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Level	Weight	Analytic conductor	Analytic rank	Dim.

Bad \(p\)	Char.	Primitive character	Character order	Is maximal/largest

Coefficient field	Self-twists	CM/RM discriminant	Inner twist count	Is self-dual

Coefficient ring index	Coefficient ring gens.	Is twist minimal	Projective image

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	19	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
513.2.a.f	$513$	$2$	513.a	1.a	$1$	$3$	$3$	$4.096$	3.3.321.1	None	✓	✓			513.2.a.e	$1$	$0$	\(1\)	\(0\)	\(5\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
513.2.a.i	$513$	$2$	513.a	1.a	$1$	$4$	$4$	$4.096$	4.4.29952.1	None	✓	✓	✓		513.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{3}q^{5}+(3+\beta _{2}+\cdots)q^{7}+\cdots\)

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