Newform orbit 816.2.cj.c

• Label: 816.2.cj.c
• Level: $816$
• Weight: $2$
• Character orbit: 816.cj
• Analytic conductor: $6.516$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.cj (of order \(16\), degree \(8\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.51579280494\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(4\) over \(\Q(\zeta_{16})\)
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{3} + 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
65.1		0		−0.802099	−	1.53513i		0		−0.0781694	−	0.116989i		0		1.47102	+	0.982905i		0		−1.71327	+	2.46266i		0
65.2		0		−0.605951	+	1.62260i		0		1.28940	+	1.92973i		0		−0.0883372	−	0.0590250i		0		−2.26565	−	1.96643i		0
65.3		0		−0.0611151	+	1.73097i		0		−1.28940	−	1.92973i		0		−0.0883372	−	0.0590250i		0		−2.99253	−	0.211577i		0
65.4		0		1.32851	−	1.11133i		0		0.0781694	+	0.116989i		0		1.47102	+	0.982905i		0		0.529896	−	2.95283i		0
113.1		0		−0.802099	+	1.53513i		0		−0.0781694	+	0.116989i		0		1.47102	−	0.982905i		0		−1.71327	−	2.46266i		0
113.2		0		−0.605951	−	1.62260i		0		1.28940	−	1.92973i		0		−0.0883372	+	0.0590250i		0		−2.26565	+	1.96643i		0
113.3		0		−0.0611151	−	1.73097i		0		−1.28940	+	1.92973i		0		−0.0883372	+	0.0590250i		0		−2.99253	+	0.211577i		0
113.4		0		1.32851	+	1.11133i		0		0.0781694	−	0.116989i		0		1.47102	−	0.982905i		0		0.529896	+	2.95283i		0
209.1		0		−1.73140	+	0.0473558i		0		−3.00087	−	0.596910i		0		0.464975	+	2.33759i		0		2.99551	−	0.163984i		0
209.2		0		−1.10321	+	1.33527i		0		2.03727	+	0.405238i		0		−0.388855	−	1.95490i		0		−0.565877	−	2.94615i		0
209.3		0		0.706330	−	1.58149i		0		3.00087	+	0.596910i		0		0.464975	+	2.33759i		0		−2.00219	−	2.23410i		0
209.4		0		1.65580	−	0.508244i		0		−2.03727	−	0.405238i		0		−0.388855	−	1.95490i		0		2.48338	−	1.68311i		0
401.1		0		−1.60072	+	0.661591i		0		0.159652	+	0.802626i		0		−0.191449	−	0.0380817i		0		2.12459	−	2.11804i		0
401.2		0		−0.00133765	−	1.73205i		0		−0.159652	−	0.802626i		0		−0.191449	−	0.0380817i		0		−3.00000	+	0.00463376i		0
401.3		0		0.961322	+	1.44078i		0		0.595296	+	2.99276i		0		2.11533	+	0.420765i		0		−1.15172	+	2.77012i		0
401.4		0		1.69899	+	0.336782i		0		−0.595296	−	2.99276i		0		2.11533	+	0.420765i		0		2.77316	+	1.14438i		0
449.1		0		−1.73140	−	0.0473558i		0		−3.00087	+	0.596910i		0		0.464975	−	2.33759i		0		2.99551	+	0.163984i		0
449.2		0		−1.10321	−	1.33527i		0		2.03727	−	0.405238i		0		−0.388855	+	1.95490i		0		−0.565877	+	2.94615i		0
449.3		0		0.706330	+	1.58149i		0		3.00087	−	0.596910i		0		0.464975	−	2.33759i		0		−2.00219	+	2.23410i		0
