Newform orbit 309.2.o.a

• Label: 309.2.o.a
• Level: $309$
• Weight: $2$
• Character orbit: 309.o
• Analytic conductor: $2.467$
• Analytic rank: $0$
• Dimension: $32$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	309.o (of order \(102\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.46737742246\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{102}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{4} + 5 q^{7} + 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1		0		0.625689	+	1.61509i	−1.81693	+	0.835921i		0			0		4.00608	+	0.496060i		0		−2.21703	+	2.02109i		0
11.1		0		−0.911807	−	1.47262i	−0.779572	+	1.84181i		0			0		−1.10983	−	3.14946i		0		−1.33722	+	2.68549i		0
20.1		0		−1.72466	−	0.159813i	−1.55816	+	1.25385i		0			0		−0.0673181	−	2.18497i		0		2.94892	+	0.551249i		0
35.1		0		−1.66593	+	0.473998i	1.10473	+	1.66720i		0			0		0.242913	+	0.171955i		0		2.55065	−	1.57930i		0
44.1		0		1.72466	−	0.159813i	−0.306783	+	1.97633i		0			0		−1.11300	+	0.597669i		0		2.94892	−	0.551249i		0
53.1		0		−1.66593	−	0.473998i	1.10473	−	1.66720i		0			0		0.242913	−	0.171955i		0		2.55065	+	1.57930i		0
62.1		0		0.625689	−	1.61509i	−1.81693	−	0.835921i		0			0		4.00608	−	0.496060i		0		−2.21703	−	2.02109i		0
65.1		0		1.66593	−	0.473998i	−1.99621	+	0.123122i		0			0		2.75651	−	1.26819i		0		2.55065	−	1.57930i		0
71.1		0		−0.625689	+	1.61509i	1.63239	−	1.15555i		0			0		−2.06121	+	4.86979i		0		−2.21703	−	2.02109i		0
74.1		0		−0.625689	−	1.61509i	1.63239	+	1.15555i		0			0		−2.06121	−	4.86979i		0		−2.21703	+	2.02109i		0
77.1		0		1.55047	−	0.772041i	1.93959	+	0.487827i		0			0		−0.104034	+	0.670197i		0		1.80790	−	2.39405i		0
86.1		0		1.16688	+	1.28000i	0.664710	+	1.88631i		0			0		−1.75994	+	1.81500i		0		−0.276805	+	2.98720i		0
101.1		0		0.911807	+	1.47262i	1.98484	−	0.245777i		0			0		−3.35907	+	3.92073i		0		−1.33722	+	2.68549i		0
143.1		0		−1.55047	+	0.772041i	−1.39227	+	1.43582i		0			0		−2.94427	−	2.36925i		0		1.80790	−	2.39405i		0
146.1		0		0.318264	+	1.70256i	0.427866	+	1.95370i		0			0		5.22320	+	0.322156i		0		−2.79742	+	1.08372i		0
170.1		0		−1.72466	+	0.159813i	−1.55816	−	1.25385i		0			0		−0.0673181	+	2.18497i		0		2.94892	−	0.551249i		0
173.1		0		−1.16688	+	1.28000i	1.30124	+	1.51881i		0			0		4.71507	−	1.18589i		0		−0.276805	−	2.98720i		0
188.1		0		−1.55047	−	0.772041i	−1.39227	−	1.43582i		0			0		−2.94427	+	2.36925i		0		1.80790	+	2.39405i		0
191.1		0		1.38221	−	1.04379i	−0.0615901	+	1.99905i		0			0		−0.940317	+	4.29362i		0		0.820989	−	2.88548i		0
212.1		0		1.16688	−	1.28000i	0.664710	−	1.88631i		0			0		−1.75994	−	1.81500i		0		−0.276805	−	2.98720i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
103.h	odd	102	1	inner
309.o	even	102	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	309.2.o.a	✓	32
3.b	odd	2	1	CM	309.2.o.a	✓	32
103.h	odd	102	1	inner	309.2.o.a	✓	32
309.o	even	102	1	inner	309.2.o.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
309.2.o.a	✓	32	1.a	even	1	1	trivial
309.2.o.a	✓	32	3.b	odd	2	1	CM
309.2.o.a	✓	32	103.h	odd	102	1	inner
309.2.o.a	✓	32	309.o	even	102	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(309, [\chi])\).