Newform orbit 111.3.f.a

• Label: 111.3.f.a
• Level: $111$
• Weight: $3$
• Character orbit: 111.f
• Analytic conductor: $3.025$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	111.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.02453093440\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 8 q^{2} + 20 q^{5} - 60 q^{8} - 72 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−2.15816	−	2.15816i			1.73205i			5.31533i	2.28543	−	2.28543i	3.73805	−	3.73805i	−10.9759			2.83869	−	2.83869i	−3.00000			−9.86464
31.2	−1.70321	−	1.70321i		−	1.73205i			1.80182i	5.66686	−	5.66686i	−2.95004	+	2.95004i	10.9534			−3.74395	+	3.74395i	−3.00000			−19.3037
31.3	−1.67294	−	1.67294i		−	1.73205i			1.59747i	−4.04481	+	4.04481i	−2.89762	+	2.89762i	−2.77594			−4.01929	+	4.01929i	−3.00000			13.5335
31.4	−0.921681	−	0.921681i			1.73205i		−	2.30101i	2.11453	−	2.11453i	1.59640	−	1.59640i	3.42289			−5.80752	+	5.80752i	−3.00000			−3.89784
31.5	−0.284993	−	0.284993i			1.73205i		−	3.83756i	−5.88958	+	5.88958i	0.493622	−	0.493622i	−7.94417			−2.23365	+	2.23365i	−3.00000			3.35698
31.6	−0.123529	−	0.123529i		−	1.73205i		−	3.96948i	1.06915	−	1.06915i	−0.213958	+	0.213958i	−7.73043			−0.984460	+	0.984460i	−3.00000			−0.264141
31.7	0.746615	+	0.746615i			1.73205i		−	2.88513i	3.97615	−	3.97615i	−1.29318	+	1.29318i	4.59754			5.14054	−	5.14054i	−3.00000			5.93731
31.8	1.06504	+	1.06504i		−	1.73205i		−	1.73137i	−0.563895	+	0.563895i	1.84471	−	1.84471i	6.70948			6.10415	−	6.10415i	−3.00000			−1.20114
31.9	1.81870	+	1.81870i		−	1.73205i			2.61536i	5.47037	−	5.47037i	3.15009	−	3.15009i	−2.07266			2.51825	−	2.51825i	−3.00000			19.8979
31.10	2.00072	+	2.00072i			1.73205i			4.00579i	−3.43593	+	3.43593i	−3.46535	+	3.46535i	2.85072			−0.0115823	+	0.0115823i	−3.00000			−13.7487
31.11	2.61593	+	2.61593i		−	1.73205i			9.68620i	−2.59767	+	2.59767i	4.53093	−	4.53093i	8.77260			−14.8747	+	14.8747i	−3.00000			−13.5907
31.12	2.61750	+	2.61750i			1.73205i			9.70258i	5.94940	−	5.94940i	−4.53364	+	4.53364i	−5.80748			−14.9265	+	14.9265i	−3.00000			31.1451
43.1	−2.15816	+	2.15816i		−	1.73205i		−	5.31533i	2.28543	+	2.28543i	3.73805	+	3.73805i	−10.9759			2.83869	+	2.83869i	−3.00000			−9.86464
43.2	−1.70321	+	1.70321i			1.73205i		−	1.80182i	5.66686	+	5.66686i	−2.95004	−	2.95004i	10.9534			−3.74395	−	3.74395i	−3.00000			−19.3037
43.3	−1.67294	+	1.67294i			1.73205i		−	1.59747i	−4.04481	−	4.04481i	−2.89762	−	2.89762i	−2.77594			−4.01929	−	4.01929i	−3.00000			13.5335
43.4	−0.921681	+	0.921681i		−	1.73205i			2.30101i	2.11453	+	2.11453i	1.59640	+	1.59640i	3.42289			−5.80752	−	5.80752i	−3.00000			−3.89784
43.5	−0.284993	+	0.284993i		−	1.73205i			3.83756i	−5.88958	−	5.88958i	0.493622	+	0.493622i	−7.94417			−2.23365	−	2.23365i	−3.00000			3.35698
43.6	−0.123529	+	0.123529i			1.73205i			3.96948i	1.06915	+	1.06915i	−0.213958	−	0.213958i	−7.73043			−0.984460	−	0.984460i	−3.00000			−0.264141
43.7	0.746615	−	0.746615i		−	1.73205i			2.88513i	3.97615	+	3.97615i	−1.29318	−	1.29318i	4.59754			5.14054	+	5.14054i	−3.00000			5.93731
43.8	1.06504	−	1.06504i			1.73205i			1.73137i	−0.563895	−	0.563895i	1.84471	+	1.84471i	6.70948			6.10415	+	6.10415i	−3.00000			−1.20114

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.d	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	111.3.f.a	✓	24
3.b	odd	2	1		333.3.i.b		24
37.d	odd	4	1	inner	111.3.f.a	✓	24
111.g	even	4	1		333.3.i.b		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
111.3.f.a	✓	24	1.a	even	1	1	trivial
111.3.f.a	✓	24	37.d	odd	4	1	inner
333.3.i.b		24	3.b	odd	2	1
333.3.i.b		24	111.g	even	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(111, [\chi])\).