Newform orbit 207.2.g.b

• Label: 207.2.g.b
• Level: $207$
• Weight: $2$
• Character orbit: 207.g
• Analytic conductor: $1.653$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	207.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.65290332184\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 6 q^{2} - 6 q^{3} + 6 q^{4} + 6 q^{6} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
68.1	−1.75416	+	1.01277i	−0.0980475	−	1.72927i	1.05139	−	1.82106i	−1.64829	+	2.85493i	1.92334	+	2.93413i	2.39846	−	1.38475i			0.208179i	−2.98077	+	0.339102i		−	6.67733i
68.2	−1.75416	+	1.01277i	−0.0980475	−	1.72927i	1.05139	−	1.82106i	1.64829	−	2.85493i	1.92334	+	2.93413i	−2.39846	+	1.38475i			0.208179i	−2.98077	+	0.339102i			6.67733i
68.3	−1.66670	+	0.962270i	−0.603119	+	1.62365i	0.851929	−	1.47558i	−1.59706	+	2.76619i	−0.557174	−	3.28651i	−1.36206	+	0.786387i		−	0.569939i	−2.27250	−	1.95851i		−	6.14721i
68.4	−1.66670	+	0.962270i	−0.603119	+	1.62365i	0.851929	−	1.47558i	1.59706	−	2.76619i	−0.557174	−	3.28651i	1.36206	−	0.786387i		−	0.569939i	−2.27250	−	1.95851i			6.14721i
68.5	−1.10357	+	0.637149i	1.52320	+	0.824536i	−0.188082	+	0.325768i	−0.456305	+	0.790343i	−2.20632	+	0.0605692i	−4.24961	+	2.45351i		−	3.02794i	1.64028	+	2.51187i		−	1.16294i
68.6	−1.10357	+	0.637149i	1.52320	+	0.824536i	−0.188082	+	0.325768i	0.456305	−	0.790343i	−2.20632	+	0.0605692i	4.24961	−	2.45351i		−	3.02794i	1.64028	+	2.51187i			1.16294i
68.7	−0.883247	+	0.509943i	−1.71838	−	0.217204i	−0.479917	+	0.831241i	−1.23454	+	2.13829i	1.62851	−	0.684429i	0.487726	−	0.281589i		−	3.01869i	2.90564	+	0.746479i		−	2.51818i
68.8	−0.883247	+	0.509943i	−1.71838	−	0.217204i	−0.479917	+	0.831241i	1.23454	−	2.13829i	1.62851	−	0.684429i	−0.487726	+	0.281589i		−	3.01869i	2.90564	+	0.746479i			2.51818i
68.9	0.107387	−	0.0619998i	−0.547823	−	1.64313i	−0.992312	+	1.71873i	−0.778677	+	1.34871i	−0.160703	−	0.142486i	−3.42634	+	1.97820i			0.494091i	−2.39978	+	1.80029i			0.193111i
68.10	0.107387	−	0.0619998i	−0.547823	−	1.64313i	−0.992312	+	1.71873i	0.778677	−	1.34871i	−0.160703	−	0.142486i	3.42634	−	1.97820i			0.494091i	−2.39978	+	1.80029i		−	0.193111i
68.11	0.378110	−	0.218302i	1.67461	−	0.442366i	−0.904689	+	1.56697i	−1.99023	+	3.44717i	0.536617	−	0.532833i	1.56036	−	0.900876i			1.66319i	2.60863	−	1.48158i			1.73788i
68.12	0.378110	−	0.218302i	1.67461	−	0.442366i	−0.904689	+	1.56697i	1.99023	−	3.44717i	0.536617	−	0.532833i	−1.56036	+	0.900876i			1.66319i	2.60863	−	1.48158i		−	1.73788i
