LMFDB - Embedded newform 736.5.e.d.689.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [736,5,Mod(689,736)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("736.689"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 736 = 2^{5} \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	736.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$76.0802928297$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.8869743.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{3} + 8 $ x^6 - 3*x^3 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 184)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			689.1
Root			$-0.261988 + 1.38973i$ of defining polynomial
Character	$\chi$	$=$	736.689
Dual form			736.5.e.d.689.6

$q$-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 $\left(\mathbb{Z}/736\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$645$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 17.9999i	− 1.99999i	−0.00320078	−	0.999995i	$-0.501019\pi$
$3$	− 17.9999i	− 1.99999i	0.00320078	−	0.999995i	$-0.498981\pi$
$4$	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
$9$	−242.997	−2.99996
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	316.603i	1.87339i	0.350148	+	0.936694i	$0.386131\pi$
$13$	316.603i	1.87339i	−0.350148	+	0.936694i	$0.613869\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	529.000	1.00000
$23$	529.000	1.00000
$24$	0	0
$25$	−625.000	−1.00000
$26$	0	0
$27$	2915.93i	3.99990i
$28$	0	0
$29$	− 939.360i	− 1.11696i	−0.829519	−	0.558478i	$-0.811386\pi$
$29$	− 939.360i	− 1.11696i	0.829519	−	0.558478i	$-0.188614\pi$
$30$	0	0
$31$	−770.227	−0.801485	−0.400742	−	0.916191i	$-0.631248\pi$
$31$	−770.227	−0.801485	−0.400742	+	0.916191i	$0.631248\pi$
$32$	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
$39$	5698.82	3.74676
$40$	0	0
$41$	3284.79	1.95407	0.977035	−	0.213081i	$-0.0683499\pi$
$41$	3284.79	1.95407	0.977035	+	0.213081i	$0.0683499\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2577.80	1.16695	0.583477	−	0.812130i	$-0.301692\pi$
$47$	2577.80	1.16695	0.583477	+	0.812130i	$0.301692\pi$
$48$	0	0
$49$	2401.00	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2992.60i	0.859695i	0.902901	+	0.429848i	$0.141433\pi$
$59$	2992.60i	0.859695i	−0.902901	+	0.429848i	$0.858567\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	− 9521.95i	− 1.99999i
$70$	0	0
$71$	2750.83	0.545691	0.272846	−	0.962058i	$-0.412035\pi$
$71$	2750.83	0.545691	0.272846	+	0.962058i	$0.412035\pi$
$72$	0	0
$73$	−10333.3	−1.93907	−0.969535	−	0.244954i	$-0.921227\pi$
$73$	−10333.3	−1.93907	−0.969535	+	0.244954i	$0.921227\pi$
$74$	0	0
$75$	11249.9i	1.99999i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	32803.7	4.99980
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−16908.4	−2.23390
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	13864.0i	1.60296i
$94$	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
$101$	19106.6i	1.87301i	0.350652	+	0.936506i	$0.385960\pi$
$101$	19106.6i	1.87301i	−0.350652	+	0.936506i	$0.614040\pi$
$102$	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		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	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	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 76933.4i	− 5.62009i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−14641.0	−1.00000
$122$	0	0
$123$	− 59125.9i	− 3.90812i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	28601.2	1.77328	0.886639	−	0.462462i	$-0.153034\pi$
$127$	28601.2	1.77328	0.886639	+	0.462462i	$0.153034\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18060.0i	1.05238i	0.850366	+	0.526192i	$0.176381\pi$
$131$	18060.0i	1.05238i	−0.850366	+	0.526192i	$0.823619\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	− 6463.20i	− 0.334517i	−0.985913	−	0.167258i	$-0.946509\pi$
$139$	− 6463.20i	− 0.334517i	0.985913	−	0.167258i	$-0.0534914\pi$
$140$	0	0
$141$	− 46400.2i	− 2.33390i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 43217.8i	− 1.99999i
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	43532.7	1.90925	0.954624	−	0.297815i	$-0.0962579\pi$
$151$	43532.7	1.90925	0.954624	+	0.297815i	$0.0962579\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 25874.5i	− 0.973859i	−0.873441	−	0.486930i	$-0.838117\pi$
