Newform orbit 387.3.j.f

• Label: 387.3.j.f
• Level: $387$
• Weight: $3$
• Character orbit: 387.j
• Analytic conductor: $10.545$
• Analytic rank: $0$
• Dimension: $28$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	387.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.5449862307\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Relative dimension:	\(14\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 84 q^{4} - 30 q^{7}+O(q^{10}) \)

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		−	3.98399i		0		−11.8722			−4.48413	−	2.58892i		0		−4.28862	+	2.47604i			31.3628i		0		−10.3142	+	17.8648i
37.2		−	3.26725i		0		−6.67493			7.64103	+	4.41155i		0		−9.63802	+	5.56452i			8.73968i		0		14.4137	−	24.9652i
37.3		−	3.03676i		0		−5.22190			−0.820197	−	0.473541i		0		6.81469	−	3.93447i			3.71061i		0		−1.43803	+	2.49074i
37.4		−	2.66461i		0		−3.10014			2.37064	+	1.36869i		0		2.06363	−	1.19144i		−	2.39778i		0		3.64702	−	6.31683i
37.5		−	1.70884i		0		1.07986			−7.97167	−	4.60245i		0		−1.33514	+	0.770846i		−	8.68068i		0		−7.86485	+	13.6223i
37.6		−	1.61273i		0		1.39911			0.333970	+	0.192817i		0		−8.76989	+	5.06330i		−	8.70729i		0		0.310962	−	0.538602i
37.7		−	0.780897i		0		3.39020			4.98050	+	2.87549i		0		7.65336	−	4.41867i		−	5.77098i		0		2.24546	−	3.88926i
37.8			0.780897i		0		3.39020			−4.98050	−	2.87549i		0		7.65336	−	4.41867i			5.77098i		0		2.24546	−	3.88926i
37.9			1.61273i		0		1.39911			−0.333970	−	0.192817i		0		−8.76989	+	5.06330i			8.70729i		0		0.310962	−	0.538602i
37.10			1.70884i		0		1.07986			7.97167	+	4.60245i		0		−1.33514	+	0.770846i			8.68068i		0		−7.86485	+	13.6223i
37.11			2.66461i		0		−3.10014			−2.37064	−	1.36869i		0		2.06363	−	1.19144i			2.39778i		0		3.64702	−	6.31683i
37.12			3.03676i		0		−5.22190			0.820197	+	0.473541i		0		6.81469	−	3.93447i		−	3.71061i		0		−1.43803	+	2.49074i
37.13			3.26725i		0		−6.67493			−7.64103	−	4.41155i		0		−9.63802	+	5.56452i		−	8.73968i		0		14.4137	−	24.9652i
37.14			3.98399i		0		−11.8722			4.48413	+	2.58892i		0		−4.28862	+	2.47604i		−	31.3628i		0		−10.3142	+	17.8648i
136.1		−	3.98399i		0		−11.8722			4.48413	−	2.58892i		0		−4.28862	−	2.47604i			31.3628i		0		−10.3142	−	17.8648i
136.2		−	3.26725i		0		−6.67493			−7.64103	+	4.41155i		0		−9.63802	−	5.56452i			8.73968i		0		14.4137	+	24.9652i
136.3		−	3.03676i		0		−5.22190			0.820197	−	0.473541i		0		6.81469	+	3.93447i			3.71061i		0		−1.43803	−	2.49074i
136.4		−	2.66461i		0		−3.10014			−2.37064	+	1.36869i		0		2.06363	+	1.19144i		−	2.39778i		0		3.64702	+	6.31683i
136.5		−	1.70884i		0		1.07986			7.97167	−	4.60245i		0		−1.33514	−	0.770846i		−	8.68068i		0		−7.86485	−	13.6223i
136.6		−	1.61273i		0		1.39911			−0.333970	+	0.192817i		0		−8.76989	−	5.06330i		−	8.70729i		0		0.310962	+	0.538602i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
43.d	odd	6	1	inner
129.h	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	387.3.j.f	✓	28
3.b	odd	2	1	inner	387.3.j.f	✓	28
43.d	odd	6	1	inner	387.3.j.f	✓	28
129.h	even	6	1	inner	387.3.j.f	✓	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
387.3.j.f	✓	28	1.a	even	1	1	trivial
387.3.j.f	✓	28	3.b	odd	2	1	inner
387.3.j.f	✓	28	43.d	odd	6	1	inner
387.3.j.f	✓	28	129.h	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 49T_{2}^{12} + 942T_{2}^{10} + 9075T_{2}^{8} + 46229T_{2}^{6} + 120479T_{2}^{4} + 142468T_{2}^{2} + 51381 \) acting on \(S_{3}^{\mathrm{new}}(387, [\chi])\).