LMFDB - Newform orbit 798.2.bx.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [798,2,Mod(13,798)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(798, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("798.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 798 = 2 \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	798.bx (of order $18$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.37206208130$
Analytic rank:	$0$
Dimension:	$84$
Relative dimension:	$14$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $q$-expansion, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) = $ $ 84 q - 6 q^{7}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 84 q - 6 q^{7} - 6 q^{10} - 12 q^{11} + 42 q^{12} + 12 q^{13} + 18 q^{14} + 12 q^{19} + 12 q^{21} - 6 q^{22} + 12 q^{23} + 6 q^{25} - 42 q^{27} + 12 q^{28} - 36 q^{29} - 6 q^{31} - 12 q^{33} + 12 q^{34} - 6 q^{35} - 12 q^{38} + 6 q^{40} + 36 q^{41} - 6 q^{42} - 60 q^{43} + 12 q^{44} - 18 q^{46} - 36 q^{47} + 12 q^{49} + 24 q^{52} - 12 q^{53} - 78 q^{55} + 6 q^{57} - 84 q^{59} + 12 q^{62} + 6 q^{63} + 42 q^{64} + 36 q^{65} + 24 q^{66} - 48 q^{67} + 12 q^{69} + 18 q^{70} + 24 q^{71} + 48 q^{73} - 132 q^{75} - 60 q^{77} + 24 q^{78} + 48 q^{79} + 6 q^{82} - 36 q^{83} + 6 q^{84} + 66 q^{85} + 24 q^{86} - 48 q^{89} - 6 q^{90} - 30 q^{91} + 24 q^{92} - 36 q^{93} + 24 q^{95} + 24 q^{97} - 60 q^{98} - 12 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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_{7} $			$ a_{8} $			$ a_{9} $			$ a_{10} $
13.1	−0.642788	−	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	−3.30663	+	0.583048i	0.342020	+	0.939693i	−0.922948	−	2.47955i	0.866025	−	0.500000i	0.766044	+	0.642788i	2.57210	+	2.15825i
13.2	−0.642788	−	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	−1.97053	+	0.347457i	0.342020	+	0.939693i	−2.62814	+	0.304744i	0.866025	−	0.500000i	0.766044	+	0.642788i	1.53280	+	1.28617i
13.3	−0.642788	−	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	−1.08654	+	0.191586i	0.342020	+	0.939693i	2.31574	+	1.27959i	0.866025	−	0.500000i	0.766044	+	0.642788i	0.845177	+	0.709187i
13.4	−0.642788	−	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	−0.181551	+	0.0320123i	0.342020	+	0.939693i	−0.838228	+	2.50946i	0.866025	−	0.500000i	0.766044	+	0.642788i	0.141222	+	0.118499i
13.5	−0.642788	−	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	2.14904	−	0.378934i	0.342020	+	0.939693i	0.736406	−	2.54120i	0.866025	−	0.500000i	0.766044	+	0.642788i	−1.67166	−	1.40269i
13.6	−0.642788	−	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	2.42151	−	0.426977i	0.342020	+	0.939693i	2.64368	+	0.104630i	0.866025	−	0.500000i	0.766044	+	0.642788i	−1.88360	−	1.58053i
13.7	−0.642788	−	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	3.82553	−	0.674544i	0.342020	+	0.939693i	−1.71181	+	2.01735i	0.866025	−	0.500000i	0.766044	+	0.642788i	−2.97573	−	2.49694i
13.8	0.642788	+	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	−3.61793	+	0.637938i	−0.342020	−	0.939693i	2.61743	−	0.386052i	−0.866025	+	0.500000i	0.766044	+	0.642788i	−2.81425	−	2.36143i
13.9	0.642788	+	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	−3.50454	+	0.617946i	−0.342020	−	0.939693i	−2.63245	+	0.264946i	−0.866025	+	0.500000i	0.766044	+	0.642788i	−2.72605	−	2.28743i
13.10	0.642788	+	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	−1.40491	+	0.247723i	−0.342020	−	0.939693i	−0.0972585	−	2.64396i	−0.866025	+	0.500000i	0.766044	+	0.642788i	−1.09282	−	0.916987i
13.11	0.642788	+	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	−0.129519	+	0.0228377i	−0.342020	−	0.939693i	−0.417096	+	2.61267i	−0.866025	+	0.500000i	0.766044	+	0.642788i	−0.100748	−	0.0845375i
13.12	0.642788	+	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	0.851173	−	0.150085i	−0.342020	−	0.939693i	−1.81371	−	1.92625i	−0.866025	+	0.500000i	0.766044	+	0.642788i	0.662095	+	0.555563i
13.13	0.642788	+	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	2.41276	−	0.425435i	−0.342020	−	0.939693i	2.16005	+	1.52780i	−0.866025	+	0.500000i	0.766044	+	0.642788i	1.87680	+	1.57482i
13.14	0.642788	+	0.766044i	−0.939693	−	0.342020i	−0.173648	+	0.984808i	3.54213	−	0.624572i	−0.342020	−	0.939693i	−2.63836	−	0.197689i	−0.866025	+	0.500000i	0.766044	+	0.642788i	2.75528	+	2.31196i
97.1	−0.984808	−	0.173648i	0.766044	+	0.642788i	0.939693	+	0.342020i	−1.32194	−	3.63201i	−0.642788	−	0.766044i	−1.91753	−	1.82293i	−0.866025	−	0.500000i	0.173648	+	0.984808i	0.671168	+	3.80638i
97.2	−0.984808	−	0.173648i	0.766044	+	0.642788i	0.939693	+	0.342020i	−1.27407	−	3.50048i	−0.642788	−	0.766044i	2.30967	−	1.29051i	−0.866025	−	0.500000i	0.173648	+	0.984808i	0.646862	+	3.66854i
97.3	−0.984808	−	0.173648i	0.766044	+	0.642788i	0.939693	+	0.342020i	−0.355755	−	0.977429i	−0.642788	−	0.766044i	−2.58360	−	0.570091i	−0.866025	−	0.500000i	0.173648	+	0.984808i	0.180621	+	1.02436i
97.4	−0.984808	−	0.173648i	0.766044	+	0.642788i	0.939693	+	0.342020i	0.226334	+	0.621848i	−0.642788	−	0.766044i	−1.98402	+	1.75033i	−0.866025	−	0.500000i	0.173648	+	0.984808i	−0.114913	−	0.651703i
97.5	−0.984808	−	0.173648i	0.766044	+	0.642788i	0.939693	+	0.342020i	0.329403	+	0.905028i	−0.642788	−	0.766044i	2.27462	−	1.35134i	−0.866025	−	0.500000i	0.173648	+	0.984808i	−0.167242	−	0.948479i
97.6	−0.984808	−	0.173648i	0.766044	+	0.642788i	0.939693	+	0.342020i	0.720513	+	1.97959i	−0.642788	−	0.766044i	0.0696210	−	2.64484i	−0.866025	−	0.500000i	0.173648	+	0.984808i	−0.365814	−	2.07463i

