LMFDB - Newform orbit 798.2.cj.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [798,2,Mod(241,798)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(798, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 3, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("798.241"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$6.37206208130$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 72 q + 12 q^{7} + 12 q^{10} - 12 q^{11} - 72 q^{12} - 30 q^{13} - 18 q^{14} + 36 q^{19} - 6 q^{21} + 12 q^{22} - 6 q^{23} - 30 q^{25} - 36 q^{27} - 12 q^{28} - 6 q^{31} - 12 q^{33} - 6 q^{34} - 6 q^{35}+ \cdots + 6 q^{99}+O(q^{100}) $

72 * q + 12 * q^7 + 12 * q^10 - 12 * q^11 - 72 * q^12 - 30 * q^13 - 18 * q^14 + 36 * q^19 - 6 * q^21 + 12 * q^22 - 6 * q^23 - 30 * q^25 - 36 * q^27 - 12 * q^28 - 6 * q^31 - 12 * q^33 - 6 * q^34 - 6 * q^35 - 18 * q^37 - 12 * q^38 - 12 * q^40 - 6 * q^42 + 6 * q^43 + 12 * q^44 - 18 * q^45 - 18 * q^47 - 24 * q^49 - 18 * q^51 + 30 * q^52 - 48 * q^53 + 30 * q^55 - 18 * q^56 + 6 * q^57 - 48 * q^59 + 6 * q^61 + 12 * q^62 + 12 * q^63 + 36 * q^64 - 72 * q^65 - 12 * q^66 + 84 * q^67 - 6 * q^69 + 54 * q^70 + 6 * q^71 + 60 * q^73 - 18 * q^74 + 6 * q^75 + 18 * q^76 + 48 * q^77 - 12 * q^78 + 60 * q^79 + 6 * q^82 - 18 * q^83 - 12 * q^84 + 84 * q^85 - 12 * q^86 + 24 * q^89 - 6 * q^90 + 24 * q^91 + 6 * q^92 - 42 * q^93 + 18 * q^94 + 60 * q^95 - 30 * q^97 - 24 * q^98 + 6 * 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_{7} $			$ a_{8} $			$ a_{9} $			$ a_{10} $
241.1	−0.342020	+	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	−2.44248	−	2.91084i	0.342020	+	0.939693i	0.986551	+	2.45494i	0.866025	−	0.500000i	0.173648	−	0.984808i	3.57067	−	1.29962i
241.2	−0.342020	+	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	−2.32511	−	2.77095i	0.342020	+	0.939693i	0.361533	−	2.62093i	0.866025	−	0.500000i	0.173648	−	0.984808i	3.39908	−	1.23716i
241.3	−0.342020	+	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.407192	−	0.485273i	0.342020	+	0.939693i	2.62355	+	0.342002i	0.866025	−	0.500000i	0.173648	−	0.984808i	0.595276	−	0.216663i
241.4	−0.342020	+	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	0.488281	+	0.581911i	0.342020	+	0.939693i	−1.91382	+	1.82683i	0.866025	−	0.500000i	0.173648	−	0.984808i	−0.713819	+	0.259809i
241.5	−0.342020	+	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	0.613531	+	0.731177i	0.342020	+	0.939693i	−1.23088	−	2.34200i	0.866025	−	0.500000i	0.173648	−	0.984808i	−0.896922	+	0.326453i
241.6	−0.342020	+	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	1.75158	+	2.08746i	0.342020	+	0.939693i	0.836062	+	2.51018i	0.866025	−	0.500000i	0.173648	−	0.984808i	−2.56064	+	0.931998i
241.7	0.342020	−	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	−1.56574	−	1.86598i	−0.342020	−	0.939693i	0.777597	−	2.52890i	−0.866025	+	0.500000i	0.173648	−	0.984808i	−2.28896	+	0.833113i
241.8	0.342020	−	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.922719	−	1.09965i	−0.342020	−	0.939693i	−2.61280	+	0.416244i	−0.866025	+	0.500000i	0.173648	−	0.984808i	−1.34892	+	0.490968i
241.9	0.342020	−	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.636249	−	0.758252i	−0.342020	−	0.939693i	2.63863	−	0.194020i	−0.866025	+	0.500000i	0.173648	−	0.984808i	−0.930134	+	0.338541i
241.10	0.342020	−	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	0.0726343	+	0.0865622i	−0.342020	−	0.939693i	0.514120	−	2.59532i	−0.866025	+	0.500000i	0.173648	−	0.984808i	0.106184	−	0.0386479i
241.11	0.342020	−	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	0.852222	+	1.01564i	−0.342020	−	0.939693i	0.520870	+	2.59397i	−0.866025	+	0.500000i	0.173648	−	0.984808i	1.24586	−	0.453458i
241.12	0.342020	−	0.939693i	0.766044	−	0.642788i	−0.766044	−	0.642788i	2.29456	+	2.73455i	−0.342020	−	0.939693i	2.60465	−	0.464526i	−0.866025	+	0.500000i	0.173648	−	0.984808i	3.35442	−	1.22091i
409.1	−0.642788	−	0.766044i	0.173648	+	0.984808i	−0.173648	+	0.984808i	−3.13360	+	0.552538i	0.642788	−	0.766044i	1.71656	+	2.01331i	0.866025	−	0.500000i	−0.939693	+	0.342020i	2.43751	+	2.04531i
409.2	−0.642788	−	0.766044i	0.173648	+	0.984808i	−0.173648	+	0.984808i	−1.73735	+	0.306342i	0.642788	−	0.766044i	−0.975065	+	2.45952i	0.866025	−	0.500000i	−0.939693	+	0.342020i	1.35142	+	1.13397i
409.3	−0.642788	−	0.766044i	0.173648	+	0.984808i	−0.173648	+	0.984808i	−1.36826	+	0.241262i	0.642788	−	0.766044i	−1.08530	−	2.41291i	0.866025	−	0.500000i	−0.939693	+	0.342020i	1.06432	+	0.893070i
409.4	−0.642788	−	0.766044i	0.173648	+	0.984808i	−0.173648	+	0.984808i	1.29426	−	0.228214i	0.642788	−	0.766044i	1.70106	−	2.02642i	0.866025	−	0.500000i	−0.939693	+	0.342020i	−1.00676	−	0.844771i
409.5	−0.642788	−	0.766044i	0.173648	+	0.984808i	−0.173648	+	0.984808i	2.30738	−	0.406854i	0.642788	−	0.766044i	−2.44133	−	1.01977i	0.866025	−	0.500000i	−0.939693	+	0.342020i	−1.79483	−	1.50604i
409.6	−0.642788	−	0.766044i	0.173648	+	0.984808i	−0.173648	+	0.984808i	2.83448	−	0.499796i	0.642788	−	0.766044i	−1.47698	+	2.19512i	0.866025	−	0.500000i	−0.939693	+	0.342020i	−2.20484	−	1.85008i
409.7	0.642788	+	0.766044i	0.173648	+	0.984808i	−0.173648	+	0.984808i	−2.20172	+	0.388223i	−0.642788	+	0.766044i	−1.18056	−	2.36776i	−0.866025	+	0.500000i	−0.939693	+	0.342020i	−1.71263	−	1.43707i
409.8	0.642788	+	0.766044i	0.173648	+	0.984808i	−0.173648	+	0.984808i	−1.40284	+	0.247359i	−0.642788	+	0.766044i	0.846253	+	2.50676i	−0.866025	+	0.500000i	−0.939693	+	0.342020i	−1.09122	−	0.915640i

