Newform orbit 252.8.t.a

• Label: 252.8.t.a
• Level: $252$
• Weight: $8$
• Character orbit: 252.t
• Analytic conductor: $78.721$
• Analytic rank: $0$
• Dimension: $36$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(78.7210264220\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(18\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2634 q^{7}+O(q^{10}) \)

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		−275.537	−	477.244i		0		877.605	−	230.982i		0			0			0
17.2		0			0			0		−219.318	−	379.869i		0		−692.530	+	586.468i		0			0			0
17.3		0			0			0		−175.339	−	303.695i		0		−205.988	−	883.805i		0			0			0
17.4		0			0			0		−155.757	−	269.780i		0		895.574	+	146.596i		0			0			0
17.5		0			0			0		−148.283	−	256.835i		0		−446.098	−	790.278i		0			0			0
17.6		0			0			0		−116.642	−	202.030i		0		−84.3560	+	903.564i		0			0			0
17.7		0			0			0		−67.6516	−	117.176i		0		−907.423	−	11.2432i		0			0			0
17.8		0			0			0		−34.8304	−	60.3281i		0		734.248	+	533.313i		0			0			0
17.9		0			0			0		−21.6450	−	37.4903i		0		487.468	−	765.452i		0			0			0
17.10		0			0			0		21.6450	+	37.4903i		0		487.468	−	765.452i		0			0			0
17.11		0			0			0		34.8304	+	60.3281i		0		734.248	+	533.313i		0			0			0
17.12		0			0			0		67.6516	+	117.176i		0		−907.423	−	11.2432i		0			0			0
17.13		0			0			0		116.642	+	202.030i		0		−84.3560	+	903.564i		0			0			0
17.14		0			0			0		148.283	+	256.835i		0		−446.098	−	790.278i		0			0			0
17.15		0			0			0		155.757	+	269.780i		0		895.574	+	146.596i		0			0			0
17.16		0			0			0		175.339	+	303.695i		0		−205.988	−	883.805i		0			0			0
17.17		0			0			0		219.318	+	379.869i		0		−692.530	+	586.468i		0			0			0
17.18		0			0			0		275.537	+	477.244i		0		877.605	−	230.982i		0			0			0
89.1		0			0			0		−275.537	+	477.244i		0		877.605	+	230.982i		0			0			0
89.2		0			0			0		−219.318	+	379.869i		0		−692.530	−	586.468i		0			0			0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	252.8.t.a	✓	36
3.b	odd	2	1	inner	252.8.t.a	✓	36
7.d	odd	6	1	inner	252.8.t.a	✓	36
21.g	even	6	1	inner	252.8.t.a	✓	36

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
252.8.t.a	✓	36	1.a	even	1	1	trivial
252.8.t.a	✓	36	3.b	odd	2	1	inner
252.8.t.a	✓	36	7.d	odd	6	1	inner
252.8.t.a	✓	36	21.g	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(252, [\chi])\).