Newform orbit 117.3.k.a

• Label: 117.3.k.a
• Level: $117$
• Weight: $3$
• Character orbit: 117.k
• 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.k (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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - q^{3} - 98 q^{4} - 6 q^{5} + 12 q^{6} - q^{7} - 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1		−	3.86461i	1.97875	−	2.25490i	−10.9352			0.468040	−	0.270223i	−8.71431	−	7.64708i	−0.276958	−	0.479705i			26.8020i	−1.16913	−	8.92374i	−1.04431	−	1.80879i
29.2		−	3.53953i	0.194193	+	2.99371i	−8.52830			−6.00755	+	3.46846i	10.5963	−	0.687351i	−2.39062	−	4.14067i			16.0281i	−8.92458	+	1.16271i	12.2767	+	21.2639i
29.3		−	3.27217i	−2.67392	−	1.36020i	−6.70711			−3.48512	+	2.01214i	−4.45081	+	8.74953i	1.40029	+	2.42538i			8.85812i	5.29971	+	7.27414i	6.58406	+	11.4039i
29.4		−	3.09261i	1.33417	+	2.68700i	−5.56427			6.70477	−	3.87100i	8.30987	−	4.12608i	6.32403	+	10.9535i			4.83767i	−5.43998	+	7.16984i	−11.9715	−	20.7353i
29.5		−	2.75995i	−1.69762	−	2.47347i	−3.61733			7.95372	−	4.59208i	−6.82666	+	4.68536i	−2.71987	−	4.71096i		−	1.05616i	−3.23614	+	8.39806i	−12.6739	−	21.9519i
29.6		−	2.57904i	2.93962	+	0.598857i	−2.65146			−0.817198	+	0.471810i	1.54448	−	7.58141i	−2.52350	−	4.37084i		−	3.47795i	8.28274	+	3.52082i	1.21682	+	2.10759i
29.7		−	2.44466i	−2.12872	+	2.11389i	−1.97636			2.73246	−	1.57759i	5.16775	+	5.20400i	−3.82542	−	6.62581i		−	4.94710i	0.0628969	−	8.99978i	−3.85666	−	6.67994i
29.8		−	1.86232i	2.37269	−	1.83585i	0.531747			0.895843	−	0.517215i	−3.41894	−	4.41872i	2.55841	+	4.43129i		−	8.43958i	2.25933	−	8.71180i	−0.963222	−	1.66835i
29.9		−	1.68712i	−2.70723	+	1.29264i	1.15363			−3.08450	+	1.78084i	2.18083	+	4.56742i	6.39840	+	11.0823i		−	8.69479i	5.65818	−	6.99893i	3.00448	+	5.20392i
29.10		−	1.67065i	0.0978207	−	2.99840i	1.20892			−8.08978	+	4.67064i	−5.00930	−	0.163425i	−4.17857	−	7.23749i		−	8.70230i	−8.98086	−	0.586612i	7.80302	+	13.5152i
29.11		−	0.630376i	2.39963	+	1.80049i	3.60263			1.96357	−	1.13367i	1.13499	−	1.51267i	−4.09731	−	7.09675i		−	4.79251i	2.51645	+	8.64103i	−0.714636	−	1.23779i
29.12		−	0.487810i	1.50104	+	2.59748i	3.76204			−6.04718	+	3.49134i	1.26708	−	0.732220i	3.63606	+	6.29784i		−	3.78640i	−4.49379	+	7.79781i	1.70311	+	2.94987i
29.13		−	0.434891i	−0.475458	−	2.96208i	3.81087			3.11857	−	1.80051i	−1.28818	+	0.206772i	3.94503	+	6.83299i		−	3.39687i	−8.54788	+	2.81669i	−0.783024	−	1.35624i
29.14		−	0.376029i	−2.92720	−	0.656887i	3.85860			0.994641	−	0.574256i	−0.247009	+	1.10071i	−2.57212	−	4.45503i		−	2.95506i	8.13700	+	3.84568i	−0.215937	−	0.374014i
29.15			0.306140i	−0.721810	+	2.91187i	3.90628			5.23834	−	3.02436i	−0.891441	−	0.220975i	0.307870	+	0.533246i			2.42043i	−7.95798	−	4.20363i	0.925877	+	1.60367i
29.16			1.13140i	−2.34643	+	1.86930i	2.71993			−7.06649	+	4.07984i	−2.11493	−	2.65475i	−3.97102	−	6.87801i			7.60294i	2.01145	−	8.77235i	−4.61594	−	7.99504i
29.17			1.15571i	2.85130	−	0.932773i	2.66432			−3.81803	+	2.20434i	1.07802	+	3.29529i	2.07846	+	3.60000i			7.70206i	7.25987	−	5.31924i	−2.54759	−	4.41256i
29.18			1.28020i	1.42433	−	2.64032i	2.36108			2.95232	−	1.70452i	3.38014	+	1.82343i	−5.94919	−	10.3043i			8.14347i	−4.94255	−	7.52139i	2.18213	+	3.77956i
29.19			1.86111i	−1.65496	−	2.50222i	0.536256			−4.76252	+	2.74964i	4.65691	−	3.08007i	2.83533	+	4.91094i			8.44249i	−3.52220	+	8.28215i	−5.11740	−	8.86359i
29.20			1.89941i	−2.99862	−	0.0909164i	0.392234			5.82785	−	3.36471i	0.172688	−	5.69562i	2.17329	+	3.76424i			8.34266i	8.98347	+	0.545248i	6.39097	+	11.0695i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
117.k	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.k.a	✓	52
3.b	odd	2	1		351.3.k.a		52
9.c	even	3	1		351.3.u.a		52
9.d	odd	6	1		117.3.u.a	yes	52
13.c	even	3	1		117.3.u.a	yes	52
39.i	odd	6	1		351.3.u.a		52
117.h	even	3	1		351.3.k.a		52
117.k	odd	6	1	inner	117.3.k.a	✓	52

    

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

Hecke kernels

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