LMFDB - Newform orbit 546.2.by.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,2,Mod(19,546)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("546.19"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 10, 5])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	546.by (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [32] 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:	$4.35983195036$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$8$ over $\Q(\zeta_{12})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

$\operatorname{Tr}(f)(q) = $ $ 32 q + 8 q^{7} - 32 q^{9} - 4 q^{11} - 16 q^{12} + 4 q^{14} + 16 q^{16} - 8 q^{17} - 12 q^{19} - 8 q^{21} - 4 q^{22} - 24 q^{23} - 24 q^{25} + 8 q^{26} - 4 q^{28} - 12 q^{29} - 28 q^{31} + 4 q^{33} - 8 q^{34}+ \cdots + 4 q^{99}+O(q^{100}) $

32 * q + 8 * q^7 - 32 * q^9 - 4 * q^11 - 16 * q^12 + 4 * q^14 + 16 * q^16 - 8 * q^17 - 12 * q^19 - 8 * q^21 - 4 * q^22 - 24 * q^23 - 24 * q^25 + 8 * q^26 - 4 * q^28 - 12 * q^29 - 28 * q^31 + 4 * q^33 - 8 * q^34 - 44 * q^35 - 36 * q^37 + 8 * q^38 - 20 * q^39 - 28 * q^41 + 12 * q^42 + 84 * q^43 + 8 * q^44 - 4 * q^46 + 16 * q^47 + 24 * q^49 - 16 * q^50 + 8 * q^52 - 4 * q^53 + 48 * q^55 + 12 * q^56 + 12 * q^57 + 24 * q^58 + 12 * q^59 - 40 * q^62 - 8 * q^63 - 4 * q^65 + 8 * q^67 - 8 * q^69 + 60 * q^70 + 20 * q^71 + 4 * q^73 + 20 * q^74 + 8 * q^75 + 36 * q^76 - 12 * q^77 - 20 * q^78 + 32 * q^81 - 48 * q^82 + 12 * q^83 - 16 * q^84 - 4 * q^85 + 52 * q^86 - 36 * q^87 - 36 * q^89 - 16 * q^92 + 4 * q^93 - 48 * q^95 + 76 * q^97 + 8 * q^98 + 4 * 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_{4} $			$ a_{5} $			$ a_{6} $			$ a_{7} $			$ a_{8} $			$ a_{9} $	$ a_{10} $
19.1	−0.965926	−	0.258819i		1.00000i	0.866025	+	0.500000i	−2.19856	+	0.589103i	0.258819	−	0.965926i	1.83682	−	1.90423i	−0.707107	−	0.707107i	−1.00000	2.27612
19.2	−0.965926	−	0.258819i		1.00000i	0.866025	+	0.500000i	−1.43558	+	0.384662i	0.258819	−	0.965926i	1.74877	+	1.98539i	−0.707107	−	0.707107i	−1.00000	1.48622
19.3	−0.965926	−	0.258819i		1.00000i	0.866025	+	0.500000i	−0.106603	+	0.0285643i	0.258819	−	0.965926i	−1.53649	+	2.15388i	−0.707107	−	0.707107i	−1.00000	0.110364
19.4	−0.965926	−	0.258819i		1.00000i	0.866025	+	0.500000i	3.74075	−	1.00233i	0.258819	−	0.965926i	−2.20451	+	1.46293i	−0.707107	−	0.707107i	−1.00000	−3.87271
19.5	0.965926	+	0.258819i		1.00000i	0.866025	+	0.500000i	−2.85793	+	0.765779i	−0.258819	+	0.965926i	0.565013	+	2.58472i	0.707107	+	0.707107i	−1.00000	−2.95874
19.6	0.965926	+	0.258819i		1.00000i	0.866025	+	0.500000i	0.128673	−	0.0344778i	−0.258819	+	0.965926i	−2.64307	+	0.119073i	0.707107	+	0.707107i	−1.00000	0.133212
19.7	0.965926	+	0.258819i		1.00000i	0.866025	+	0.500000i	0.421063	−	0.112824i	−0.258819	+	0.965926i	2.44466	+	1.01175i	0.707107	+	0.707107i	−1.00000	0.435917
19.8	0.965926	+	0.258819i		1.00000i	0.866025	+	0.500000i	2.30819	−	0.618478i	−0.258819	+	0.965926i	1.78879	−	1.94942i	0.707107	+	0.707107i	−1.00000	2.38962
115.1	−0.965926	+	0.258819i	−	1.00000i	0.866025	−	0.500000i	−2.19856	−	0.589103i	0.258819	+	0.965926i	1.83682	+	1.90423i	−0.707107	+	0.707107i	−1.00000	2.27612
115.2	−0.965926	+	0.258819i	−	1.00000i	0.866025	−	0.500000i	−1.43558	−	0.384662i	0.258819	+	0.965926i	1.74877	−	1.98539i	−0.707107	+	0.707107i	−1.00000	1.48622
115.3	−0.965926	+	0.258819i	−	1.00000i	0.866025	−	0.500000i	−0.106603	−	0.0285643i	0.258819	+	0.965926i	−1.53649	−	2.15388i	−0.707107	+	0.707107i	−1.00000	0.110364
115.4	−0.965926	+	0.258819i	−	1.00000i	0.866025	−	0.500000i	3.74075	+	1.00233i	0.258819	+	0.965926i	−2.20451	−	1.46293i	−0.707107	+	0.707107i	−1.00000	−3.87271
115.5	0.965926	−	0.258819i	−	1.00000i	0.866025	−	0.500000i	−2.85793	−	0.765779i	−0.258819	−	0.965926i	0.565013	−	2.58472i	0.707107	−	0.707107i	−1.00000	−2.95874
115.6	0.965926	−	0.258819i	−	1.00000i	0.866025	−	0.500000i	0.128673	+	0.0344778i	−0.258819	−	0.965926i	−2.64307	−	0.119073i	0.707107	−	0.707107i	−1.00000	0.133212
115.7	0.965926	−	0.258819i	−	1.00000i	0.866025	−	0.500000i	0.421063	+	0.112824i	−0.258819	−	0.965926i	2.44466	−	1.01175i	0.707107	−	0.707107i	−1.00000	0.435917
115.8	0.965926	−	0.258819i	−	1.00000i	0.866025	−	0.500000i	2.30819	+	0.618478i	−0.258819	−	0.965926i	1.78879	+	1.94942i	0.707107	−	0.707107i	−1.00000	2.38962
397.1	−0.258819	+	0.965926i	−	1.00000i	−0.866025	−	0.500000i	−0.995376	−	3.71479i	0.965926	+	0.258819i	2.54668	−	0.717240i	0.707107	−	0.707107i	−1.00000	3.84584
397.2	−0.258819	+	0.965926i	−	1.00000i	−0.866025	−	0.500000i	−0.166844	−	0.622671i	0.965926	+	0.258819i	−2.38407	+	1.14726i	0.707107	−	0.707107i	−1.00000	0.644637
397.3	−0.258819	+	0.965926i	−	1.00000i	−0.866025	−	0.500000i	0.379698	+	1.41705i	0.965926	+	0.258819i	1.63697	−	2.07854i	0.707107	−	0.707107i	−1.00000	−1.46704
397.4	−0.258819	+	0.965926i	−	1.00000i	−0.866025	−	0.500000i	0.782522	+	2.92041i	0.965926	+	0.258819i	1.58056	+	2.12175i	0.707107	−	0.707107i	−1.00000	−3.02343

