Newform orbit 693.2.bg.b

• Label: 693.2.bg.b
• Level: $693$
• Weight: $2$
• Character orbit: 693.bg
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.bg (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	no (minimal twist has level 231)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 12 q^{4} + 12 q^{5}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
10.1	−2.23994	+	1.29323i		0		2.34487	−	4.06144i	1.30115	−	0.751219i		0		−1.55637	−	2.13956i			6.95692i		0		−1.94300	+	3.36537i
10.2	−2.16769	+	1.25152i		0		2.13259	−	3.69375i	0.849508	−	0.490464i		0		−1.62716	+	2.08623i			5.66981i		0		−1.22765	+	2.12635i
10.3	−1.71613	+	0.990810i		0		0.963407	−	1.66867i	−0.315938	+	0.182407i		0		−0.143176	−	2.64187i		−	0.145025i		0		0.361461	−	0.626069i
10.4	−1.49489	+	0.863074i		0		0.489794	−	0.848348i	−1.43342	+	0.827587i		0		2.00786	−	1.72293i		−	1.76138i		0		1.42854	−	2.47430i
10.5	−0.931635	+	0.537879i		0		−0.421371	+	0.729836i	2.91250	−	1.68153i		0		−0.879349	+	2.49534i		−	3.05811i		0		−1.80893	+	3.13315i
10.6	−0.567337	+	0.327552i		0		−0.785419	+	1.36039i	2.94602	−	1.70089i		0		−2.64073	−	0.162880i		−	2.33927i		0		−1.11426	+	1.92995i
10.7	−0.533044	+	0.307753i		0		−0.810576	+	1.40396i	−2.84562	+	1.64292i		0		−0.613452	+	2.57365i		−	2.22884i		0		1.01123	−	1.75150i
10.8	−0.360631	+	0.208210i		0		−0.913297	+	1.58188i	−0.414204	+	0.239141i		0		2.50100	+	0.863126i		−	1.59347i		0		0.0995832	−	0.172483i
10.9	0.360631	−	0.208210i		0		−0.913297	+	1.58188i	−0.414204	+	0.239141i		0		−2.50100	−	0.863126i			1.59347i		0		−0.0995832	+	0.172483i
10.10	0.533044	−	0.307753i		0		−0.810576	+	1.40396i	−2.84562	+	1.64292i		0		0.613452	−	2.57365i			2.22884i		0		−1.01123	+	1.75150i
10.11	0.567337	−	0.327552i		0		−0.785419	+	1.36039i	2.94602	−	1.70089i		0		2.64073	+	0.162880i			2.33927i		0		1.11426	−	1.92995i
10.12	0.931635	−	0.537879i		0		−0.421371	+	0.729836i	2.91250	−	1.68153i		0		0.879349	−	2.49534i			3.05811i		0		1.80893	−	3.13315i
10.13	1.49489	−	0.863074i		0		0.489794	−	0.848348i	−1.43342	+	0.827587i		0		−2.00786	+	1.72293i			1.76138i		0		−1.42854	+	2.47430i
10.14	1.71613	−	0.990810i		0		0.963407	−	1.66867i	−0.315938	+	0.182407i		0		0.143176	+	2.64187i			0.145025i		0		−0.361461	+	0.626069i
10.15	2.16769	−	1.25152i		0		2.13259	−	3.69375i	0.849508	−	0.490464i		0		1.62716	−	2.08623i		−	5.66981i		0		1.22765	−	2.12635i
10.16	2.23994	−	1.29323i		0		2.34487	−	4.06144i	1.30115	−	0.751219i		0		1.55637	+	2.13956i		−	6.95692i		0		1.94300	−	3.36537i
208.1	−2.23994	−	1.29323i		0		2.34487	+	4.06144i	1.30115	+	0.751219i		0		−1.55637	+	2.13956i		−	6.95692i		0		−1.94300	−	3.36537i
208.2	−2.16769	−	1.25152i		0		2.13259	+	3.69375i	0.849508	+	0.490464i		0		−1.62716	−	2.08623i		−	5.66981i		0		−1.22765	−	2.12635i
208.3	−1.71613	−	0.990810i		0		0.963407	+	1.66867i	−0.315938	−	0.182407i		0		−0.143176	+	2.64187i			0.145025i		0		0.361461	+	0.626069i
208.4	−1.49489	−	0.863074i		0		0.489794	+	0.848348i	−1.43342	−	0.827587i		0		2.00786	+	1.72293i			1.76138i		0		1.42854	+	2.47430i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
11.b	odd	2	1	inner
77.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.bg.b		32
3.b	odd	2	1		231.2.p.a	✓	32
7.d	odd	6	1	inner	693.2.bg.b		32
11.b	odd	2	1	inner	693.2.bg.b		32
21.g	even	6	1		231.2.p.a	✓	32
21.g	even	6	1		1617.2.c.a		32
21.h	odd	6	1		1617.2.c.a		32
33.d	even	2	1		231.2.p.a	✓	32
77.i	even	6	1	inner	693.2.bg.b		32
231.k	odd	6	1		231.2.p.a	✓	32
231.k	odd	6	1		1617.2.c.a		32
231.l	even	6	1		1617.2.c.a		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
231.2.p.a	✓	32	3.b	odd	2	1
231.2.p.a	✓	32	21.g	even	6	1
231.2.p.a	✓	32	33.d	even	2	1
231.2.p.a	✓	32	231.k	odd	6	1
693.2.bg.b		32	1.a	even	1	1	trivial
693.2.bg.b		32	7.d	odd	6	1	inner
693.2.bg.b		32	11.b	odd	2	1	inner
693.2.bg.b		32	77.i	even	6	1	inner
1617.2.c.a		32	21.g	even	6	1
1617.2.c.a		32	21.h	odd	6	1
1617.2.c.a		32	231.k	odd	6	1
1617.2.c.a		32	231.l	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^32 - 22*T2^30 + 297*T2^28 - 2562*T2^26 + 16250*T2^24 - 74382*T2^22 + 257705*T2^20 - 647554*T2^18 + 1208706*T2^16 - 1539134*T2^14 + 1427009*T2^12 - 877842*T2^10 + 395177*T2^8 - 119784*T2^6 + 25968*T2^4 - 3200*T2^2 + 256