449.4		0		1.65580	+	0.508244i		0		−2.03727	+	0.405238i		0		−0.388855	+	1.95490i		0		2.48338	+	1.68311i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
17.e	odd	16	1	inner
51.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	816.2.cj.c		32
3.b	odd	2	1	inner	816.2.cj.c		32
4.b	odd	2	1		51.2.i.a	✓	32
12.b	even	2	1		51.2.i.a	✓	32
17.e	odd	16	1	inner	816.2.cj.c		32
51.i	even	16	1	inner	816.2.cj.c		32
68.d	odd	2	1		867.2.i.h		32
68.f	odd	4	1		867.2.i.c		32
68.f	odd	4	1		867.2.i.d		32
68.g	odd	8	1		867.2.i.b		32
68.g	odd	8	1		867.2.i.f		32
68.g	odd	8	1		867.2.i.g		32
68.g	odd	8	1		867.2.i.i		32
68.i	even	16	1		51.2.i.a	✓	32
68.i	even	16	1		867.2.i.b		32
68.i	even	16	1		867.2.i.c		32
68.i	even	16	1		867.2.i.d		32
68.i	even	16	1		867.2.i.f		32
68.i	even	16	1		867.2.i.g		32
68.i	even	16	1		867.2.i.h		32
68.i	even	16	1		867.2.i.i		32
204.h	even	2	1		867.2.i.h		32
204.l	even	4	1		867.2.i.c		32
204.l	even	4	1		867.2.i.d		32
204.p	even	8	1		867.2.i.b		32
204.p	even	8	1		867.2.i.f		32
204.p	even	8	1		867.2.i.g		32
204.p	even	8	1		867.2.i.i		32
204.t	odd	16	1		51.2.i.a	✓	32
204.t	odd	16	1		867.2.i.b		32
204.t	odd	16	1		867.2.i.c		32
204.t	odd	16	1		867.2.i.d		32
204.t	odd	16	1		867.2.i.f		32
204.t	odd	16	1		867.2.i.g		32
204.t	odd	16	1		867.2.i.h		32
204.t	odd	16	1		867.2.i.i		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
51.2.i.a	✓	32	4.b	odd	2	1
51.2.i.a	✓	32	12.b	even	2	1
51.2.i.a	✓	32	68.i	even	16	1
51.2.i.a	✓	32	204.t	odd	16	1
816.2.cj.c		32	1.a	even	1	1	trivial
816.2.cj.c		32	3.b	odd	2	1	inner
816.2.cj.c		32	17.e	odd	16	1	inner
816.2.cj.c		32	51.i	even	16	1	inner
867.2.i.b		32	68.g	odd	8	1
867.2.i.b		32	68.i	even	16	1
867.2.i.b		32	204.p	even	8	1
867.2.i.b		32	204.t	odd	16	1
867.2.i.c		32	68.f	odd	4	1
867.2.i.c		32	68.i	even	16	1
867.2.i.c		32	204.l	even	4	1
867.2.i.c		32	204.t	odd	16	1
867.2.i.d		32	68.f	odd	4	1
867.2.i.d		32	68.i	even	16	1
867.2.i.d		32	204.l	even	4	1
867.2.i.d		32	204.t	odd	16	1
867.2.i.f		32	68.g	odd	8	1
867.2.i.f		32	68.i	even	16	1
867.2.i.f		32	204.p	even	8	1
867.2.i.f		32	204.t	odd	16	1
867.2.i.g		32	68.g	odd	8	1
867.2.i.g		32	68.i	even	16	1
867.2.i.g		32	204.p	even	8	1
867.2.i.g		32	204.t	odd	16	1
867.2.i.h		32	68.d	odd	2	1
867.2.i.h		32	68.i	even	16	1
867.2.i.h		32	204.h	even	2	1
867.2.i.h		32	204.t	odd	16	1
867.2.i.i		32	68.g	odd	8	1
867.2.i.i		32	68.i	even	16	1
867.2.i.i		32	204.p	even	8	1
867.2.i.i		32	204.t	odd	16	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^32 - 8*T5^30 - 52*T5^28 + 656*T5^26 + 456*T5^24 - 27912*T5^22 + 352752*T5^20 - 1443704*T5^18 + 10227589*T5^16 - 61607408*T5^14 + 108290404*T5^12 + 183041688*T5^10 + 61746760*T5^8 - 4108752*T5^6 + 2895864*T5^4 + 42704*T5^2 + 1156