Newform orbit 117.3.u.a

• Label: 117.3.u.a
• Level: $117$
• Weight: $3$
• Character orbit: 117.u
• Analytic conductor: $3.188$
• Analytic rank: $0$
• Dimension: $52$
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.18801909302\)
Analytic rank:	\(0\)
Dimension:	\(52\)
Relative dimension:	\(26\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 3 q^{2} - q^{3} + 49 q^{4} - 6 q^{5} - 3 q^{6} + 2 q^{7} - 3 q^{9}+O(q^{10}) \)

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	−3.34685	−	1.93231i	−2.94217	+	0.586194i	5.46761	+	9.47019i	0.468040	+	0.270223i	10.9797	+	3.72327i	0.553915				−	26.8020i	8.31275	−	3.44937i	−1.04431	−	1.80879i
68.2	−3.06533	−	1.76977i	2.49553	+	1.66503i	4.26415	+	7.38572i	−6.00755	−	3.46846i	−4.70290	−	9.52037i	4.78124				−	16.0281i	3.45535	+	8.31027i	12.2767	+	21.2639i
68.3	−2.83378	−	1.63609i	0.158993	−	2.99578i	3.35355	+	5.80853i	−3.48512	−	2.01214i	−5.35191	+	8.22928i	−2.80059				−	8.85812i	−8.94944	−	0.952618i	6.58406	+	11.4039i
68.4	−2.67828	−	1.54631i	1.65993	+	2.49893i	2.78213	+	4.81880i	6.70477	+	3.87100i	−0.581647	−	9.25959i	−12.6481				−	4.83767i	−3.48928	+	8.29608i	−11.9715	−	20.7353i
68.5	−2.39019	−	1.37998i	−1.29328	−	2.70692i	1.80866	+	3.13270i	7.95372	+	4.59208i	−0.644309	+	8.25474i	5.43975					1.05616i	−5.65486	+	7.00161i	−12.6739	−	21.9519i
68.6	−2.23352	−	1.28952i	−0.951185	+	2.84521i	1.32573	+	2.29623i	−0.817198	−	0.471810i	5.79345	−	5.12826i	5.04701					3.47795i	−7.19049	−	5.41265i	1.21682	+	2.10759i
68.7	−2.11714	−	1.22233i	2.89505	−	0.786578i	0.988182	+	1.71158i	2.73246	+	1.57759i	−7.09067	−	1.87341i	7.65083					4.94710i	7.76259	−	4.55436i	−3.85666	−	6.67994i
68.8	−1.61282	−	0.931162i	−2.77624	+	1.13689i	−0.265873	−	0.460506i	0.895843	+	0.517215i	5.53620	+	0.751532i	−5.11682					8.43958i	6.41497	−	6.31253i	−0.963222	−	1.66835i
68.9	−1.46109	−	0.843560i	2.47307	−	1.69821i	−0.576814	−	0.999071i	−3.08450	−	1.78084i	−5.04592	+	0.395052i	−12.7968					8.69479i	3.23216	−	8.39959i	3.00448	+	5.20392i
68.10	−1.44683	−	0.835327i	−2.64561	−	1.41449i	−0.604458	−	1.04695i	−8.08978	−	4.67064i	2.64618	+	4.25647i	8.35713					8.70230i	4.99845	+	7.48435i	7.80302	+	13.5152i
68.11	−0.545922	−	0.315188i	0.359457	+	2.97839i	−1.80131	−	3.11997i	1.96357	+	1.13367i	0.742517	−	1.73926i	8.19462					4.79251i	−8.74158	+	2.14120i	−0.714636	−	1.23779i
68.12	−0.422456	−	0.243905i	1.49896	+	2.59867i	−1.88102	−	3.25802i	−6.04718	−	3.49134i	0.000583198	−	1.46343i	−7.27212					3.78640i	−4.50621	+	7.79064i	1.70311	+	2.94987i
68.13	−0.376626	−	0.217445i	−2.32751	−	1.89280i	−1.90544	−	3.30031i	3.11857	+	1.80051i	0.465021	+	1.21898i	−7.89006					3.39687i	1.83461	+	8.81103i	−0.783024	−	1.35624i
68.14	−0.325651	−	0.188014i	0.894719	−	2.86347i	−1.92930	−	3.34165i	0.994641	+	0.574256i	−0.829740	+	0.764272i	5.14423					2.95506i	−7.39896	−	5.12401i	−0.215937	−	0.374014i
68.15	0.265125	+	0.153070i	2.88266	+	0.830829i	−1.95314	−	3.38294i	5.23834	+	3.02436i	0.637090	+	0.661523i	−0.615739				−	2.42043i	7.61944	+	4.79000i	0.925877	+	1.60367i
68.16	0.979822	+	0.565700i	2.79207	−	1.09742i	−1.35997	−	2.35553i	−7.06649	−	4.07984i	3.35654	+	0.504204i	7.94204				−	7.60294i	6.59135	−	6.12814i	−4.61594	−	7.99504i
68.17	1.00088	+	0.577857i	−2.23346	+	2.00292i	−1.33216	−	2.30737i	−3.81803	−	2.20434i	−3.39282	+	0.714055i	−4.15692				−	7.70206i	0.976660	−	8.94685i	−2.54759	−	4.41256i
68.18	1.10869	+	0.640101i	−2.99875	−	0.0866492i	−1.18054	−	2.04476i	2.95232	+	1.70452i	−3.26921	−	2.01557i	11.8984				−	8.14347i	8.98498	+	0.519678i	2.18213	+	3.77956i
68.19	1.61177	+	0.930557i	−1.33950	−	2.68435i	−0.268128	−	0.464411i	−4.76252	−	2.74964i	0.338963	−	5.57304i	−5.67067				−	8.44249i	−5.41146	+	7.19139i	−5.11740	−	8.86359i
68.20	1.64494	+	0.949706i	1.42058	−	2.64234i	−0.196117	−	0.339685i	5.82785	+	3.36471i	4.84621	−	2.99736i	−4.34657				−	8.34266i	−4.96393	−	7.50729i	6.39097	+	11.0695i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	117.3.u.a	yes	52
3.b	odd	2	1		351.3.u.a		52
9.c	even	3	1		351.3.k.a		52
9.d	odd	6	1		117.3.k.a	✓	52
13.c	even	3	1		117.3.k.a	✓	52
39.i	odd	6	1		351.3.k.a		52
117.f	even	3	1		351.3.u.a		52
117.u	odd	6	1	inner	117.3.u.a	yes	52

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
117.3.k.a	✓	52	9.d	odd	6	1
117.3.k.a	✓	52	13.c	even	3	1
117.3.u.a	yes	52	1.a	even	1	1	trivial
117.3.u.a	yes	52	117.u	odd	6	1	inner
351.3.k.a		52	9.c	even	3	1
351.3.k.a		52	39.i	odd	6	1
351.3.u.a		52	3.b	odd	2	1
351.3.u.a		52	117.f	even	3	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(117, [\chi])\).