LMFDB - Newform orbit 816.2.u.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [816,2,Mod(205,816)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(816, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 3, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("816.205"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$816 = 2^{4} \cdot 3 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	816.u (of order $4$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [64,0,0,4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)

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

Self dual:	no
Analytic conductor:	$6.51579280494$
Analytic rank:	$0$
Dimension:	$64$
Relative dimension:	$32$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -expansion

The algebraic $q$ -expansion of this newform has not been computed, but we have computed the trace expansion.

\operatorname{Tr}(f)(q) =

64 q + 4 q^{4} + 12 q^{10} + 8 q^{11} - 8 q^{12} + 20 q^{14} + 24 q^{15} + 28 q^{16} - 64 q^{17} - 4 q^{18} - 48 q^{22} - 20 q^{26} + 16 q^{29} + 8 q^{30} - 24 q^{31} + 40 q^{32} - 24 q^{35} + 16 q^{37}+ \cdots + 8 q^{99}+O(q^{100})

64 * q + 4 * q^4 + 12 * q^10 + 8 * q^11 - 8 * q^12 + 20 * q^14 + 24 * q^15 + 28 * q^16 - 64 * q^17 - 4 * q^18 - 48 * q^22 - 20 * q^26 + 16 * q^29 + 8 * q^30 - 24 * q^31 + 40 * q^32 - 24 * q^35 + 16 * q^37 - 12 * q^38 - 56 * q^40 + 8 * q^42 + 8 * q^43 + 24 * q^44 + 52 * q^46 + 64 * q^47 - 64 * q^49 + 68 * q^50 + 44 * q^52 - 16 * q^53 - 56 * q^56 + 4 * q^58 + 4 * q^60 + 76 * q^62 - 8 * q^63 + 52 * q^64 + 8 * q^66 + 16 * q^67 - 4 * q^68 - 136 * q^70 + 4 * q^72 - 84 * q^74 - 16 * q^75 - 20 * q^76 + 32 * q^77 + 24 * q^78 + 40 * q^79 + 56 * q^80 - 64 * q^81 + 64 * q^83 - 52 * q^84 - 44 * q^86 - 96 * q^88 - 20 * q^90 - 56 * q^91 + 32 * q^92 + 88 * q^94 - 48 * q^95 + 72 * q^98 + 8 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$					$a_{8}$				$a_{10}$
205.1	−1.41228	+	0.0739715i	−0.707107	−	0.707107i	1.98906	−	0.208937i	−2.01039	+	2.01039i	1.05094	+	0.946325i	−	3.19495i	−2.79364	+	0.442210i	1.00000i	2.69052	−	2.98794i
205.2	−1.40827	+	0.129521i	0.707107	+	0.707107i	1.96645	−	0.364802i	−0.644405	+	0.644405i	−1.08738	−	0.904212i	−	1.28446i	−2.72204	+	0.768437i	1.00000i	0.824032	−	0.990961i
205.3	−1.38749	−	0.273605i	−0.707107	−	0.707107i	1.85028	+	0.759252i	1.87129	−	1.87129i	0.787638	+	1.17457i	−	4.49580i	−2.35952	−	1.55970i	1.00000i	−3.10841	+	2.08441i
205.4	−1.32847	+	0.484949i	0.707107	+	0.707107i	1.52965	−	1.28848i	−0.0879778	+	0.0879778i	−1.28228	−	0.596458i		3.84329i	−1.40724	+	2.45350i	1.00000i	0.0742109	−	0.159540i
205.5	−1.29863	+	0.559954i	−0.707107	−	0.707107i	1.37290	−	1.45435i	0.593934	−	0.593934i	1.31422	+	0.522326i		1.08628i	−0.968528	+	2.65743i	1.00000i	−0.438728	+	1.10388i
205.6	−1.29298	−	0.572901i	0.707107	+	0.707107i	1.34357	+	1.48149i	2.98172	−	2.98172i	−0.509169	−	1.31937i		1.17350i	−0.888453	−	2.68527i	1.00000i	−5.56351	+	2.14706i
205.7	−1.16815	−	0.797139i	−0.707107	−	0.707107i	0.729140	+	1.86235i	−0.639271	+	0.639271i	0.262343	+	1.38967i		2.85654i	0.632809	−	2.75673i	1.00000i	1.25635	−	0.237175i
205.8	−1.13900	−	0.838255i	0.707107	+	0.707107i	0.594656	+	1.90955i	−0.565655	+	0.565655i	−0.212661	−	1.39813i	−	1.93055i	0.923376	−	2.67346i	1.00000i	1.11845	−	0.170119i
205.9	−0.860813	+	1.12205i	0.707107	+	0.707107i	−0.518001	−	1.93175i	0.163974	−	0.163974i	−1.40210	+	0.184723i	−	4.22726i	2.61343	+	1.08166i	1.00000i	0.0428363	+	0.325138i
205.10	−0.850495	+	1.12989i	−0.707107	−	0.707107i	−0.553316	−	1.92194i	2.29734	−	2.29734i	1.40035	−	0.197564i		4.90695i	2.64218	+	1.00941i	1.00000i	0.641871	+	4.54962i
205.11	−0.780809	+	1.17913i	−0.707107	−	0.707107i	−0.780674	−	1.84134i	−2.45315	+	2.45315i	1.38588	−	0.281652i	−	3.71241i	2.78073	+	0.517226i	1.00000i	−0.977128	−	4.80801i
205.12	−0.606058	−	1.27777i	0.707107	+	0.707107i	−1.26539	+	1.54880i	−2.11034	+	2.11034i	0.474971	−	1.33207i		1.09709i	2.74591	+	0.678206i	1.00000i	3.97552	+	1.41754i
205.13	−0.596675	−	1.28218i	−0.707107	−	0.707107i	−1.28796	+	1.53009i	0.102668	−	0.102668i	−0.484724	+	1.32855i	−	0.792539i	2.73033	+	0.738428i	1.00000i	−0.192897	−	0.0703790i
205.14	−0.359024	+	1.36788i	0.707107	+	0.707107i	−1.74220	−	0.982204i	2.64915	−	2.64915i	−1.22111	+	0.713371i		1.00268i	1.96903	−	2.03050i	1.00000i	2.67262	+	4.57484i
205.15	0.0155033	+	1.41413i	0.707107	+	0.707107i	−1.99952	+	0.0438473i	−0.192924	+	0.192924i	−0.988977	+	1.01090i	−	4.31863i	−0.0930049	−	2.82690i	1.00000i	−0.275810	−	0.269828i
205.16	0.0612118	−	1.41289i	0.707107	+	0.707107i	−1.99251	−	0.172971i	−0.807173	+	0.807173i	1.04235	−	0.955780i		0.0213332i	−0.366353	+	2.80460i	1.00000i	1.09104	+	1.18985i
205.17	0.0694285	−	1.41251i	−0.707107	−	0.707107i	−1.99036	−	0.196137i	−2.79966	+	2.79966i	−1.04789	+	0.949701i		3.41523i	−0.415232	+	2.79778i	1.00000i	3.76017	+	4.14892i
205.18	0.158748	+	1.40528i	−0.707107	−	0.707107i	−1.94960	+	0.446169i	−2.32351	+	2.32351i	0.881428	−	1.10593i		2.75370i	−0.936484	−	2.66889i	1.00000i	−3.63402	−	2.89632i
205.19	0.326049	+	1.37611i	−0.707107	−	0.707107i	−1.78738	+	0.897362i	2.69913	−	2.69913i	0.742509	−	1.20361i	−	1.55925i	−1.81765	−	2.16706i	1.00000i	4.59437	+	2.83427i
205.20	0.487868	−	1.32740i	0.707107	+	0.707107i	−1.52397	−	1.29519i	0.905261	−	0.905261i	1.28359	−	0.593637i		5.09628i	−2.46273	+	1.39103i	1.00000i	−0.759993	−	1.64329i

