Newform orbit 308.2.w.a

• Label: 308.2.w.a
• Level: $308$
• Weight: $2$
• Character orbit: 308.w
• Analytic conductor: $2.459$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	308.w (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.45939238226\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 5 q^{7} + 14 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1		0		−1.89651	−	2.61033i		0		−1.37174	+	0.445705i		0		−1.29885	+	2.30499i		0		−2.28999	+	7.04787i		0
13.2		0		−1.09677	−	1.50958i		0		4.07202	−	1.32308i		0		1.58132	+	2.12119i		0		−0.148869	+	0.458171i		0
13.3		0		−0.525576	−	0.723393i		0		−2.65575	+	0.862904i		0		2.56006	−	0.667889i		0		0.679983	−	2.09277i		0
13.4		0		−0.488294	−	0.672078i		0		0.892371	−	0.289949i		0		−2.59757	−	0.502635i		0		0.713792	−	2.19683i		0
13.5		0		0.488294	+	0.672078i		0		−0.892371	+	0.289949i		0		1.28073	+	2.31511i		0		0.713792	−	2.19683i		0
13.6		0		0.525576	+	0.723393i		0		2.65575	−	0.862904i		0		−0.155903	−	2.64115i		0		0.679983	−	2.09277i		0
13.7		0		1.09677	+	1.50958i		0		−4.07202	+	1.32308i		0		−2.50602	−	0.848437i		0		−0.148869	+	0.458171i		0
13.8		0		1.89651	+	2.61033i		0		1.37174	−	0.445705i		0		−1.79081	+	1.94756i		0		−2.28999	+	7.04787i		0
41.1		0		−2.74247	+	0.891082i		0		1.27787	−	1.75883i		0		−0.297393	+	2.62898i		0		4.30005	−	3.12417i		0
41.2		0		−2.44572	+	0.794664i		0		−1.36739	+	1.88205i		0		−2.17692	−	1.50366i		0		2.92302	−	2.12370i		0
41.3		0		−1.51783	+	0.493173i		0		−0.260986	+	0.359216i		0		2.51498	−	0.821504i		0		−0.366459	+	0.266248i		0
41.4		0		−0.359392	+	0.116773i		0		1.61768	−	2.22654i		0		0.904543	−	2.48632i		0		−2.31152	+	1.67942i		0
41.5		0		0.359392	−	0.116773i		0		−1.61768	+	2.22654i		0		−0.729634	+	2.54315i		0		−2.31152	+	1.67942i		0
41.6		0		1.51783	−	0.493173i		0		0.260986	−	0.359216i		0		1.55179	+	2.14288i		0		−0.366459	+	0.266248i		0
41.7		0		2.44572	−	0.794664i		0		1.36739	−	1.88205i		0		−2.64500	−	0.0630766i		0		2.92302	−	2.12370i		0
41.8		0		2.74247	−	0.891082i		0		−1.27787	+	1.75883i		0		1.30468	−	2.30170i		0		4.30005	−	3.12417i		0
237.1		0		−1.89651	+	2.61033i		0		−1.37174	−	0.445705i		0		−1.29885	−	2.30499i		0		−2.28999	−	7.04787i		0
237.2		0		−1.09677	+	1.50958i		0		4.07202	+	1.32308i		0		1.58132	−	2.12119i		0		−0.148869	−	0.458171i		0
237.3		0		−0.525576	+	0.723393i		0		−2.65575	−	0.862904i		0		2.56006	+	0.667889i		0		0.679983	+	2.09277i		0
237.4		0		−0.488294	+	0.672078i		0		0.892371	+	0.289949i		0		−2.59757	+	0.502635i		0		0.713792	+	2.19683i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
11.d	odd	10	1	inner
77.l	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	308.2.w.a	✓	32
7.b	odd	2	1	inner	308.2.w.a	✓	32
11.c	even	5	1		3388.2.c.c		32
11.d	odd	10	1	inner	308.2.w.a	✓	32
11.d	odd	10	1		3388.2.c.c		32
77.j	odd	10	1		3388.2.c.c		32
77.l	even	10	1	inner	308.2.w.a	✓	32
77.l	even	10	1		3388.2.c.c		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.2.w.a	✓	32	1.a	even	1	1	trivial
308.2.w.a	✓	32	7.b	odd	2	1	inner
308.2.w.a	✓	32	11.d	odd	10	1	inner
308.2.w.a	✓	32	77.l	even	10	1	inner
3388.2.c.c		32	11.c	even	5	1
3388.2.c.c		32	11.d	odd	10	1
3388.2.c.c		32	77.j	odd	10	1
3388.2.c.c		32	77.l	even	10	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(308, [\chi])\).