Newform orbit 309.3.n.a

• Label: 309.3.n.a
• Level: $309$
• Weight: $3$
• Character orbit: 309.n
• Analytic conductor: $8.420$
• Analytic rank: $0$
• Dimension: $32$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(8.41964016873\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 6 q^{3} + 4 q^{4} + 11 q^{7} - 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1		0		2.55065	−	1.57930i	3.96968	−	0.491553i		0			0		−3.11864	+	3.64010i		0		4.01165	−	8.05647i		0
17.1		0		−2.21703	+	2.02109i	1.32942	+	3.77262i		0			0		−9.72045	+	10.0245i		0		0.830415	−	8.96161i		0
26.1		0		−1.33722	−	2.68549i	3.87919	+	0.975655i		0			0		−1.94124	+	12.5057i		0		−5.42371	+	7.18216i		0
29.1		0		−2.79742	+	1.08372i	3.26479	+	2.31110i		0			0		−2.77869	−	6.56491i		0		6.65108	−	6.06326i		0
32.1		0		−2.79742	−	1.08372i	3.26479	−	2.31110i		0			0		−2.77869	+	6.56491i		0		6.65108	+	6.06326i		0
38.1		0		0.820989	+	2.88548i	−3.99241	+	0.246244i		0			0		−4.83246	+	2.22328i		0		−7.65195	+	4.73789i		0
41.1		0		−2.79742	−	1.08372i	−3.63386	−	1.67184i		0			0		13.2842	−	1.64494i		0		6.65108	+	6.06326i		0
50.1		0		0.820989	−	2.88548i	2.20946	−	3.33441i		0			0		−9.25563	+	6.55192i		0		−7.65195	−	4.73789i		0
59.1		0		−0.276805	−	2.98720i	−0.613567	+	3.95266i		0			0		8.49143	−	4.55980i		0		−8.84676	+	1.65375i		0
68.1		0		0.820989	+	2.88548i	2.20946	+	3.33441i		0			0		−9.25563	−	6.55192i		0		−7.65195	+	4.73789i		0
83.1		0		−0.276805	+	2.98720i	−3.11632	+	2.50770i		0			0		0.234300	+	7.60475i		0		−8.84676	−	1.65375i		0
92.1		0		2.55065	−	1.57930i	−1.55914	+	3.68362i		0			0		−4.62353	−	13.1206i		0		4.01165	−	8.05647i		0
98.1		0		−2.79742	+	1.08372i	−3.63386	+	1.67184i		0			0		13.2842	+	1.64494i		0		6.65108	−	6.06326i		0
107.1		0		−1.33722	+	2.68549i	3.87919	−	0.975655i		0			0		−1.94124	−	12.5057i		0		−5.42371	−	7.18216i		0
110.1		0		−0.276805	+	2.98720i	−0.613567	−	3.95266i		0			0		8.49143	+	4.55980i		0		−8.84676	−	1.65375i		0
119.1		0		1.80790	+	2.39405i	3.52405	−	1.89237i		0			0		12.9028	−	4.10469i		0		−2.46297	+	8.65643i		0
122.1		0		0.820989	−	2.88548i	−3.99241	−	0.246244i		0			0		−4.83246	−	2.22328i		0		−7.65195	−	4.73789i		0
128.1		0		−2.21703	+	2.02109i	2.60247	−	3.03762i		0			0		2.22542	+	0.559714i		0		0.830415	−	8.96161i		0
131.1		0		2.55065	+	1.57930i	−1.55914	−	3.68362i		0			0		−4.62353	+	13.1206i		0		4.01165	+	8.05647i		0
152.1		0		2.94892	+	0.551249i	−3.81177	+	1.21261i		0			0		−6.85155	−	10.3400i		0		8.39225	+	3.25117i		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.g	even	51	1	inner
309.n	odd	102	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	309.3.n.a	✓	32
3.b	odd	2	1	CM	309.3.n.a	✓	32
103.g	even	51	1	inner	309.3.n.a	✓	32
309.n	odd	102	1	inner	309.3.n.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
309.3.n.a	✓	32	1.a	even	1	1	trivial
309.3.n.a	✓	32	3.b	odd	2	1	CM
309.3.n.a	✓	32	103.g	even	51	1	inner
309.3.n.a	✓	32	309.n	odd	102	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{3}^{\mathrm{new}}(309, [\chi])\).