Newform orbit 1620.2.e.a

• Label: 1620.2.e.a
• Level: $1620$
• Weight: $2$
• Character orbit: 1620.e
• Analytic conductor: $12.936$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1620.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.9357651274\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
971.1	−1.41417	−	0.0114552i		0		1.99974	+	0.0323992i			1.00000i		0			−	2.33067i	−2.82759	−	0.0687254i		0		0.0114552	−	1.41417i
971.2	−1.41417	+	0.0114552i		0		1.99974	−	0.0323992i		−	1.00000i		0				2.33067i	−2.82759	+	0.0687254i		0		0.0114552	+	1.41417i
971.3	−1.38301	−	0.295458i		0		1.82541	+	0.817240i		−	1.00000i		0				1.21861i	−2.28309	−	1.66958i		0		−0.295458	+	1.38301i
971.4	−1.38301	+	0.295458i		0		1.82541	−	0.817240i			1.00000i		0			−	1.21861i	−2.28309	+	1.66958i		0		−0.295458	−	1.38301i
971.5	−1.31128	−	0.529663i		0		1.43891	+	1.38907i			1.00000i		0			−	0.139793i	−1.15108	−	2.58361i		0		0.529663	−	1.31128i
971.6	−1.31128	+	0.529663i		0		1.43891	−	1.38907i		−	1.00000i		0				0.139793i	−1.15108	+	2.58361i		0		0.529663	+	1.31128i
971.7	−1.19110	−	0.762415i		0		0.837446	+	1.81623i		−	1.00000i		0			−	4.47781i	0.387236	−	2.80179i		0		−0.762415	+	1.19110i
971.8	−1.19110	+	0.762415i		0		0.837446	−	1.81623i			1.00000i		0				4.47781i	0.387236	+	2.80179i		0		−0.762415	−	1.19110i
971.9	−1.16189	−	0.806231i		0		0.699983	+	1.87351i		−	1.00000i		0			−	1.36783i	0.697175	−	2.74116i		0		−0.806231	+	1.16189i
971.10	−1.16189	+	0.806231i		0		0.699983	−	1.87351i			1.00000i		0				1.36783i	0.697175	+	2.74116i		0		−0.806231	−	1.16189i
971.11	−1.03403	−	0.964765i		0		0.138457	+	1.99520i		−	1.00000i		0				3.54880i	1.78173	−	2.19669i		0		−0.964765	+	1.03403i
971.12	−1.03403	+	0.964765i		0		0.138457	−	1.99520i			1.00000i		0			−	3.54880i	1.78173	+	2.19669i		0		−0.964765	−	1.03403i
971.13	−0.990125	−	1.00978i		0		−0.0393069	+	1.99961i			1.00000i		0				2.09532i	2.05809	−	1.94018i		0		1.00978	−	0.990125i
971.14	−0.990125	+	1.00978i		0		−0.0393069	−	1.99961i		−	1.00000i		0			−	2.09532i	2.05809	+	1.94018i		0		1.00978	+	0.990125i
971.15	−0.821961	−	1.15082i		0		−0.648762	+	1.89185i			1.00000i		0			−	1.74129i	2.71043	−	0.808422i		0		1.15082	−	0.821961i
971.16	−0.821961	+	1.15082i		0		−0.648762	−	1.89185i		−	1.00000i		0				1.74129i	2.71043	+	0.808422i		0		1.15082	+	0.821961i
971.17	−0.570331	−	1.29411i		0		−1.34945	+	1.47614i		−	1.00000i		0			−	3.80698i	2.67992	+	0.904444i		0		−1.29411	+	0.570331i
971.18	−0.570331	+	1.29411i		0		−1.34945	−	1.47614i			1.00000i		0				3.80698i	2.67992	−	0.904444i		0		−1.29411	−	0.570331i
971.19	−0.531562	−	1.31051i		0		−1.43488	+	1.39324i		−	1.00000i		0				1.38206i	2.58858	+	1.13984i		0		−1.31051	+	0.531562i
971.20	−0.531562	+	1.31051i		0		−1.43488	−	1.39324i			1.00000i		0			−	1.38206i	2.58858	−	1.13984i		0		−1.31051	−	0.531562i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
4.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1620.2.e.a	✓	48
3.b	odd	2	1	inner	1620.2.e.a	✓	48
4.b	odd	2	1	inner	1620.2.e.a	✓	48
12.b	even	2	1	inner	1620.2.e.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1620.2.e.a	✓	48	1.a	even	1	1	trivial
1620.2.e.a	✓	48	3.b	odd	2	1	inner
1620.2.e.a	✓	48	4.b	odd	2	1	inner
1620.2.e.a	✓	48	12.b	even	2	1	inner