See all 72 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
133.bf	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.cj.a	yes	72
7.d	odd	6	1		798.2.ca.a	✓	72
19.f	odd	18	1		798.2.ca.a	✓	72
133.bf	even	18	1	inner	798.2.cj.a	yes	72

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
798.2.ca.a	✓	72	7.d	odd	6	1
798.2.ca.a	✓	72	19.f	odd	18	1
798.2.cj.a	yes	72	1.a	even	1	1	trivial
798.2.cj.a	yes	72	133.bf	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{72} + 15 T_{5}^{70} - 90 T_{5}^{69} + 213 T_{5}^{68} - 876 T_{5}^{67} + 2715 T_{5}^{66} + \cdots + 6080412529 $ T5^72 + 15*T5^70 - 90*T5^69 + 213*T5^68 - 876*T5^67 + 2715*T5^66 - 14058*T5^65 - 1902*T5^64 + 69642*T5^63 - 92910*T5^62 + 1410792*T5^61 - 4159264*T5^60 + 19399686*T5^59 - 15143103*T5^58 - 94751682*T5^57 + 82413729*T5^56 - 1528713300*T5^55 + 6828243428*T5^54 - 21134089554*T5^53 + 35230251183*T5^52 + 18093956274*T5^51 - 12934185597*T5^50 + 438195502404*T5^49 - 2357297731427*T5^48 + 5775331796904*T5^47 - 5302504822623*T5^46 - 22778362320834*T5^45 + 8354136219132*T5^44 - 6918013225620*T5^43 + 431031170021534*T5^42 - 903466551320286*T5^41 + 768570938591610*T5^40 + 4613721351177780*T5^39 - 6003711886937478*T5^38 - 13403451775072302*T5^37 - 47992704792574515*T5^36 + 132626074175695386*T5^35 + 156881497761975900*T5^34 - 808008420665845254*T5^33 + 753271603618809984*T5^32 + 2806009244450561508*T5^31 - 4794259945646280024*T5^30 - 7665209111709593208*T5^29 + 13204570604086268007*T5^28 + 17338634427704711790*T5^27 - 30499037473186500009*T5^26 - 27997204848087003414*T5^25 + 58669276589126009453*T5^24 + 31631519345617440180*T5^23 - 70478558811101174427*T5^22 - 29665401352388141076*T5^21 + 55212193558403441214*T5^20 + 24784067924442923196*T5^19 - 48943449420327602140*T5^18 - 13137289219635020214*T5^17 + 30072596089680545871*T5^16 + 5510676305274197586*T5^15 + 4379324757804780333*T5^14 - 4495262159427677166*T5^13 + 8114190965715076779*T5^12 + 1708915754229724392*T5^11 + 2629921911730719627*T5^10 - 339059139043796946*T5^9 + 150012448072853376*T5^8 - 70692767379911904*T5^7 + 12635220826407768*T5^6 - 2498118637592094*T5^5 + 513120729052959*T5^4 - 67014450085512*T5^3 + 5422799662842*T5^2 - 192528955896*T5 + 6080412529 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 $\F_{q}$
Belyi maps

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

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 241.12
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more