Newform orbit 756.3.p.a

• Label: 756.3.p.a
• Level: $756$
• Weight: $3$
• Character orbit: 756.p
• Analytic conductor: $20.600$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(20.5995079856\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 32 * q + q^7 + 12 * q^11 + 15 * q^13 + 27 * q^17 - 30 * q^23 - 160 * q^25 - 24 * q^29 - 24 * q^31 - 141 * q^35 + 11 * q^37 + 90 * q^41 - 16 * q^43 - 108 * q^47 - 61 * q^49 - 54 * q^53 - 45 * q^59 - 165 * q^61 + 33 * q^65 - 35 * q^67 - 138 * q^71 + 123 * q^77 + 37 * q^79 - 378 * q^83 + 30 * q^85 + 261 * q^89 + 45 * q^91 + 48 * q^95 - 57 * q^97

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
397.1		0			0			0			−	8.86566i		0		4.97462	−	4.92475i		0			0			0
397.2		0			0			0			−	7.47619i		0		−1.51829	−	6.83336i		0			0			0
397.3		0			0			0			−	6.76413i		0		−6.54281	−	2.48830i		0			0			0
397.4		0			0			0			−	5.81748i		0		2.61032	+	6.49509i		0			0			0
397.5		0			0			0			−	4.03768i		0		2.36524	+	6.58829i		0			0			0
397.6		0			0			0			−	3.08422i		0		6.74289	+	1.87973i		0			0			0
397.7		0			0			0			−	2.13702i		0		−6.60308	+	2.32365i		0			0			0
397.8		0			0			0				0.196075i		0		−5.97368	−	3.64898i		0			0			0
397.9		0			0			0				0.963344i		0		6.86265	−	1.37988i		0			0			0
397.10		0			0			0				2.30326i		0		−1.15122	+	6.90469i		0			0			0
397.11		0			0			0				3.16374i		0		4.69832	−	5.18900i		0			0			0
397.12		0			0			0				4.01620i		0		1.61222	−	6.81181i		0			0			0
397.13		0			0			0				4.90249i		0		−6.05364	+	3.51475i		0			0			0
397.14		0			0			0				5.22643i		0		−2.36655	−	6.58783i		0			0			0
397.15		0			0			0				8.39011i		0		5.61921	+	4.17426i		0			0			0
397.16		0			0			0				9.02073i		0		−4.77619	+	5.11742i		0			0			0
577.1		0			0			0			−	9.02073i		0		−4.77619	−	5.11742i		0			0			0
577.2		0			0			0			−	8.39011i		0		5.61921	−	4.17426i		0			0			0
577.3		0			0			0			−	5.22643i		0		−2.36655	+	6.58783i		0			0			0
577.4		0			0			0			−	4.90249i		0		−6.05364	−	3.51475i		0			0			0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	756.3.p.a		32
3.b	odd	2	1		252.3.p.a	✓	32
7.d	odd	6	1		756.3.bd.a		32
9.c	even	3	1		756.3.bd.a		32
9.d	odd	6	1		252.3.bd.a	yes	32
21.g	even	6	1		252.3.bd.a	yes	32
63.k	odd	6	1	inner	756.3.p.a		32
63.s	even	6	1		252.3.p.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.3.p.a	✓	32	3.b	odd	2	1
252.3.p.a	✓	32	63.s	even	6	1
252.3.bd.a	yes	32	9.d	odd	6	1
252.3.bd.a	yes	32	21.g	even	6	1
756.3.p.a		32	1.a	even	1	1	trivial
756.3.p.a		32	63.k	odd	6	1	inner
756.3.bd.a		32	7.d	odd	6	1
756.3.bd.a		32	9.c	even	3	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(756, [\chi])\).