$163$	− 25874.5i	− 0.973859i	0.873441	−	0.486930i	$-0.161883\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2786.00	−0.0998960	−0.0499480	−	0.998752i	$-0.515906\pi$
$167$	−2786.00	−0.0998960	−0.0499480	+	0.998752i	$0.515906\pi$
$168$	0	0
$169$	−71676.2	−2.50958
$170$	0	0
$171$	0	0
$172$	0	0
$173$	7596.60i	0.253821i	0.991914	+	0.126910i	$0.0405060\pi$
$173$	7596.60i	0.253821i	−0.991914	+	0.126910i	$0.959494\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	53866.5	1.71938
$178$	0	0
$179$	− 46622.5i	− 1.45509i	−0.686060	−	0.727545i	$-0.740661\pi$
$179$	− 46622.5i	− 1.45509i	0.686060	−	0.727545i	$-0.259339\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	24566.7	0.659528	0.329764	−	0.944063i	$-0.393031\pi$
$193$	24566.7	0.659528	0.329764	+	0.944063i	$0.393031\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	61414.7i	1.58249i	0.611502	+	0.791243i	$0.290566\pi$
$197$	61414.7i	1.58249i	−0.611502	+	0.791243i	$0.709434\pi$
$198$	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
$207$	−128545.	−2.99996
$208$	0	0
$209$	0	0
$210$	0	0
$211$	46730.6i	1.04963i	0.851216	+	0.524815i	$0.175866\pi$
$211$	46730.6i	1.04963i	−0.851216	+	0.524815i	$0.824134\pi$
$212$	0	0
$213$	− 49514.7i	− 1.09138i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	185998.i	3.87812i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	46466.0	0.934384	0.467192	−	0.884156i	$-0.345266\pi$
$223$	46466.0	0.934384	0.467192	+	0.884156i	$0.345266\pi$
$224$	0	0
$225$	151873.	2.99996
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	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
$233$	72635.2	1.33794	0.668968	−	0.743291i	$-0.266736\pi$
$233$	72635.2	1.33794	0.668968	+	0.743291i	$0.266736\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	113646.	1.98956	0.994781	−	0.102030i	$-0.0325337\pi$
$239$	113646.	1.98956	0.994781	+	0.102030i	$0.0325337\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	− 354273.i	− 5.99964i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−112877.	−1.70899	−0.854497	−	0.519457i	$-0.826134\pi$
$257$	−112877.	−1.70899	−0.854497	+	0.519457i	$0.826134\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	228261.i	3.35082i
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 112955.i	− 1.56100i	−0.625158	−	0.780498i	$-0.714965\pi$
$269$	− 112955.i	− 1.56100i	0.625158	−	0.780498i	$-0.285035\pi$
$270$	0	0
$271$	65086.0	0.886235	0.443118	−	0.896463i	$-0.353872\pi$
$271$	65086.0	0.886235	0.443118	+	0.896463i	$0.353872\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	107882.i	1.40601i	0.711182	+	0.703007i	$0.248160\pi$
$277$	107882.i	1.40601i	−0.711182	+	0.703007i	$0.751840\pi$
$278$	0	0
$279$	187163.	2.40442
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	83521.0	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	167483.i	1.87339i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	343917.	3.74600
$304$	0	0
$305$	0	0
$306$	0	0
$307$	123157.i	1.30672i	0.757048	+	0.653359i	$0.226641\pi$
$307$	123157.i	1.30672i	−0.757048	+	0.653359i	$0.773359\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−185012.	−1.91284	−0.956420	−	0.291996i	$-0.905681\pi$
$311$	−185012.	−1.91284	−0.956420	+	0.291996i	$0.905681\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	153083.i	1.52338i	0.647942	+	0.761690i	$0.275630\pi$
$317$	153083.i	1.52338i	−0.647942	+	0.761690i	$0.724370\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 197877.i	− 1.87339i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	189415.i	1.72886i	0.502756	+	0.864428i	$0.332319\pi$
$331$	189415.i	1.72886i	−0.502756	+	0.864428i	$0.667681\pi$
$332$	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
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 207871.i	− 1.72637i	−0.504888	−	0.863185i	$-0.668466\pi$
$347$	− 207871.i	− 1.72637i	0.504888	−	0.863185i	$-0.331534\pi$
$348$	0	0
$349$	201294.i	1.65264i	0.563198	+	0.826322i	$0.309571\pi$
$349$	201294.i	1.65264i	−0.563198	+	0.826322i	$0.690429\pi$
$350$	0	0
$351$	−923190.	−7.49336
$352$	0	0
$353$	−43457.9	−0.348754	−0.174377	−	0.984679i	$-0.555791\pi$
$353$	−43457.9	−0.348754	−0.174377	+	0.984679i	$0.555791\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−130321.	−1.00000