See all 64 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	816.2.u.a	✓	64
16.e	even	4	1	inner	816.2.u.a	✓	64

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
816.2.u.a	✓	64	1.a	even	1	1	trivial
816.2.u.a	✓	64	16.e	even	4	1	inner

This newform subspace can be constructed as the kernel of the linear operator $T_{5}^{64} + 16 T_{5}^{61} + 1056 T_{5}^{60} + 336 T_{5}^{59} + 128 T_{5}^{58} + 11536 T_{5}^{57} + \cdots + 982540877824$ T5^64 + 16*T5^61 + 1056*T5^60 + 336*T5^59 + 128*T5^58 + 11536*T5^57 + 467948*T5^56 + 275600*T5^55 + 105856*T5^54 + 2959504*T5^53 + 112577320*T5^52 + 89381840*T5^51 + 34812032*T5^50 + 273668944*T5^49 + 15923581094*T5^48 + 14590382160*T5^47 + 5823013248*T5^46 - 9343354512*T5^45 + 1352977021816*T5^44 + 1265742827440*T5^43 + 528360832384*T5^42 - 3520343514320*T5^41 + 69223934086876*T5^40 + 57316699001520*T5^39 + 26120107007104*T5^38 - 245164906466768*T5^37 + 2152361995757272*T5^36 + 1264771079703280*T5^35 + 693599629977984*T5^34 - 7450575292376464*T5^33 + 40842053369958689*T5^32 + 10483196395953776*T5^31 + 9492065390720128*T5^30 - 105177483237417024*T5^29 + 451042188479641320*T5^28 - 30058201046373312*T5^27 + 60566539291731968*T5^26 - 612457470867843584*T5^25 + 2549348309105768688*T5^24 - 698920823713825280*T5^23 + 157025385001365504*T5^22 - 1029980651682480128*T5^21 + 6203327028263088896*T5^20 - 1860762097433446400*T5^19 + 183821843036045312*T5^18 - 214831515175997440*T5^17 + 7068859786989219584*T5^16 - 1586230807798403072*T5^15 + 97329953047871488*T5^14 + 498251797527379968*T5^13 + 3822258329925445632*T5^12 - 377773532468133888*T5^11 + 18495571830505472*T5^10 + 208320585835479040*T5^9 + 776652446983786496*T5^8 + 18934518942007296*T5^7 - 190384240066560*T5^6 - 1402804757856256*T5^5 + 3343729163239424*T5^4 - 75285164720128*T5^3 + 491270438912*T5^2 + 982540877824*T5 + 982540877824 acting on $S_{2}^{\mathrm{new}}(816, [\chi])$ .

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

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 205.32
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more