−112.072	−1.53524	−0.767618	−	0.640907i	$-0.778558\pi$
$73$	−112.072	−1.53524	−0.767618	+	0.640907i	$0.778558\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−114.734	−1.49005
$78$	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
$85$	281.722	3.31437
$86$	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
$95$	189.500	1.99474
$96$	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
$115$	−347.794	−3.02430
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−390.501	−3.28152
$120$	0	0
$121$	−52.1238	−0.430775
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−493.440	−3.94752
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	108.183	0.825822	0.412911	−	0.910771i	$-0.364512\pi$
$131$	108.183	0.825822	0.412911	+	0.910771i	$0.364512\pi$
$132$	0	0
$133$	−262.670	−1.97496
$134$	0	0
$135$	0	0
$136$	0	0
$137$	270.964	1.97784	0.988919	−	0.148456i	$-0.0474303\pi$
$137$	270.964	1.97784	0.988919	+	0.148456i	$0.0474303\pi$
$138$	0	0
$139$	−71.3713	−0.513462	−0.256731	−	0.966483i	$-0.582646\pi$
$139$	−71.3713	−0.513462	−0.256731	+	0.966483i	$0.582646\pi$
$140$	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
$149$	33.4168	0.224274	0.112137	−	0.993693i	$-0.464231\pi$
$149$	33.4168	0.224274	0.112137	+	0.993693i	$0.464231\pi$
$150$	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
$161$	482.086	2.99432
$162$	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
$175$	1029.59	5.88336
$176$	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
$187$	234.423	1.25360
$188$	0	0
$189$	0	0
$190$	0	0
$191$	265.794	1.39159	0.695795	−	0.718241i	$-0.255052\pi$
$191$	265.794	1.39159	0.695795	+	0.718241i	$0.255052\pi$
$192$	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
$199$	−169.619	−0.852356	−0.426178	−	0.904639i	$-0.640140\pi$
$199$	−169.619	−0.852356	−0.426178	+	0.904639i	$0.640140\pi$
$200$	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
$209$	157.684	0.754470
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	536.831	2.49689
$216$	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
$229$	387.866	1.69374	0.846870	−	0.531800i	$-0.178484\pi$
$229$	387.866	1.69374	0.846870	+	0.531800i	$0.178484\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	387.446	1.66286	0.831428	−	0.555632i	$-0.187524\pi$
$233$	387.446	1.66286	0.831428	+	0.555632i	$0.187524\pi$
$234$	0	0
$235$	365.227	1.55416
$236$	0	0
$237$	0	0
$238$	0	0
$239$	316.029	1.32230	0.661148	−	0.750255i	$-0.270070\pi$
$239$	316.029	1.32230	0.661148	+	0.750255i	$0.270070\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1417.50	−5.78570
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	274.019	1.09171	0.545855	−	0.837879i	$-0.316205\pi$
$251$	274.019	1.09171	0.545855	+	0.837879i	$0.316205\pi$
$252$	0	0
$253$	−289.402	−1.14388
$254$	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
$263$	−518.558	−1.97170	−0.985852	−	0.167621i	$-0.946392\pi$
$263$	−518.558	−1.97170	−0.985852	+	0.167621i	$0.946392\pi$
$264$	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
$275$	−618.075	−2.24754
$276$	0	0
$277$	392.072	1.41542	0.707712	−	0.706501i	$-0.249728\pi$
$277$	392.072	1.41542	0.707712	+	0.706501i	$0.249728\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	548.567	1.93840	0.969200	−	0.246274i	$-0.0792064\pi$
$283$	548.567	1.93840	0.969200	+	0.246274i	$0.0792064\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	508.866	1.76078
$290$	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
$301$	−744.114	−2.47214
$302$	0	0
$303$	0	0
$304$	0	0
$305$	51.1027	0.167550
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−598.494	−1.92442	−0.962209	−	0.272312i	$-0.912212\pi$
$311$	−598.494	−1.92442	−0.962209	+	0.272312i	$0.912212\pi$
$312$	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
$323$	536.684	1.66156
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−506.249	−1.53875
$330$	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
$343$	1287.41	3.75339
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−503.928	−1.45224	−0.726120	−	0.687568i	$-0.758678\pi$
$347$	−503.928	−1.45224	−0.726120	+	0.687568i	$0.758678\pi$
$348$	0	0
$349$	−659.866	−1.89073	−0.945367	−	0.326007i	$-0.894297\pi$
$349$	−659.866	−1.89073	−0.945367	+	0.326007i	$0.894297\pi$
$350$	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
$359$	−548.259	−1.52718	−0.763592	−	0.645699i	$-0.776566\pi$
$359$	−548.259	−1.52718	−0.763592	+	0.645699i	$0.776566\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1117.77	3.06239
$366$	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
$385$	1144.32	2.97226
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−248.902	−0.639850	−0.319925	−	0.947443i	$-0.603658\pi$
$389$	−248.902	−0.639850	−0.319925	+	0.947443i	$0.603658\pi$