$362$	0	0
$363$	263537.i	1.99999i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	−798193.	−5.86213
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	297404.	2.09249
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	− 514819.i	− 3.54654i
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	325078.	2.10476
$394$	0	0
$395$	0	0
$396$	0	0
$397$	187802.i	1.19157i	0.803145	+	0.595784i	$0.203158\pi$
$397$	187802.i	1.19157i	−0.803145	+	0.595784i	$0.796842\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	− 243856.i	− 1.50149i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	306892.	1.83459	0.917296	−	0.398207i	$-0.130367\pi$
$409$	306892.	1.83459	0.917296	+	0.398207i	$0.130367\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−116337.	−0.669030
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	−626397.	−3.50081
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	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
$437$	0	0
$438$	0	0
$439$	337501.	1.75124	0.875622	−	0.482998i	$-0.160452\pi$
$439$	337501.	1.75124	0.875622	+	0.482998i	$0.160452\pi$
$440$	0	0
$441$	−583435.	−2.99996
$442$	0	0
$443$	235205.i	1.19850i	0.800561	+	0.599252i	$0.204535\pi$
$443$	235205.i	1.19850i	−0.800561	+	0.599252i	$0.795465\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	73726.0	0.365703	0.182851	−	0.983141i	$-0.441467\pi$
$449$	73726.0	0.365703	0.182851	+	0.983141i	$0.441467\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	− 783585.i	− 3.81847i
$454$	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
$461$	− 89198.8i	− 0.419718i	−0.977732	−	0.209859i	$-0.932700\pi$
$461$	− 89198.8i	− 0.419718i	0.977732	−	0.209859i	$-0.0673005\pi$
$462$	0	0
$463$	419134.	1.95520	0.977599	−	0.210474i	$-0.0675009\pi$
$463$	419134.	1.95520	0.977599	+	0.210474i	$0.0675009\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−403641.	−1.70191	−0.850956	−	0.525237i	$-0.823977\pi$
$487$	−403641.	−1.70191	−0.850956	+	0.525237i	$0.823977\pi$
$488$	0	0
$489$	−465738.	−1.94771
$490$	0	0
$491$	479703.i	1.98980i	0.100861	+	0.994901i	$0.467840\pi$
$491$	479703.i	1.98980i	−0.100861	+	0.994901i	$0.532160\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 431788.i	− 1.73408i	−0.498238	−	0.867040i	$-0.666019\pi$
$499$	− 431788.i	− 1.73408i	0.498238	−	0.867040i	$-0.333981\pi$
$500$	0	0
$501$	50147.7i	0.199791i
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.29017e6i	5.01914i
$508$	0	0
$509$	281870.i	1.08796i	0.839098	+	0.543980i	$0.183083\pi$
$509$	281870.i	1.08796i	−0.839098	+	0.543980i	$0.816917\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	136738.	0.507639
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	279841.	1.00000
$530$	0	0
$531$	− 727192.i	− 2.57905i
$532$	0	0
$533$	1.03997e6i	3.66073i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−839201.	−2.91016
$538$	0	0
$539$	0	0
$540$	0	0
$541$	520798.i	1.77940i	0.456541	+	0.889702i	$0.349088\pi$
$541$	520798.i	1.77940i	−0.456541	+	0.889702i	$0.650912\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 158012.i	− 0.528099i	−0.964509	−	0.264050i	$-0.914942\pi$
$547$	− 158012.i	− 0.528099i	0.964509	−	0.264050i	$-0.0850583\pi$
$548$	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
$557$	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
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−330625.	−1.00000
$576$	0	0
$577$	−419399.	−1.25973	−0.629863	−	0.776706i	$-0.716889\pi$
$577$	−419399.	−1.25973	−0.629863	+	0.776706i	$0.716889\pi$
$578$	0	0
$579$	− 442199.i	− 1.31905i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 69337.8i	− 0.201231i	−0.994925	−	0.100615i	$-0.967919\pi$
$587$	− 69337.8i	− 0.201231i	0.994925	−	0.100615i	$-0.0320811\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	1.10546e6	3.16495
$592$	0	0
$593$	621502.	1.76739	0.883697	−	0.468060i	$-0.155047\pi$
$593$	621502.	1.76739	0.883697	+	0.468060i	$0.155047\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	505634.	1.40923	0.704616	−	0.709589i	$-0.251119\pi$
$599$	505634.	1.40923	0.704616	+	0.709589i	$0.251119\pi$
$600$	0	0
$601$	351933.	0.974343	0.487171	−	0.873306i	$-0.338029\pi$
