Newform orbit 756.4.i.a

• Label: 756.4.i.a
• Level: $756$
• Weight: $4$
• Character orbit: 756.i
• 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.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(44.6054439643\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(24\) over \(\Q(\zeta_{3})\)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 20 q^{5} - 6 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1		0			0			0		−10.8301	−	18.7582i		0		−12.6645	−	13.5134i		0			0			0
37.2		0			0			0		−10.2794	−	17.8044i		0		18.5152	+	0.432294i		0			0			0
37.3		0			0			0		−8.50761	−	14.7356i		0		18.0631	−	4.08939i		0			0			0
37.4		0			0			0		−7.81356	−	13.5335i		0		6.71488	+	17.2601i		0			0			0
37.5		0			0			0		−6.02006	−	10.4271i		0		−8.82278	+	16.2837i		0			0			0
37.6		0			0			0		−4.93620	−	8.54976i		0		−6.20989	−	17.4481i		0			0			0
37.7		0			0			0		−4.53303	−	7.85144i		0		−15.6380	−	9.92235i		0			0			0
37.8		0			0			0		−4.29795	−	7.44428i		0		−11.6662	+	14.3840i		0			0			0
37.9		0			0			0		−3.23563	−	5.60427i		0		4.58917	−	17.9427i		0			0			0
37.10		0			0			0		−3.12929	−	5.42009i		0		−9.69674	+	15.7789i		0			0			0
37.11		0			0			0		−2.84184	−	4.92220i		0		15.2960	−	10.4419i		0			0			0
37.12		0			0			0		−1.32728	−	2.29891i		0		−14.0290	−	12.0908i		0			0			0
37.13		0			0			0		−0.863703	−	1.49598i		0		−16.5772	+	8.25811i		0			0			0
37.14		0			0			0		0.275008	+	0.476327i		0		18.2696	+	3.03661i		0			0			0
37.15		0			0			0		0.310848	+	0.538404i		0		11.0807	+	14.8397i		0			0			0
37.16		0			0			0		3.16904	+	5.48893i		0		6.46303	−	17.3560i		0			0			0
37.17		0			0			0		4.98332	+	8.63137i		0		−18.3914	−	2.18061i		0			0			0
37.18		0			0			0		5.16462	+	8.94539i		0		8.74826	+	16.3238i		0			0			0
37.19		0			0			0		5.68089	+	9.83959i		0		18.4438	−	1.68138i		0			0			0
37.20		0			0			0		6.20694	+	10.7507i		0		−18.4402	−	1.72023i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
63.h	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.4.i.a		48
3.b	odd	2	1		252.4.i.a	✓	48
7.c	even	3	1		756.4.l.a		48
9.c	even	3	1		756.4.l.a		48
9.d	odd	6	1		252.4.l.a	yes	48
21.h	odd	6	1		252.4.l.a	yes	48
63.h	even	3	1	inner	756.4.i.a		48
63.j	odd	6	1		252.4.i.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.4.i.a	✓	48	3.b	odd	2	1
252.4.i.a	✓	48	63.j	odd	6	1
252.4.l.a	yes	48	9.d	odd	6	1
252.4.l.a	yes	48	21.h	odd	6	1
756.4.i.a		48	1.a	even	1	1	trivial
756.4.i.a		48	63.h	even	3	1	inner
756.4.l.a		48	7.c	even	3	1
756.4.l.a		48	9.c	even	3	1

Hecke kernels

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