Newform orbit 352.2.s.b

• Label: 352.2.s.b
• Level: $352$
• Weight: $2$
• Character orbit: 352.s
• Analytic conductor: $2.811$
• Analytic rank: $0$
• Dimension: $32$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 352 = 2^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	352.s (of order \(10\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.81073415115\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{10})\)
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{3} - 10 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
79.1		0		−1.70387	+	1.23794i		0		−1.49538	+	0.485879i		0		−1.63043	−	1.18458i		0		0.443645	−	1.36540i		0
79.2		0		−1.70387	+	1.23794i		0		1.49538	−	0.485879i		0		1.63043	+	1.18458i		0		0.443645	−	1.36540i		0
79.3		0		−0.0248408	+	0.0180479i		0		−1.78547	+	0.580134i		0		−0.623146	−	0.452742i		0		−0.926760	+	2.85227i		0
79.4		0		−0.0248408	+	0.0180479i		0		1.78547	−	0.580134i		0		0.623146	+	0.452742i		0		−0.926760	+	2.85227i		0
79.5		0		0.903665	−	0.656551i		0		−3.74056	+	1.21538i		0		−2.25832	−	1.64076i		0		−0.541500	+	1.66657i		0
79.6		0		0.903665	−	0.656551i		0		3.74056	−	1.21538i		0		2.25832	+	1.64076i		0		−0.541500	+	1.66657i		0
79.7		0		1.63407	−	1.18722i		0		−1.62415	+	0.527718i		0		3.70164	+	2.68940i		0		0.333632	−	1.02681i		0
79.8		0		1.63407	−	1.18722i		0		1.62415	−	0.527718i		0		−3.70164	−	2.68940i		0		0.333632	−	1.02681i		0
239.1		0		−0.852681	−	2.62428i		0		−1.35292	+	1.86213i		0		−1.08979	+	3.35402i		0		−3.73274	+	2.71199i		0
239.2		0		−0.852681	−	2.62428i		0		1.35292	−	1.86213i		0		1.08979	−	3.35402i		0		−3.73274	+	2.71199i		0
239.3		0		−0.385688	−	1.18702i		0		−2.03454	+	2.80031i		0		0.442181	−	1.36089i		0		1.16678	−	0.847714i		0
239.4		0		−0.385688	−	1.18702i		0		2.03454	−	2.80031i		0		−0.442181	+	1.36089i		0		1.16678	−	0.847714i		0
239.5		0		0.303809	+	0.935028i		0		−0.398383	+	0.548327i		0		−1.40393	+	4.32085i		0		1.64507	−	1.19522i		0
239.6		0		0.303809	+	0.935028i		0		0.398383	−	0.548327i		0		1.40393	−	4.32085i		0		1.64507	−	1.19522i		0
239.7		0		0.625543	+	1.92522i		0		−1.75152	+	2.41076i		0		−0.216882	+	0.667493i		0		−0.888128	+	0.645263i		0
239.8		0		0.625543	+	1.92522i		0		1.75152	−	2.41076i		0		0.216882	−	0.667493i		0		−0.888128	+	0.645263i		0
271.1		0		−0.852681	+	2.62428i		0		−1.35292	−	1.86213i		0		−1.08979	−	3.35402i		0		−3.73274	−	2.71199i		0
271.2		0		−0.852681	+	2.62428i		0		1.35292	+	1.86213i		0		1.08979	+	3.35402i		0		−3.73274	−	2.71199i		0
271.3		0		−0.385688	+	1.18702i		0		−2.03454	−	2.80031i		0		0.442181	+	1.36089i		0		1.16678	+	0.847714i		0
271.4		0		−0.385688	+	1.18702i		0		2.03454	+	2.80031i		0		−0.442181	−	1.36089i		0		1.16678	+	0.847714i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
11.d	odd	10	1	inner
88.k	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	352.2.s.b		32
4.b	odd	2	1		88.2.k.b	✓	32
8.b	even	2	1		88.2.k.b	✓	32
8.d	odd	2	1	inner	352.2.s.b		32
11.c	even	5	1		3872.2.g.d		32
11.d	odd	10	1	inner	352.2.s.b		32
11.d	odd	10	1		3872.2.g.d		32
12.b	even	2	1		792.2.bp.b		32
24.h	odd	2	1		792.2.bp.b		32
44.c	even	2	1		968.2.k.h		32
44.g	even	10	1		88.2.k.b	✓	32
44.g	even	10	1		968.2.g.e		32
44.g	even	10	1		968.2.k.e		32
44.g	even	10	1		968.2.k.i		32
44.h	odd	10	1		968.2.g.e		32
44.h	odd	10	1		968.2.k.e		32
44.h	odd	10	1		968.2.k.h		32
44.h	odd	10	1		968.2.k.i		32
88.b	odd	2	1		968.2.k.h		32
88.k	even	10	1	inner	352.2.s.b		32
88.k	even	10	1		3872.2.g.d		32
88.l	odd	10	1		3872.2.g.d		32
88.o	even	10	1		968.2.g.e		32
88.o	even	10	1		968.2.k.e		32
88.o	even	10	1		968.2.k.h		32
88.o	even	10	1		968.2.k.i		32
88.p	odd	10	1		88.2.k.b	✓	32
88.p	odd	10	1		968.2.g.e		32
88.p	odd	10	1		968.2.k.e		32
88.p	odd	10	1		968.2.k.i		32
132.n	odd	10	1		792.2.bp.b		32
264.u	even	10	1		792.2.bp.b		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.2.k.b	✓	32	4.b	odd	2	1
88.2.k.b	✓	32	8.b	even	2	1
88.2.k.b	✓	32	44.g	even	10	1
88.2.k.b	✓	32	88.p	odd	10	1
352.2.s.b		32	1.a	even	1	1	trivial
352.2.s.b		32	8.d	odd	2	1	inner
352.2.s.b		32	11.d	odd	10	1	inner
352.2.s.b		32	88.k	even	10	1	inner
792.2.bp.b		32	12.b	even	2	1
792.2.bp.b		32	24.h	odd	2	1
792.2.bp.b		32	132.n	odd	10	1
792.2.bp.b		32	264.u	even	10	1
968.2.g.e		32	44.g	even	10	1
968.2.g.e		32	44.h	odd	10	1
968.2.g.e		32	88.o	even	10	1
968.2.g.e		32	88.p	odd	10	1
968.2.k.e		32	44.g	even	10	1
968.2.k.e		32	44.h	odd	10	1
968.2.k.e		32	88.o	even	10	1
968.2.k.e		32	88.p	odd	10	1
968.2.k.h		32	44.c	even	2	1
968.2.k.h		32	44.h	odd	10	1
968.2.k.h		32	88.b	odd	2	1
968.2.k.h		32	88.o	even	10	1
968.2.k.i		32	44.g	even	10	1
968.2.k.i		32	44.h	odd	10	1
968.2.k.i		32	88.o	even	10	1
968.2.k.i		32	88.p	odd	10	1
3872.2.g.d		32	11.c	even	5	1
3872.2.g.d		32	11.d	odd	10	1
3872.2.g.d		32	88.k	even	10	1
3872.2.g.d		32	88.l	odd	10	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^16 - T3^15 + 9*T3^14 - 16*T3^13 + 51*T3^12 - 58*T3^11 + 181*T3^10 - 288*T3^9 + 1012*T3^8 - 1432*T3^7 + 2281*T3^6 - 2517*T3^5 + 2741*T3^4 - 1659*T3^3 + 974*T3^2 + 51*T3 + 1