Newform orbit 756.4.bm.a

• Label: 756.4.bm.a
• Level: $756$
• Weight: $4$
• Character orbit: 756.bm
• Analytic conductor: $44.605$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.bm (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(44.6054439643\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 252)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 6 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
17.1		0			0			0		−18.4760				0		−4.17729	−	18.0430i		0			0			0
17.2		0			0			0		−18.3795				0		−11.4907	+	14.5246i		0			0			0
17.3		0			0			0		−17.7911				0		−5.94198	+	17.5412i		0			0			0
17.4		0			0			0		−16.5659				0		17.5228	+	5.99607i		0			0			0
17.5		0			0			0		−13.0880				0		15.3663	+	10.3381i		0			0			0
17.6		0			0			0		−10.4059				0		15.5309	−	10.0892i		0			0			0
17.7		0			0			0		−10.3118				0		−12.2405	−	13.8985i		0			0			0
17.8		0			0			0		−8.75818				0		−10.7333	−	15.0930i		0			0			0
17.9		0			0			0		−6.67787				0		−16.9163	+	7.53913i		0			0			0
17.10		0			0			0		−5.50907				0		−15.5186	−	10.1081i		0			0			0
17.11		0			0			0		−3.45780				0		13.4196	−	12.7638i		0			0			0
17.12		0			0			0		1.90049				0		3.04981	+	18.2674i		0			0			0
17.13		0			0			0		2.27383				0		17.2174	−	6.82357i		0			0			0
17.14		0			0			0		3.95888				0		0.649207	+	18.5089i		0			0			0
17.15		0			0			0		6.03810				0		10.8562	+	15.0047i		0			0			0
17.16		0			0			0		6.49181				0		18.4099	+	2.01918i		0			0			0
17.17		0			0			0		7.54500				0		−17.0316	+	7.27489i		0			0			0
17.18		0			0			0		7.63023				0		−0.685930	−	18.5076i		0			0			0
17.19		0			0			0		9.81402				0		−17.8827	+	4.81749i		0			0			0
17.20		0			0			0		12.5831				0		−6.65587	+	17.2829i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.s	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.4.bm.a		48
3.b	odd	2	1		252.4.bm.a	yes	48
7.d	odd	6	1		756.4.w.a		48
9.c	even	3	1		252.4.w.a	✓	48
9.d	odd	6	1		756.4.w.a		48
21.g	even	6	1		252.4.w.a	✓	48
63.k	odd	6	1		252.4.bm.a	yes	48
63.s	even	6	1	inner	756.4.bm.a		48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.4.w.a	✓	48	9.c	even	3	1
252.4.w.a	✓	48	21.g	even	6	1
252.4.bm.a	yes	48	3.b	odd	2	1
252.4.bm.a	yes	48	63.k	odd	6	1
756.4.w.a		48	7.d	odd	6	1
756.4.w.a		48	9.d	odd	6	1
756.4.bm.a		48	1.a	even	1	1	trivial
756.4.bm.a		48	63.s	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(756, [\chi])\).