Newform orbit 322.3.b.a

• Label: 322.3.b.a
• Level: $322$
• Weight: $3$
• Character orbit: 322.b
• Analytic conductor: $8.774$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 322 = 2 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	322.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.77386451240\)
Analytic rank:	\(0\)
Dimension:	\(32\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64 q^{4} - 12 q^{7} - 128 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
139.1	−1.41421				−	5.78135i	2.00000				−	5.81041i			8.17607i	2.60526	+	6.49712i	−2.82843			−24.4240					8.21716i
139.2	−1.41421				−	5.04789i	2.00000					0.579996i			7.13880i	5.70567	−	4.05528i	−2.82843			−16.4812				−	0.820238i
139.3	−1.41421				−	3.82674i	2.00000					7.51239i			5.41183i	−4.12978	+	5.65198i	−2.82843			−5.64396				−	10.6241i
139.4	−1.41421				−	3.46826i	2.00000					2.85039i			4.90486i	−6.91362	+	1.09631i	−2.82843			−3.02883				−	4.03106i
139.5	−1.41421				−	3.38816i	2.00000				−	5.67755i			4.79158i	−6.01256	−	3.58456i	−2.82843			−2.47960					8.02927i
139.6	−1.41421				−	2.45755i	2.00000					1.94523i			3.47550i	6.91966	+	1.05748i	−2.82843			2.96047				−	2.75098i
139.7	−1.41421				−	0.808444i	2.00000				−	8.92170i			1.14331i	0.0318752	+	6.99993i	−2.82843			8.34642					12.6172i
139.8	−1.41421				−	0.499241i	2.00000					1.45903i			0.706034i	0.207708	−	6.99692i	−2.82843			8.75076				−	2.06338i
139.9	−1.41421					0.499241i	2.00000				−	1.45903i		−	0.706034i	0.207708	+	6.99692i	−2.82843			8.75076					2.06338i
139.10	−1.41421					0.808444i	2.00000					8.92170i		−	1.14331i	0.0318752	−	6.99993i	−2.82843			8.34642				−	12.6172i
139.11	−1.41421					2.45755i	2.00000				−	1.94523i		−	3.47550i	6.91966	−	1.05748i	−2.82843			2.96047					2.75098i
139.12	−1.41421					3.38816i	2.00000					5.67755i		−	4.79158i	−6.01256	+	3.58456i	−2.82843			−2.47960				−	8.02927i
139.13	−1.41421					3.46826i	2.00000				−	2.85039i		−	4.90486i	−6.91362	−	1.09631i	−2.82843			−3.02883					4.03106i
139.14	−1.41421					3.82674i	2.00000				−	7.51239i		−	5.41183i	−4.12978	−	5.65198i	−2.82843			−5.64396					10.6241i
139.15	−1.41421					5.04789i	2.00000				−	0.579996i		−	7.13880i	5.70567	+	4.05528i	−2.82843			−16.4812					0.820238i
139.16	−1.41421					5.78135i	2.00000					5.81041i		−	8.17607i	2.60526	−	6.49712i	−2.82843			−24.4240				−	8.21716i
139.17	1.41421				−	5.72743i	2.00000					1.63680i		−	8.09981i	−6.95198	+	0.818518i	2.82843			−23.8034					2.31478i
139.18	1.41421				−	4.95679i	2.00000					9.92413i		−	7.00996i	6.78730	+	1.71249i	2.82843			−15.5698					14.0348i
139.19	1.41421				−	4.52542i	2.00000				−	5.90217i		−	6.39991i	4.60838	−	5.26905i	2.82843			−11.4794				−	8.34694i
139.20	1.41421				−	3.31400i	2.00000				−	9.29762i		−	4.68671i	−4.96652	+	4.93292i	2.82843			−1.98262				−	13.1488i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	322.3.b.a	✓	32
7.b	odd	2	1	inner	322.3.b.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
322.3.b.a	✓	32	1.a	even	1	1	trivial
322.3.b.a	✓	32	7.b	odd	2	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(322, [\chi])\).