LMFDB - Newform orbit 798.2.bc.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [798,2,Mod(31,798)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(798, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 1, 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("798.31"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [28,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$6.37206208130$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 28 q + 28 q^{3} + 14 q^{4} + 6 q^{5} - 8 q^{7} + 28 q^{9} + 2 q^{10} + 14 q^{12} + 2 q^{13} - 8 q^{14} + 6 q^{15} - 14 q^{16} + 12 q^{19} - 8 q^{21} - 12 q^{22} + 24 q^{23} + 14 q^{25} + 24 q^{26} + 28 q^{27}+ \cdots + 28 q^{98}+O(q^{100}) $

28 * q + 28 * q^3 + 14 * q^4 + 6 * q^5 - 8 * q^7 + 28 * q^9 + 2 * q^10 + 14 * q^12 + 2 * q^13 - 8 * q^14 + 6 * q^15 - 14 * q^16 + 12 * q^19 - 8 * q^21 - 12 * q^22 + 24 * q^23 + 14 * q^25 + 24 * q^26 + 28 * q^27 + 2 * q^28 + 18 * q^29 + 2 * q^30 + 4 * q^31 - 4 * q^34 - 10 * q^35 + 14 * q^36 - 24 * q^37 - 6 * q^38 + 2 * q^39 - 2 * q^40 + 4 * q^41 - 8 * q^42 - 14 * q^43 + 6 * q^45 - 12 * q^46 - 14 * q^48 - 24 * q^49 + 4 * q^52 + 24 * q^53 + 18 * q^55 - 4 * q^56 + 12 * q^57 - 2 * q^58 + 44 * q^59 - 12 * q^62 - 8 * q^63 - 28 * q^64 - 24 * q^65 - 12 * q^66 - 18 * q^67 + 6 * q^68 + 24 * q^69 + 6 * q^70 - 6 * q^71 + 12 * q^74 + 14 * q^75 + 40 * q^77 + 24 * q^78 - 24 * q^79 - 6 * q^80 + 28 * q^81 + 2 * q^84 + 28 * q^85 + 18 * q^87 - 12 * q^88 + 16 * q^89 + 2 * q^90 - 20 * q^91 + 12 * q^92 + 4 * q^93 - 18 * q^94 + 26 * q^95 - 40 * q^97 + 28 * q^98

