Newform orbit 117.2.l.b

• Label: 117.2.l.b
• Level: $117$
• Weight: $2$
• Character orbit: 117.l
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $22$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.l (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(22\)
Relative dimension:	\(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - q^{3} - 20 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} + 7 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1		−	2.59035i	1.72707	−	0.131225i	−4.70993			1.18696	+	0.685292i	−0.339918	−	4.47373i	−3.20825	−	1.85228i			7.01967i	2.96556	−	0.453269i	1.77515	−	3.07465i
4.2		−	2.31664i	−0.644502	−	1.60767i	−3.36682			−1.09505	−	0.632228i	−3.72440	+	1.49308i	3.61589	+	2.08764i			3.16642i	−2.16923	+	2.07230i	−1.46464	+	2.53684i
4.3		−	1.40574i	−1.72346	+	0.172331i	0.0239006			−2.61504	−	1.50979i	0.242252	+	2.42273i	−2.76663	−	1.59731i		−	2.84507i	2.94060	−	0.594010i	−2.12237	+	3.67605i
4.4		−	1.05773i	1.56468	+	0.742818i	0.881215			−2.71101	−	1.56520i	0.785698	−	1.65500i	0.784891	+	0.453157i		−	3.04754i	1.89644	+	2.32454i	−1.65555	+	2.86750i
4.5		−	0.968164i	−0.767957	−	1.55250i	1.06266			3.54737	+	2.04808i	−1.50307	+	0.743509i	−3.06763	−	1.77110i		−	2.96516i	−1.82048	+	2.38450i	1.98287	−	3.43444i
4.6			0.391710i	0.608332	−	1.62171i	1.84656			−1.60580	−	0.927107i	0.635238	+	0.238289i	−0.0712374	−	0.0411289i			1.50674i	−2.25987	−	1.97307i	0.363157	−	0.629006i
4.7			0.571953i	−0.656000	+	1.60302i	1.67287			0.796103	+	0.459630i	−0.916851	−	0.375201i	−1.67386	−	0.966405i			2.10071i	−2.13933	−	2.10316i	−0.262887	+	0.455333i
4.8			0.782672i	−1.67281	−	0.449109i	1.38742			0.0536139	+	0.0309540i	0.351505	−	1.30926i	3.25250	+	1.87783i			2.65124i	2.59660	+	1.50255i	−0.0242268	+	0.0419621i
4.9			1.93463i	0.820210	+	1.52553i	−1.74278			−2.26677	−	1.30872i	−2.95134	+	1.58680i	2.01692	+	1.16447i			0.497616i	−1.65451	+	2.50252i	2.53189	−	4.38536i
4.10			2.00617i	1.62924	−	0.587859i	−2.02472			0.778411	+	0.449416i	1.17935	+	3.26853i	−2.10878	−	1.21751i		−	0.0495935i	2.30884	−	1.91553i	−0.901605	+	1.56163i
4.11			2.65149i	−1.38481	+	1.04034i	−5.03038			2.43120	+	1.40366i	−2.75846	−	3.67179i	0.226187	+	0.130589i		−	8.03502i	0.835370	−	2.88135i	−3.72178	+	6.44631i
88.1		−	2.65149i	−1.38481	−	1.04034i	−5.03038			2.43120	−	1.40366i	−2.75846	+	3.67179i	0.226187	−	0.130589i			8.03502i	0.835370	+	2.88135i	−3.72178	−	6.44631i
88.2		−	2.00617i	1.62924	+	0.587859i	−2.02472			0.778411	−	0.449416i	1.17935	−	3.26853i	−2.10878	+	1.21751i			0.0495935i	2.30884	+	1.91553i	−0.901605	−	1.56163i
88.3		−	1.93463i	0.820210	−	1.52553i	−1.74278			−2.26677	+	1.30872i	−2.95134	−	1.58680i	2.01692	−	1.16447i		−	0.497616i	−1.65451	−	2.50252i	2.53189	+	4.38536i
88.4		−	0.782672i	−1.67281	+	0.449109i	1.38742			0.0536139	−	0.0309540i	0.351505	+	1.30926i	3.25250	−	1.87783i		−	2.65124i	2.59660	−	1.50255i	−0.0242268	−	0.0419621i
88.5		−	0.571953i	−0.656000	−	1.60302i	1.67287			0.796103	−	0.459630i	−0.916851	+	0.375201i	−1.67386	+	0.966405i		−	2.10071i	−2.13933	+	2.10316i	−0.262887	−	0.455333i
88.6		−	0.391710i	0.608332	+	1.62171i	1.84656			−1.60580	+	0.927107i	0.635238	−	0.238289i	−0.0712374	+	0.0411289i		−	1.50674i	−2.25987	+	1.97307i	0.363157	+	0.629006i
88.7			0.968164i	−0.767957	+	1.55250i	1.06266			3.54737	−	2.04808i	−1.50307	−	0.743509i	−3.06763	+	1.77110i			2.96516i	−1.82048	−	2.38450i	1.98287	+	3.43444i
88.8			1.05773i	1.56468	−	0.742818i	0.881215			−2.71101	+	1.56520i	0.785698	+	1.65500i	0.784891	−	0.453157i			3.04754i	1.89644	−	2.32454i	−1.65555	−	2.86750i
88.9			1.40574i	−1.72346	−	0.172331i	0.0239006			−2.61504	+	1.50979i	0.242252	−	2.42273i	−2.76663	+	1.59731i			2.84507i	2.94060	+	0.594010i	−2.12237	−	3.67605i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.l	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.2.l.b	✓	22
3.b	odd	2	1		351.2.l.b		22
9.c	even	3	1		117.2.r.b	yes	22
9.d	odd	6	1		351.2.r.b		22
13.e	even	6	1		117.2.r.b	yes	22
39.h	odd	6	1		351.2.r.b		22
117.l	even	6	1	inner	117.2.l.b	✓	22
117.v	odd	6	1		351.2.l.b		22

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.2.l.b	✓	22	1.a	even	1	1	trivial
117.2.l.b	✓	22	117.l	even	6	1	inner
117.2.r.b	yes	22	9.c	even	3	1
117.2.r.b	yes	22	13.e	even	6	1
351.2.l.b		22	3.b	odd	2	1
351.2.l.b		22	117.v	odd	6	1
351.2.r.b		22	9.d	odd	6	1
351.2.r.b		22	39.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^22 + 32*T2^20 + 432*T2^18 + 3212*T2^16 + 14428*T2^14 + 40524*T2^12 + 71556*T2^10 + 78408*T2^8 + 51660*T2^6 + 19224*T2^4 + 3564*T2^2 + 243