Newform orbit 503.3.b.a

• Label: 503.3.b.a
• Level: $503$
• Weight: $3$
• Character orbit: 503.b
• Self dual: yes
• Analytic conductor: $13.706$
• Analytic rank: $0$
• Dimension: $21$
• CM discriminant: -503
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 503 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	503.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(13.7057572973\)
Analytic rank:	\(0\)
Dimension:	\(21\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 21 q + 84 q^{4} + 189 q^{9}+O(q^{10}) \)

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.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
502.1	−3.99711			−4.89501			11.9769				0		19.5659			2.92442			−31.8846			14.9611				0
502.2	−3.99711			−0.557353			11.9769				0		2.22780			−13.3191			−31.8846			−8.68936				0
502.3	−3.99711			5.45236			11.9769				0		−21.7937			10.3947			−31.8846			20.7282				0
502.4	−2.61095			−5.76472			2.81704				0		15.0514			3.30555			3.08865			24.2320				0
502.5	−2.61095			1.44151			2.81704				0		−3.76372			−13.4343			3.08865			−6.92204				0
502.6	−2.61095			4.32320			2.81704				0		−11.2876			10.1288			3.08865			9.69008				0
502.7	−2.37337			−5.01821			1.63290				0		11.9101			13.8714			5.61801			16.1824				0
502.8	−2.37337			−0.339260			1.63290				0		0.805189			−8.57519			5.61801			−8.88490				0
502.9	−2.37337			5.35747			1.63290				0		−12.7153			−5.29623			5.61801			19.7025				0
502.10	0.741317			−3.65482			−3.45045				0		−2.70938			13.8131			−5.52314			4.35773				0
502.11	0.741317			−2.29348			−3.45045				0		−1.70019			−8.88083			−5.52314			−3.73997				0
502.12	0.741317			5.94830			−3.45045				0		4.40957			−4.93232			−5.52314			26.3822				0
502.13	1.03757			−5.70025			−2.92346				0		−5.91439			−11.6763			−7.18354			23.4929				0
502.14	1.03757			1.22829			−2.92346				0		1.27444			−0.851200			−7.18354			−7.49130				0
502.15	1.03757			4.47196			−2.92346				0		4.63995			12.5275			−7.18354			10.9984				0
502.16	3.53535			−5.99900			8.49872				0		−21.2086			−11.4561			15.9046			26.9880				0
502.17	3.53535			2.90480			8.49872				0		10.2695			12.6971			15.9046			−0.562138				0
502.18	3.53535			3.09420			8.49872				0		10.9391			−1.24104			15.9046			0.574096				0
502.19	3.66720			−3.82581			9.44833				0		−14.0300			6.83004			19.9801			5.63684				0
502.20	3.66720			−2.08989			9.44833				0		−7.66403			7.16860			19.9801			−4.63236				0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
503.b	odd	2	1	CM by \(\Q(\sqrt{-503}) \)

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	503.3.b.a	✓	21
503.b	odd	2	1	CM	503.3.b.a	✓	21

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
503.3.b.a	✓	21	1.a	even	1	1	trivial
503.3.b.a	✓	21	503.b	odd	2	1	CM