68.13	1.27130	−	0.733984i	−1.57305	+	0.724930i	0.0774660	−	0.134175i	−1.34029	+	2.32145i	−1.46772	+	2.07619i	−3.08525	+	1.78127i			2.70850i	1.94895	−	2.28070i			3.93501i
68.14	1.27130	−	0.733984i	−1.57305	+	0.724930i	0.0774660	−	0.134175i	1.34029	−	2.32145i	−1.46772	+	2.07619i	3.08525	−	1.78127i			2.70850i	1.94895	−	2.28070i		−	3.93501i
68.15	2.15089	−	1.24182i	−0.157396	+	1.72488i	2.08422	−	3.60997i	−1.77102	+	3.06750i	1.80345	+	3.90549i	3.95624	−	2.28414i		−	5.38559i	−2.95045	−	0.542979i			8.79713i
68.16	2.15089	−	1.24182i	−0.157396	+	1.72488i	2.08422	−	3.60997i	1.77102	−	3.06750i	1.80345	+	3.90549i	−3.95624	+	2.28414i		−	5.38559i	−2.95045	−	0.542979i		−	8.79713i
137.1	−1.75416	−	1.01277i	−0.0980475	+	1.72927i	1.05139	+	1.82106i	−1.64829	−	2.85493i	1.92334	−	2.93413i	2.39846	+	1.38475i		−	0.208179i	−2.98077	−	0.339102i			6.67733i
137.2	−1.75416	−	1.01277i	−0.0980475	+	1.72927i	1.05139	+	1.82106i	1.64829	+	2.85493i	1.92334	−	2.93413i	−2.39846	−	1.38475i		−	0.208179i	−2.98077	−	0.339102i		−	6.67733i
137.3	−1.66670	−	0.962270i	−0.603119	−	1.62365i	0.851929	+	1.47558i	−1.59706	−	2.76619i	−0.557174	+	3.28651i	−1.36206	−	0.786387i			0.569939i	−2.27250	+	1.95851i			6.14721i
137.4	−1.66670	−	0.962270i	−0.603119	−	1.62365i	0.851929	+	1.47558i	1.59706	+	2.76619i	−0.557174	+	3.28651i	1.36206	+	0.786387i			0.569939i	−2.27250	+	1.95851i		−	6.14721i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.d	odd	6	1	inner
23.b	odd	2	1	inner
207.g	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	207.2.g.b	✓	32
3.b	odd	2	1		621.2.g.b		32
9.c	even	3	1		621.2.g.b		32
9.c	even	3	1		1863.2.c.b		32
9.d	odd	6	1	inner	207.2.g.b	✓	32
9.d	odd	6	1		1863.2.c.b		32
23.b	odd	2	1	inner	207.2.g.b	✓	32
69.c	even	2	1		621.2.g.b		32
207.f	odd	6	1		621.2.g.b		32
207.f	odd	6	1		1863.2.c.b		32
207.g	even	6	1	inner	207.2.g.b	✓	32
207.g	even	6	1		1863.2.c.b		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
207.2.g.b	✓	32	1.a	even	1	1	trivial
207.2.g.b	✓	32	9.d	odd	6	1	inner
207.2.g.b	✓	32	23.b	odd	2	1	inner
207.2.g.b	✓	32	207.g	even	6	1	inner
621.2.g.b		32	3.b	odd	2	1
621.2.g.b		32	9.c	even	3	1
621.2.g.b		32	69.c	even	2	1
621.2.g.b		32	207.f	odd	6	1
1863.2.c.b		32	9.c	even	3	1
1863.2.c.b		32	9.d	odd	6	1
1863.2.c.b		32	207.f	odd	6	1
1863.2.c.b		32	207.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^16 + 3*T2^15 - 5*T2^14 - 24*T2^13 + 25*T2^12 + 165*T2^11 + 94*T2^10 - 312*T2^9 - 278*T2^8 + 489*T2^7 + 760*T2^6 + 78*T2^5 - 266*T2^4 - 15*T2^3 + 76*T2^2 - 15*T2 + 1