Newform orbit 861.2.e.a

• Label: 861.2.e.a
• Level: $861$
• Weight: $2$
• Character orbit: 861.e
• Analytic conductor: $6.875$
• Analytic rank: $0$
• Dimension: $28$
• CM discriminant: -287
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 861 = 3 \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	861.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.87511961403\)
Analytic rank:	\(0\)
Dimension:	\(28\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 56 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
860.1		−	2.81328i	−0.318219	+	1.70257i	−5.91456				0		4.78980	+	0.895239i	2.64575					11.0128i	−2.79747	−	1.08358i		0
860.2		−	2.81328i	0.318219	−	1.70257i	−5.91456				0		−4.78980	−	0.895239i	−2.64575					11.0128i	−2.79747	−	1.08358i		0
860.3		−	2.66150i	−1.52953	−	0.812740i	−5.08361				0		−2.16311	+	4.07084i	2.64575					8.20704i	1.67891	+	2.48622i		0
860.4		−	2.66150i	1.52953	+	0.812740i	−5.08361				0		2.16311	−	4.07084i	−2.64575					8.20704i	1.67891	+	2.48622i		0
860.5		−	2.40786i	−1.13271	+	1.31033i	−3.79777				0		3.15508	+	2.72741i	−2.64575					4.32877i	−0.433914	−	2.96845i		0
860.6		−	2.40786i	1.13271	−	1.31033i	−3.79777				0		−3.15508	−	2.72741i	2.64575					4.32877i	−0.433914	−	2.96845i		0
860.7		−	1.98258i	−1.58907	−	0.689097i	−1.93064				0		−1.36619	+	3.15046i	2.64575				−	0.137518i	2.05029	+	2.19005i		0
860.8		−	1.98258i	1.58907	+	0.689097i	−1.93064				0		1.36619	−	3.15046i	−2.64575				−	0.137518i	2.05029	+	2.19005i		0
860.9		−	1.52552i	−1.73069	+	0.0686167i	−0.327220				0		0.104676	+	2.64021i	−2.64575				−	2.55186i	2.99058	−	0.237509i		0
860.10		−	1.52552i	1.73069	−	0.0686167i	−0.327220				0		−0.104676	−	2.64021i	2.64575				−	2.55186i	2.99058	−	0.237509i		0
860.11		−	0.910987i	−0.452011	+	1.67203i	1.17010				0		1.52320	+	0.411776i	2.64575				−	2.88792i	−2.59137	−	1.51155i		0
860.12		−	0.910987i	0.452011	−	1.67203i	1.17010				0		−1.52320	−	0.411776i	−2.64575				−	2.88792i	−2.59137	−	1.51155i		0
860.13		−	0.341042i	−1.02542	−	1.39589i	1.88369				0		−0.476057	+	0.349712i	−2.64575				−	1.32450i	−0.897021	+	2.86275i		0
860.14		−	0.341042i	1.02542	+	1.39589i	1.88369				0		0.476057	−	0.349712i	2.64575				−	1.32450i	−0.897021	+	2.86275i		0
860.15			0.341042i	−1.02542	+	1.39589i	1.88369				0		−0.476057	−	0.349712i	−2.64575					1.32450i	−0.897021	−	2.86275i		0
860.16			0.341042i	1.02542	−	1.39589i	1.88369				0		0.476057	+	0.349712i	2.64575					1.32450i	−0.897021	−	2.86275i		0
860.17			0.910987i	−0.452011	−	1.67203i	1.17010				0		1.52320	−	0.411776i	2.64575					2.88792i	−2.59137	+	1.51155i		0
860.18			0.910987i	0.452011	+	1.67203i	1.17010				0		−1.52320	+	0.411776i	−2.64575					2.88792i	−2.59137	+	1.51155i		0
860.19			1.52552i	−1.73069	−	0.0686167i	−0.327220				0		0.104676	−	2.64021i	−2.64575					2.55186i	2.99058	+	0.237509i		0
860.20			1.52552i	1.73069	+	0.0686167i	−0.327220				0		−0.104676	+	2.64021i	2.64575					2.55186i	2.99058	+	0.237509i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
287.d	odd	2	1	CM by \(\Q(\sqrt{-287}) \)
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner
41.b	even	2	1	inner
123.b	odd	2	1	inner
861.e	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	861.2.e.a	✓	28
3.b	odd	2	1	inner	861.2.e.a	✓	28
7.b	odd	2	1	inner	861.2.e.a	✓	28
21.c	even	2	1	inner	861.2.e.a	✓	28
41.b	even	2	1	inner	861.2.e.a	✓	28
123.b	odd	2	1	inner	861.2.e.a	✓	28
287.d	odd	2	1	CM	861.2.e.a	✓	28
861.e	even	2	1	inner	861.2.e.a	✓	28

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
861.2.e.a	✓	28	1.a	even	1	1	trivial
861.2.e.a	✓	28	3.b	odd	2	1	inner
861.2.e.a	✓	28	7.b	odd	2	1	inner
861.2.e.a	✓	28	21.c	even	2	1	inner
861.2.e.a	✓	28	41.b	even	2	1	inner
861.2.e.a	✓	28	123.b	odd	2	1	inner
861.2.e.a	✓	28	287.d	odd	2	1	CM
861.2.e.a	✓	28	861.e	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 28T_{2}^{12} + 308T_{2}^{10} + 1680T_{2}^{8} + 4704T_{2}^{6} + 6272T_{2}^{4} + 3136T_{2}^{2} + 287 \) acting on \(S_{2}^{\mathrm{new}}(861, [\chi])\).