See all 32 embeddings

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
91.w	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	546.2.by.a	✓	32
7.d	odd	6	1		546.2.cg.a	yes	32
13.f	odd	12	1		546.2.cg.a	yes	32
91.w	even	12	1	inner	546.2.by.a	✓	32

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
546.2.by.a	✓	32	1.a	even	1	1	trivial
546.2.by.a	✓	32	91.w	even	12	1	inner
546.2.cg.a	yes	32	7.d	odd	6	1
546.2.cg.a	yes	32	13.f	odd	12	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{32} + 12 T_{5}^{30} - 36 T_{5}^{29} - 156 T_{5}^{28} - 124 T_{5}^{27} - 1800 T_{5}^{26} + \cdots + 9409 $ T5^32 + 12*T5^30 - 36*T5^29 - 156*T5^28 - 124*T5^27 - 1800*T5^26 + 5688*T5^25 + 25227*T5^24 + 55924*T5^23 + 350906*T5^22 + 12920*T5^21 - 385232*T5^20 + 114404*T5^19 - 12298432*T5^18 - 8010972*T5^17 + 30207436*T5^16 + 14075776*T5^15 + 201325146*T5^14 + 426371364*T5^13 + 426053236*T5^12 + 774387260*T5^11 + 697001302*T5^10 + 260538016*T5^9 + 291652167*T5^8 - 323424356*T5^7 + 68593920*T5^6 - 95145272*T5^5 + 46256392*T5^4 + 552780*T5^3 - 1137166*T5^2 + 6596*T5 + 9409 acting on $S_{2}^{\mathrm{new}}(546, [\chi])$.

Overview	Random
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Classical	Maass
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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:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	546.by (of order \(12\), degree \(4\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 19.8
Significant digits:
Format:

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