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_{9} $	$ a_{10} $
31.1	−0.866025	−	0.500000i	1.00000	0.500000	+	0.866025i	−2.99811	−	1.73096i	−0.866025	−	0.500000i	2.47235	−	0.942075i	−	1.00000i	1.00000	1.73096	+	2.99811i
31.2	−0.866025	−	0.500000i	1.00000	0.500000	+	0.866025i	−2.02521	−	1.16925i	−0.866025	−	0.500000i	−0.397795	+	2.61568i	−	1.00000i	1.00000	1.16925	+	2.02521i
31.3	−0.866025	−	0.500000i	1.00000	0.500000	+	0.866025i	−0.656557	−	0.379064i	−0.866025	−	0.500000i	−1.59981	−	2.10727i	−	1.00000i	1.00000	0.379064	+	0.656557i
31.4	−0.866025	−	0.500000i	1.00000	0.500000	+	0.866025i	−0.317873	−	0.183524i	−0.866025	−	0.500000i	−1.05442	−	2.42656i	−	1.00000i	1.00000	0.183524	+	0.317873i
31.5	−0.866025	−	0.500000i	1.00000	0.500000	+	0.866025i	0.999902	+	0.577293i	−0.866025	−	0.500000i	2.64324	−	0.115227i	−	1.00000i	1.00000	−0.577293	−	0.999902i
31.6	−0.866025	−	0.500000i	1.00000	0.500000	+	0.866025i	2.10204	+	1.21361i	−0.866025	−	0.500000i	−0.919460	+	2.48085i	−	1.00000i	1.00000	−1.21361	−	2.10204i
31.7	−0.866025	−	0.500000i	1.00000	0.500000	+	0.866025i	3.52979	+	2.03792i	−0.866025	−	0.500000i	−1.41205	−	2.23744i	−	1.00000i	1.00000	−2.03792	−	3.52979i
31.8	0.866025	+	0.500000i	1.00000	0.500000	+	0.866025i	−2.82315	−	1.62995i	0.866025	+	0.500000i	−1.88485	+	1.85670i		1.00000i	1.00000	−1.62995	−	2.82315i
31.9	0.866025	+	0.500000i	1.00000	0.500000	+	0.866025i	−2.35515	−	1.35975i	0.866025	+	0.500000i	−2.09808	−	1.61185i		1.00000i	1.00000	−1.35975	−	2.35515i
31.10	0.866025	+	0.500000i	1.00000	0.500000	+	0.866025i	−0.184175	−	0.106333i	0.866025	+	0.500000i	0.768966	−	2.53154i		1.00000i	1.00000	−0.106333	−	0.184175i
31.11	0.866025	+	0.500000i	1.00000	0.500000	+	0.866025i	0.479759	+	0.276989i	0.866025	+	0.500000i	2.21615	+	1.44523i		1.00000i	1.00000	0.276989	+	0.479759i
31.12	0.866025	+	0.500000i	1.00000	0.500000	+	0.866025i	2.04913	+	1.18307i	0.866025	+	0.500000i	1.13558	−	2.38966i		1.00000i	1.00000	1.18307	+	2.04913i
31.13	0.866025	+	0.500000i	1.00000	0.500000	+	0.866025i	2.56801	+	1.48264i	0.866025	+	0.500000i	−2.64115	+	0.155916i		1.00000i	1.00000	1.48264	+	2.56801i
31.14	0.866025	+	0.500000i	1.00000	0.500000	+	0.866025i	2.63161	+	1.51936i	0.866025	+	0.500000i	−1.22866	+	2.34316i		1.00000i	1.00000	1.51936	+	2.63161i
103.1	−0.866025	+	0.500000i	1.00000	0.500000	−	0.866025i	−2.99811	+	1.73096i	−0.866025	+	0.500000i	2.47235	+	0.942075i		1.00000i	1.00000	1.73096	−	2.99811i
103.2	−0.866025	+	0.500000i	1.00000	0.500000	−	0.866025i	−2.02521	+	1.16925i	−0.866025	+	0.500000i	−0.397795	−	2.61568i		1.00000i	1.00000	1.16925	−	2.02521i
103.3	−0.866025	+	0.500000i	1.00000	0.500000	−	0.866025i	−0.656557	+	0.379064i	−0.866025	+	0.500000i	−1.59981	+	2.10727i		1.00000i	1.00000	0.379064	−	0.656557i
103.4	−0.866025	+	0.500000i	1.00000	0.500000	−	0.866025i	−0.317873	+	0.183524i	−0.866025	+	0.500000i	−1.05442	+	2.42656i		1.00000i	1.00000	0.183524	−	0.317873i
103.5	−0.866025	+	0.500000i	1.00000	0.500000	−	0.866025i	0.999902	−	0.577293i	−0.866025	+	0.500000i	2.64324	+	0.115227i		1.00000i	1.00000	−0.577293	+	0.999902i
103.6	−0.866025	+	0.500000i	1.00000	0.500000	−	0.866025i	2.10204	−	1.21361i	−0.866025	+	0.500000i	−0.919460	−	2.48085i		1.00000i	1.00000	−1.21361	+	2.10204i

See all 28 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
133.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	798.2.bc.b	yes	28
7.d	odd	6	1		798.2.m.b	✓	28
19.d	odd	6	1		798.2.m.b	✓	28
133.s	even	6	1	inner	798.2.bc.b	yes	28

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
798.2.m.b	✓	28	7.d	odd	6	1
798.2.m.b	✓	28	19.d	odd	6	1
798.2.bc.b	yes	28	1.a	even	1	1	trivial
798.2.bc.b	yes	28	133.s	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{28} - 6 T_{5}^{27} - 24 T_{5}^{26} + 216 T_{5}^{25} + 392 T_{5}^{24} - 4878 T_{5}^{23} + \cdots + 328329 $ T5^28 - 6*T5^27 - 24*T5^26 + 216*T5^25 + 392*T5^24 - 4878*T5^23 - 1888*T5^22 + 68346*T5^21 - 17519*T5^20 - 701838*T5^19 + 587168*T5^18 + 4941138*T5^17 - 6069588*T5^16 - 25481544*T5^15 + 42625485*T5^14 + 81083076*T5^13 - 172490515*T5^12 - 163668462*T5^11 + 482169200*T5^10 + 6278544*T5^9 - 432402066*T5^8 + 12970998*T5^7 + 276955962*T5^6 + 8823882*T5^5 - 61287680*T5^4 - 6016560*T5^3 + 10131612*T5^2 + 3252348*T5 + 328329 acting on $S_{2}^{\mathrm{new}}(798, [\chi])$.

Overview	Random
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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.bc (of order \(6\), degree \(2\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 31.14
Significant digits:
Format:

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