$601$	351933.	0.974343	0.487171	+	0.873306i	$0.338029\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−587902.	−1.59561	−0.797806	−	0.602914i	$-0.794006\pi$
$607$	−587902.	−1.59561	−0.797806	+	0.602914i	$0.794006\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	816138.i	2.18616i
$612$	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
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	1.54252e6i	3.99990i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	390625.	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
$633$	841146.	2.09925
$634$	0	0
$635$	0	0
$636$	0	0
$637$	760163.i	1.87339i
$638$	0	0
$639$	−668442.	−1.63705
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	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
$647$	645850.	1.54285	0.771424	−	0.636321i	$-0.219545\pi$
$647$	645850.	1.54285	0.771424	+	0.636321i	$0.219545\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 772505.i	− 1.81165i	−0.423650	−	0.905826i	$-0.639251\pi$
$653$	− 772505.i	− 1.81165i	0.423650	−	0.905826i	$-0.360749\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.51096e6	5.81713
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 496921.i	− 1.11696i
$668$	0	0
$669$	− 836384.i	− 1.86876i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	705679.	1.55803	0.779017	−	0.627002i	$-0.215718\pi$
$673$	705679.	1.55803	0.779017	+	0.627002i	$0.215718\pi$
$674$	0	0
$675$	− 1.82245e6i	− 3.99990i
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	169707.i	0.363796i	0.983317	+	0.181898i	$0.0582241\pi$
$683$	169707.i	0.363796i	−0.983317	+	0.181898i	$0.941776\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	701419.i	1.46900i	0.678609	+	0.734500i	$0.262583\pi$
$691$	701419.i	1.46900i	−0.678609	+	0.734500i	$0.737417\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 1.30743e6i	− 2.67586i
$700$	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
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−407450.	−0.801485
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 2.04561e6i	− 3.97911i
$718$	0	0
$719$	−290878.	−0.562669	−0.281335	−	0.959610i	$-0.590777\pi$
$719$	−290878.	−0.562669	−0.281335	+	0.959610i	$0.590777\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	587100.i	1.11696i
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−3.71978e6	−6.99943
$730$	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
$739$	− 113939.i	− 0.208632i	−0.994544	−	0.104316i	$-0.966735\pi$
$739$	− 113939.i	− 0.208632i	0.994544	−	0.104316i	$-0.0332654\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	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
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−33128.9	−0.0572055	−0.0286027	−	0.999591i	$-0.509106\pi$
$761$	−33128.9	−0.0572055	−0.0286027	+	0.999591i	$0.509106\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−947465.	−1.61054
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	2.03178e6i	3.41797i
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	481392.	0.801485
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	2.73910e6	4.46771
$784$	0	0
$785$	0	0
$786$	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
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.03318e6	−3.12198
$808$	0	0
$809$	−1.28765e6	−1.96743	−0.983715	−	0.179733i	$-0.942476\pi$
$809$	−1.28765e6	−1.96743	−0.983715	+	0.179733i	$0.942476\pi$
$810$	0	0
$811$	− 1.19765e6i	− 1.82091i	−0.413605	−	0.910457i	$-0.635730\pi$
$811$	− 1.19765e6i	− 1.82091i	0.413605	−	0.910457i	$-0.364270\pi$
$812$	0	0
$813$	− 1.17154e6i	− 1.77246i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.31974e6i	− 1.95795i	−0.203989	−	0.978973i	$-0.565391\pi$
$821$	− 1.31974e6i	− 1.95795i	0.203989	−	0.978973i	$-0.434609\pi$
$822$	0	0
$823$	1.18309e6	1.74669	0.873347	−	0.487099i	$-0.161945\pi$
$823$	1.18309e6	1.74669	0.873347	+	0.487099i	$0.161945\pi$
$824$	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
$829$	− 190375.i	− 0.277014i	−0.990361	−	0.138507i	$-0.955770\pi$
$829$	− 190375.i	− 0.277014i	0.990361	−	0.138507i	$-0.0442303\pi$
$830$	0	0
$831$	1.94187e6	2.81202
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 2.24592e6i	− 3.20586i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−175115.	−0.247590