$390$	0	0
$391$	−984.990	−2.51916
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	610.320	1.53733	0.768665	−	0.639652i	$-0.220921\pi$
$397$	610.320	1.53733	0.768665	+	0.639652i	$0.220921\pi$
$398$	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
$415$	−1391.18	−3.35223
$416$	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
$425$	−2103.64	−4.94974
$426$	0	0
$427$	−70.8347	−0.165889
$428$	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
$443$	−403.457	−0.910738	−0.455369	−	0.890303i	$-0.650493\pi$
$443$	−403.457	−0.910738	−0.455369	+	0.890303i	$0.650493\pi$
$444$	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
$457$	−265.062	−0.580005	−0.290003	−	0.957026i	$-0.593656\pi$
$457$	−265.062	−0.580005	−0.290003	+	0.957026i	$0.593656\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.0848	0.0348911	0.0174455	−	0.999848i	$-0.494447\pi$
$461$	16.0848	0.0348911	0.0174455	+	0.999848i	$0.494447\pi$
$462$	0	0
$463$	−86.8148	−0.187505	−0.0937525	−	0.995596i	$-0.529886\pi$
$463$	−86.8148	−0.187505	−0.0937525	+	0.995596i	$0.529886\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	886.130	1.89749	0.948747	−	0.316037i	$-0.102352\pi$
$467$	886.130	1.89749	0.948747	+	0.316037i	$0.102352\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	446.701	0.944399
$474$	0	0
$475$	−1415.01	−2.97897
$476$	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
$499$	474.609	0.951120	0.475560	−	0.879683i	$-0.342245\pi$
$499$	474.609	0.951120	0.475560	+	0.879683i	$0.342245\pi$
$500$	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
$505$	−1738.97	−3.44351
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−1549.37	−3.03204
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	303.908	0.587830
$518$	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
$539$	−1179.51	−2.18833
$540$	0	0
$541$	−620.856	−1.14761	−0.573804	−	0.818992i	$-0.694533\pi$
$541$	−620.856	−1.14761	−0.573804	+	0.818992i	$0.694533\pi$
$542$	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
$557$	1050.73	1.88641	0.943206	−	0.332207i	$-0.107793\pi$
$557$	1050.73	1.88641	0.943206	+	0.332207i	$0.107793\pi$
$558$	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
$575$	2597.01	4.51653
$576$	0	0
$577$	−447.928	−0.776305	−0.388152	−	0.921595i	$-0.626887\pi$
$577$	−447.928	−0.776305	−0.388152	+	0.921595i	$0.626887\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1928.34	3.31901
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1142.10	−1.94565	−0.972825	−	0.231543i	$-0.925623\pi$
$587$	−1142.10	−1.94565	−0.972825	+	0.231543i	$0.925623\pi$
$588$	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
$595$	3894.73	6.54577
$596$	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
$605$	519.866	0.859282
$606$	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
$613$	1178.05	1.92178	0.960891	−	0.276927i	$-0.0893159\pi$
$613$	1178.05	1.92178	0.960891	+	0.276927i	$0.0893159\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−610.656	−0.989718	−0.494859	−	0.868973i	$-0.664780\pi$
$617$	−610.656	−0.989718	−0.494859	+	0.868973i	$0.664780\pi$
$618$	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
$625$	3059.56	4.89529
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−104.361	−0.165390	−0.0826952	−	0.996575i	$-0.526353\pi$
$631$	−104.361	−0.165390	−0.0826952	+	0.996575i	$0.526353\pi$
$632$	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
$643$	2.58717	0.00402359	0.00201180	−	0.999998i	$-0.499360\pi$
$643$	2.58717	0.00402359	0.00201180	+	0.999998i	$0.499360\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−462.798	−0.715299	−0.357650	−	0.933856i	$-0.616422\pi$
$647$	−462.798	−0.715299	−0.357650	+	0.933856i	$0.616422\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	950.262	1.45522	0.727612	−	0.685989i	$-0.240630\pi$
$653$	950.262	1.45522	0.727612	+	0.685989i	$0.240630\pi$
$654$	0	0
$655$	−1078.98	−1.64730
$656$	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
$665$	2619.79	3.93953
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	42.5229	0.0633725
$672$	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
$685$	−2702.51	−3.94526
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1267.60	−1.83444	−0.917221	−	0.398380i	$-0.869573\pi$
$691$	−1267.60	−1.83444	−0.917221	+	0.398380i	$0.869573\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	711.834	1.02422
$696$	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
$707$	2410.43	3.40937
$708$	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
$719$	300.017	0.417270	0.208635	−	0.977994i	$-0.433098\pi$
$719$	300.017	0.417270	0.208635	+	0.977994i	$0.433098\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1214.55	1.67063	0.835315	−	0.549772i	$-0.185285\pi$
