Newform orbit 324.4.i.a

• Label: 324.4.i.a
• Level: $324$
• Weight: $4$
• Character orbit: 324.i
• Analytic conductor: $19.117$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.i (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.1166188419\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(9\) over \(\Q(\zeta_{9})\)
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 12 q^{5}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1		0			0			0		−19.7528	+	7.18944i		0		0.723940	+	4.10567i		0			0			0
37.2		0			0			0		−13.2490	+	4.82225i		0		3.17294	+	17.9946i		0			0			0
37.3		0			0			0		−11.1609	+	4.06223i		0		−4.28132	−	24.2806i		0			0			0
37.4		0			0			0		−5.21695	+	1.89881i		0		1.58868	+	9.00985i		0			0			0
37.5		0			0			0		4.27250	−	1.55506i		0		−4.69568	−	26.6305i		0			0			0
37.6		0			0			0		4.80495	−	1.74886i		0		5.74341	+	32.5725i		0			0			0
37.7		0			0			0		6.50317	−	2.36696i		0		1.34685	+	7.63837i		0			0			0
37.8		0			0			0		10.0092	−	3.64305i		0		−2.90933	−	16.4997i		0			0			0
37.9		0			0			0		13.0875	−	4.76347i		0		−0.689494	−	3.91032i		0			0			0
73.1		0			0			0		−15.7004	+	13.1742i		0		−8.81657	+	3.20897i		0			0			0
73.2		0			0			0		−8.18465	+	6.86774i		0		−24.4722	+	8.90714i		0			0			0
73.3		0			0			0		−7.52940	+	6.31792i		0		24.9076	−	9.06562i		0			0			0
73.4		0			0			0		−6.54095	+	5.48851i		0		11.9650	−	4.35489i		0			0			0
73.5		0			0			0		1.73722	−	1.45770i		0		−5.32680	+	1.93880i		0			0			0
73.6		0			0			0		5.18194	−	4.34817i		0		16.8240	−	6.12342i		0			0			0
73.7		0			0			0		5.19451	−	4.35871i		0		−17.8009	+	6.47901i		0			0			0
73.8		0			0			0		12.8040	−	10.7439i		0		−20.8033	+	7.57178i		0			0			0
73.9		0			0			0		14.9393	−	12.5356i		0		23.5233	−	8.56177i		0			0			0
145.1		0			0			0		−3.27839	+	18.5927i		0		4.82725	−	4.05055i		0			0			0
145.2		0			0			0		−2.26368	+	12.8380i		0		−3.04053	+	2.55131i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.4.i.a		54
3.b	odd	2	1		108.4.i.a	✓	54
27.e	even	9	1	inner	324.4.i.a		54
27.f	odd	18	1		108.4.i.a	✓	54

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.4.i.a	✓	54	3.b	odd	2	1
108.4.i.a	✓	54	27.f	odd	18	1
324.4.i.a		54	1.a	even	1	1	trivial
324.4.i.a		54	27.e	even	9	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(324, [\chi])\).