$842$	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
$853$	924253.i	1.27026i	0.772405	+	0.635130i	$0.219053\pi$
$853$	924253.i	1.27026i	−0.772405	+	0.635130i	$0.780947\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	354901.	0.483221	0.241610	−	0.970373i	$-0.422324\pi$
$857$	354901.	0.483221	0.241610	+	0.970373i	$0.422324\pi$
$858$	0	0
$859$	− 25862.4i	− 0.0350495i	−0.999846	−	0.0175248i	$-0.994421\pi$
$859$	− 25862.4i	− 0.0350495i	0.999846	−	0.0175248i	$-0.00557859\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.45639e6	−1.95549	−0.977743	−	0.209805i	$-0.932717\pi$
$863$	−1.45639e6	−1.95549	−0.977743	+	0.209805i	$0.932717\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.50337e6i	− 1.99999i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.25528e6i	1.63208i	0.577995	+	0.816040i	$0.303835\pi$
$877$	1.25528e6i	1.63208i	−0.577995	+	0.816040i	$0.696165\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 1.40353e6i	− 1.80011i	−0.435772	−	0.900057i	$-0.643525\pi$
$883$	− 1.40353e6i	− 1.80011i	0.435772	−	0.900057i	$-0.356475\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.57082e6	−1.99655	−0.998274	−	0.0587350i	$-0.981293\pi$
$887$	−1.57082e6	−1.99655	−0.998274	+	0.0587350i	$0.981293\pi$
$888$	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
$897$	3.01468e6	3.74676
$898$	0	0
$899$	723520.i	0.895223i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	− 4.64284e6i	− 5.61896i
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	2.21681e6	2.61342
$922$	0	0
$923$	870920.i	1.02229i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−266089.	−0.308316	−0.154158	−	0.988046i	$-0.549266\pi$
$929$	−266089.	−0.308316	−0.154158	+	0.988046i	$0.549266\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.33019e6i	3.82566i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	1.73765e6	1.95407
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.78848e6i	1.99427i	0.0756540	+	0.997134i	$0.475896\pi$
$947$	1.78848e6i	1.99427i	−0.0756540	+	0.997134i	$0.524104\pi$
$948$	0	0
$949$	− 3.27155e6i	− 3.63263i
$950$	0	0
$951$	2.75548e6	3.04674
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−330272.	−0.357623
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.22322e6	−1.30813	−0.654066	−	0.756438i	$-0.726938\pi$
$967$	−1.22322e6	−1.30813	−0.654066	+	0.756438i	$0.726938\pi$
$968$	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
$975$	−3.56176e6	−3.74676
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−632446.	−0.643986	−0.321993	−	0.946742i	$-0.604353\pi$
$991$	−632446.	−0.643986	−0.321993	+	0.946742i	$0.604353\pi$
$992$	0	0
$993$	3.40946e6	3.45770
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.97212e6i	1.98401i	0.126203	+	0.992004i	$0.459721\pi$
$997$	1.97212e6i	1.98401i	−0.126203	+	0.992004i	$0.540279\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	736.5.e.d.689.1		6
4.3	odd	2		184.5.e.d.45.1	✓	6
8.3	odd	2		184.5.e.d.45.2	yes	6
8.5	even	2	inner	736.5.e.d.689.6		6
23.22	odd	2	CM	736.5.e.d.689.1		6
92.91	even	2		184.5.e.d.45.1	✓	6
184.45	odd	2	inner	736.5.e.d.689.6		6
184.91	even	2		184.5.e.d.45.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.5.e.d.45.1	✓	6	4.3	odd	2
184.5.e.d.45.1	✓	6	92.91	even	2
184.5.e.d.45.2	yes	6	8.3	odd	2
184.5.e.d.45.2	yes	6	184.91	even	2
736.5.e.d.689.1		6	1.1	even	1	trivial
736.5.e.d.689.1		6	23.22	odd	2	CM
736.5.e.d.689.6		6	8.5	even	2	inner
736.5.e.d.689.6		6	184.45	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	736.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-17.9999i q^{3} -242.997 q^{9} +316.603i q^{13} +529.000 q^{23} -625.000 q^{25} +2915.93i q^{27} -939.360i q^{29} -770.227 q^{31} +5698.82 q^{39} +3284.79 q^{41} +2577.80 q^{47} +2401.00 q^{49} +2992.60i q^{59} -9521.95i q^{69} +2750.83 q^{71} -10333.3 q^{73} +11249.9i q^{75} +32803.7 q^{81} -16908.4 q^{87} +13864.0i q^{93} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 486 q^{9} + 3174 q^{23} - 3750 q^{25} + 17076 q^{39} + 14406 q^{49} + 39366 q^{81} - 50484 q^{87}+O(q^{100}) \) 6 * q - 486 * q^9 + 3174 * q^23 - 3750 * q^25 + 17076 * q^39 + 14406 * q^49 + 39366 * q^81 - 50484 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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