See all 84 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
133.ba	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	798.2.bx.a	✓	84
7.b	odd	2	1		798.2.bx.b	yes	84
19.f	odd	18	1		798.2.bx.b	yes	84
133.ba	even	18	1	inner	798.2.bx.a	✓	84

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
798.2.bx.a	✓	84	1.a	even	1	1	trivial
798.2.bx.a	✓	84	133.ba	even	18	1	inner
798.2.bx.b	yes	84	7.b	odd	2	1
798.2.bx.b	yes	84	19.f	odd	18	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{84} - 3 T_{5}^{82} + 150 T_{5}^{80} + 564 T_{5}^{79} - 7842 T_{5}^{78} - 12168 T_{5}^{77} + \cdots + 21\!\cdots\!21 $ T5^84 - 3*T5^82 + 150*T5^80 + 564*T5^79 - 7842*T5^78 - 12168*T5^77 + 9969*T5^76 + 116028*T5^75 - 524019*T5^74 - 3692496*T5^73 + 32835200*T5^72 + 64695804*T5^71 - 16092135*T5^70 - 373917408*T5^69 + 1545469209*T5^68 + 13609779720*T5^67 - 82430215927*T5^66 - 251736353916*T5^65 + 22358119461*T5^64 + 1693887080796*T5^63 - 32006315475*T5^62 - 24442726746672*T5^61 + 109859313206515*T5^60 + 356760583433004*T5^59 - 98524718023002*T5^58 - 2361201770479656*T5^57 + 480494527562082*T5^56 + 34880862220716276*T5^55 - 80781787158887647*T5^54 - 381096971742070068*T5^53 + 126331881224993580*T5^52 + 2375208341285171868*T5^51 + 138439386970951512*T5^50 - 17262843959892618504*T5^49 + 32367774477590815206*T5^48 + 93654368872430999076*T5^47 - 9463090063667964297*T5^46 - 401732782848572772648*T5^45 + 46303310341079504556*T5^44 + 150553488637577449464*T5^43 - 6474489380480706887115*T5^42 - 4233952507506406575120*T5^41 + 13129763861869796647329*T5^40 + 42497148581499139677696*T5^39 + 55177149949798412155638*T5^38 + 37925104074198544358688*T5^37 + 181489580956422781777523*T5^36 - 141546776562970274498220*T5^35 - 863168721517460849753601*T5^34 - 1993664071287163814415552*T5^33 - 2011696523633728221603006*T5^32 - 1889690526078167183102088*T5^31 - 4030850857892170579327480*T5^30 + 4583305737085285612395192*T5^29 + 23258961656897012136212610*T5^28 + 50517974224413628792087236*T5^27 + 85229923706231821709848872*T5^26 + 121199556135992102317887360*T5^25 + 155246289341335455722748186*T5^24 + 134306915236729820471730012*T5^23 + 107424037386072509708472375*T5^22 + 49218605296399227614900364*T5^21 - 19487943384337445958092625*T5^20 - 46230613085906130927032268*T5^19 - 91565800319325140104443501*T5^18 - 61609122368025607781094084*T5^17 - 32563472304031396275642537*T5^16 - 18906004682219904553400124*T5^15 + 24355323854643308107823490*T5^14 + 20325634021377624025609536*T5^13 + 31511710933240592327642038*T5^12 + 18105862254073763094786564*T5^11 + 14703509270688312483189225*T5^10 + 7274959936596257620754772*T5^9 + 4013845023334850859315645*T5^8 + 1550045488066335030364416*T5^7 + 645696810976649891039019*T5^6 + 284517547004058174511092*T5^5 + 81889224002698933010052*T5^4 + 13917487832100838230012*T5^3 + 1410071008288720806693*T5^2 + 82066090635529725276*T5 + 2187631023088649121 acting on $S_{2}^{\mathrm{new}}(798, [\chi])$.

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	798.bx (of order \(18\), degree \(6\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.14
Significant digits:
Format:

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