$727$	1214.55	1.67063	0.835315	+	0.549772i	$0.185285\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1520.36	2.07984
$732$	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
$739$	915.599	1.23897	0.619485	−	0.785008i	$-0.287341\pi$
$739$	915.599	1.23897	0.619485	+	0.785008i	$0.287341\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−333.288	−0.447367
$746$	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
$757$	728.650	0.962550	0.481275	−	0.876570i	$-0.340174\pi$
$757$	728.650	0.962550	0.481275	+	0.876570i	$0.340174\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1421.51	−1.86794	−0.933972	−	0.357345i	$-0.883682\pi$
$761$	−1421.51	−1.86794	−0.933972	+	0.357345i	$0.883682\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−431.104	−0.560603	−0.280302	−	0.959912i	$-0.590434\pi$
$769$	−431.104	−0.560603	−0.280302	+	0.959912i	$0.590434\pi$
$770$	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
$785$	99.7368	0.127053
$786$	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
$799$	1034.36	1.29457
$800$	0	0
$801$	0	0
$802$	0	0
$803$	930.107	1.15829
$804$	0	0
$805$	−4808.17	−5.97288
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1521.24	1.88040	0.940199	−	0.340624i	$-0.110638\pi$
$809$	1521.24	1.88040	0.940199	+	0.340624i	$0.110638\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2493.42	−3.05941
$816$	0	0
$817$	1022.67	1.25174
$818$	0	0
$819$	0	0
$820$	0	0
$821$	433.098	0.527524	0.263762	−	0.964588i	$-0.415037\pi$
$821$	433.098	0.527524	0.263762	+	0.964588i	$0.415037\pi$
$822$	0	0
$823$	340.835	0.414137	0.207068	−	0.978326i	$-0.433608\pi$
$823$	340.835	0.414137	0.207068	+	0.978326i	$0.433608\pi$
$824$	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
$833$	−4014.50	−4.81933
$834$	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
$845$	−1685.55	−1.99474
$846$	0	0
$847$	−720.598	−0.850765
$848$	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
$859$	1482.61	1.72597	0.862985	−	0.505229i	$-0.168592\pi$
$859$	1482.61	1.72597	0.862985	+	0.505229i	$0.168592\pi$
$860$	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
$875$	−6821.69	−7.79622
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1304.14	−1.48030	−0.740150	−	0.672442i	$-0.765245\pi$
$881$	−1304.14	−1.48030	−0.740150	+	0.672442i	$0.765245\pi$
$882$	0	0
$883$	−1765.15	−1.99903	−0.999516	−	0.0311055i	$-0.990097\pi$
$883$	−1765.15	−1.99903	−0.999516	+	0.0311055i	$0.990097\pi$
$884$	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
$893$	695.762	0.779129
$894$	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
$913$	−1157.61	−1.26792
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1495.60	1.63097
$918$	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
$931$	−2700.35	−2.90048
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2338.06	−2.50060
$936$	0	0
$937$	1429.29	1.52539	0.762695	−	0.646759i	$-0.223876\pi$
$937$	1429.29	1.52539	0.762695	+	0.646759i	$0.223876\pi$
$938$	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
$955$	−2650.94	−2.77585
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3746.01	3.90616
$960$	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
$973$	−986.690	−1.01407
$974$	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
$985$	−3825.73	−3.88399
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1876.93	−1.89781
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1691.72	1.70022
$996$	0	0
$997$	749.680	0.751936	0.375968	−	0.926633i	$-0.377310\pi$
$997$	749.680	0.751936	0.375968	+	0.926633i	$0.377310\pi$
$998$	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.721.1		4
3.2	odd	2	inner	2736.3.o.k.721.4		4
4.3	odd	2		684.3.h.d.37.1	✓	4
12.11	even	2		684.3.h.d.37.4	yes	4
19.18	odd	2	CM	2736.3.o.k.721.1		4
57.56	even	2	inner	2736.3.o.k.721.4		4
76.75	even	2		684.3.h.d.37.1	✓	4
228.227	odd	2		684.3.h.d.37.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.h.d.37.1	✓	4	4.3	odd	2
684.3.h.d.37.1	✓	4	76.75	even	2
684.3.h.d.37.4	yes	4	12.11	even	2
684.3.h.d.37.4	yes	4	228.227	odd	2
2736.3.o.k.721.1		4	1.1	even	1	trivial
2736.3.o.k.721.1		4	19.18	odd	2	CM
2736.3.o.k.721.4		4	3.2	odd	2	inner
2736.3.o.k.721.4		4	57.56	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.97368 q^{5} +13.8248 q^{7} +O(q^{10})\)	\(q-9.97368 q^{5} +13.8248 q^{7} -8.29917 q^{11} -28.2465 q^{17} -19.0000 q^{19} +34.8712 q^{23} +74.4743 q^{25} -137.884 q^{35} -53.8248 q^{43} -36.6191 q^{47} +142.124 q^{49} +82.7733 q^{55} -5.12376 q^{61} -112.072 q^{73} -114.734 q^{77} +139.485 q^{83} +281.722 q^{85} +189.500 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{7}+O(q^{10}) \) 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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