Newform orbit 324.3.k.a

• Label: 324.3.k.a
• Level: $324$
• Weight: $3$
• Character orbit: 324.k
• Analytic conductor: $8.828$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.k (of order \(18\), degree \(6\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.82836056527\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 9 q^{5}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0			0			0		−7.10446	−	1.25271i		0		−3.36440	+	2.82306i		0			0			0
17.2		0			0			0		−3.29223	−	0.580508i		0		3.84473	−	3.22611i		0			0			0
17.3		0			0			0		−2.92426	−	0.515626i		0		0.715829	−	0.600652i		0			0			0
17.4		0			0			0		4.32861	+	0.763252i		0		−2.73772	+	2.29722i		0			0			0
17.5		0			0			0		5.13754	+	0.905886i		0		−8.93347	+	7.49607i		0			0			0
17.6		0			0			0		7.65293	+	1.34942i		0		10.4750	−	8.78960i		0			0			0
89.1		0			0			0		−5.64904	−	6.73227i		0		4.05297	−	1.47516i		0			0			0
89.2		0			0			0		−2.68656	−	3.20172i		0		−4.88621	+	1.77844i		0			0			0
89.3		0			0			0		−0.980262	−	1.16823i		0		3.23920	−	1.17897i		0			0			0
89.4		0			0			0		0.298552	+	0.355800i		0		−10.1488	+	3.69384i		0			0			0
89.5		0			0			0		2.69546	+	3.21232i		0		11.1367	−	4.05342i		0			0			0
89.6		0			0			0		5.00278	+	5.96208i		0		−3.39388	+	1.23527i		0			0			0
125.1		0			0			0		−1.65461	−	4.54600i		0		1.68621	+	9.56295i		0			0			0
125.2		0			0			0		−1.26650	−	3.47969i		0		−0.0728181	−	0.412972i		0			0			0
125.3		0			0			0		−0.740753	−	2.03520i		0		1.08248	+	6.13906i		0			0			0
125.4		0			0			0		−0.0686711	−	0.188672i		0		−1.47862	−	8.38565i		0			0			0
125.5		0			0			0		2.50118	+	6.87194i		0		−1.62729	−	9.22884i		0			0			0
125.6		0			0			0		3.25030	+	8.93012i		0		0.410040	+	2.32545i		0			0			0
197.1		0			0			0		−1.65461	+	4.54600i		0		1.68621	−	9.56295i		0			0			0
197.2		0			0			0		−1.26650	+	3.47969i		0		−0.0728181	+	0.412972i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.f	odd	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.3.k.a		36
3.b	odd	2	1		108.3.k.a	✓	36
12.b	even	2	1		432.3.bc.b		36
27.e	even	9	1		108.3.k.a	✓	36
27.e	even	9	1		2916.3.c.b		36
27.f	odd	18	1	inner	324.3.k.a		36
27.f	odd	18	1		2916.3.c.b		36
108.j	odd	18	1		432.3.bc.b		36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
108.3.k.a	✓	36	3.b	odd	2	1
108.3.k.a	✓	36	27.e	even	9	1
324.3.k.a		36	1.a	even	1	1	trivial
324.3.k.a		36	27.f	odd	18	1	inner
432.3.bc.b		36	12.b	even	2	1
432.3.bc.b		36	108.j	odd	18	1
2916.3.c.b		36	27.e	even	9	1
2916.3.c.b		36	27.f	odd	18	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(324, [\chi])\).