Newform orbit 139.2.g.a

• Label: 139.2.g.a
• Level: $139$
• Weight: $2$
• Character orbit: 139.g
• Analytic conductor: $1.110$
• Analytic rank: $0$
• Dimension: $484$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	139.g (of order \(69\), degree \(44\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.10992058810\)
Analytic rank:	\(0\)
Dimension:	\(484\)
Relative dimension:	\(11\) over \(\Q(\zeta_{69})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{69}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 484 q - 46 q^{2} - 48 q^{3} - 32 q^{4} - 44 q^{5} - 52 q^{6} - 45 q^{7} - 46 q^{8} - 41 q^{9}+O(q^{10}) \)

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	−2.78541	−	0.126908i	0.322459	−	1.05759i	5.75070	+	0.525114i	0.579526	+	0.566481i	−1.03240	+	2.90489i	−0.696466	+	0.817339i	−10.4268	−	1.43312i	1.47515	+	0.991747i	−1.54233	−	1.65143i
4.2	−2.11602	−	0.0964097i	−0.914274	+	2.99859i	2.47653	+	0.226140i	2.14153	+	2.09332i	2.22371	−	6.25693i	−0.821629	+	0.964225i	−1.02161	−	0.140417i	−5.66600	−	3.80927i	−4.32970	−	4.63598i
4.3	−1.81355	−	0.0826288i	−0.336923	+	1.10502i	1.29044	+	0.117834i	−1.77714	−	1.73714i	0.702334	−	1.97618i	1.52979	−	1.79528i	1.26651	+	0.174078i	1.38210	+	0.929188i	3.07941	+	3.29724i
4.4	−1.57692	−	0.0718474i	0.823398	−	2.70054i	0.489805	+	0.0447257i	0.959934	+	0.938326i	−1.49246	+	4.19938i	2.06108	−	2.41878i	2.35854	+	0.324174i	−4.12527	−	2.77343i	−1.44632	−	1.54863i
4.5	−0.657902	−	0.0299752i	0.0342028	−	0.112177i	−1.55978	−	0.142428i	2.20801	+	2.15830i	−0.0258646	+	0.0727760i	−0.635631	+	0.745946i	2.32681	+	0.319813i	2.47824	+	1.66613i	−1.38796	−	1.48614i
4.6	−0.116135	−	0.00529132i	−0.491792	+	1.61296i	−1.97825	−	0.180641i	−1.86708	−	1.82506i	0.0656490	−	0.184719i	−2.58098	+	3.02892i	0.459134	+	0.0631066i	0.129889	+	0.0873245i	0.207177	+	0.221832i
4.7	0.529763	+	0.0241370i	0.454922	−	1.49203i	−1.71165	−	0.156296i	−1.75410	−	1.71462i	0.277014	−	0.779443i	2.01345	−	2.36289i	−1.95374	−	0.268536i	0.470450	+	0.316285i	−0.887872	−	0.950679i
4.8	1.33401	+	0.0607800i	0.929973	−	3.05008i	−0.215819	−	0.0197071i	1.85782	+	1.81600i	1.42598	−	4.01232i	−2.30102	+	2.70037i	−2.93262	−	0.403080i	−5.94848	−	3.99918i	2.36798	+	2.53548i
4.9	1.57533	+	0.0717750i	−0.285764	+	0.937234i	0.484807	+	0.0442693i	1.10653	+	1.08162i	−0.517443	+	1.45594i	1.53903	−	1.80613i	−2.36400	−	0.324925i	1.69291	+	1.13815i	1.66552	+	1.78334i
4.10	2.11442	+	0.0963370i	0.227249	−	0.745319i	2.46980	+	0.225525i	−0.797949	−	0.779988i	0.552302	−	1.55403i	−1.82988	+	2.14746i	1.00666	+	0.138363i	1.98580	+	1.33506i	−1.61206	−	1.72610i
4.11	2.51552	+	0.114612i	−0.888838	+	2.91517i	4.32298	+	0.394745i	−2.80273	−	2.73964i	−2.57000	+	7.23129i	1.00773	−	1.18262i	5.83994	+	0.802681i	−5.21852	−	3.50842i	−6.73631	−	7.21283i
5.1	−1.05774	+	2.59529i	0.237258	−	1.47623i	−4.18648	−	4.09225i	0.542067	−	1.77785i	3.58028	+	2.17721i	−2.92014	−	1.59844i	9.90768	−	4.30351i	0.725963	+	0.239539i	4.04065	+	3.28731i
5.2	−0.808674	+	1.98418i	−0.351241	+	2.18544i	−1.85279	−	1.81108i	−0.253138	+	0.830231i	−4.05226	−	2.46423i	0.866854	+	0.474503i	1.16130	−	0.504424i	−1.80385	−	0.595201i	−1.44262	−	1.17366i
5.3	−0.690750	+	1.69484i	0.134385	−	0.836149i	−0.965116	−	0.943392i	−0.856860	+	2.81029i	1.32431	+	0.805330i	−0.903824	−	0.494740i	−1.09180	+	0.474235i	2.16783	+	0.715300i	−4.17110	−	3.39344i
5.4	−0.493971	+	1.21202i	0.0160969	−	0.100156i	0.205243	+	0.200623i	1.21057	−	3.97036i	0.113439	+	0.0689837i	2.64906	+	1.45006i	−2.74546	+	1.19252i	2.83915	+	0.936808i	4.21416	+	3.42847i
5.5	−0.300788	+	0.738018i	0.501401	−	3.11974i	0.976022	+	0.954052i	0.560808	−	1.83931i	2.15161	+	1.30842i	−2.54545	−	1.39334i	−2.45964	+	1.06837i	−6.63244	−	2.18845i	1.18876	+	0.967128i
5.6	−0.0711969	+	0.174690i	0.214794	−	1.33646i	1.40477	+	1.37315i	−0.543392	+	1.78219i	0.218173	+	0.132674i	1.20686	+	0.660618i	−0.685939	+	0.297946i	1.10894	+	0.365907i	−0.272643	−	0.221812i
5.7	−0.0576037	+	0.141338i	−0.433530	+	2.69744i	1.41356	+	1.38174i	0.403457	−	1.32324i	−0.356277	−	0.216657i	−0.864406	−	0.473163i	−0.556698	+	0.241808i	−4.23934	−	1.39882i	0.163783	+	0.133247i
5.8	0.456268	−	1.11951i	0.0153297	−	0.0953820i	0.385103	+	0.376435i	0.442597	−	1.45161i	−0.0997864	−	0.0606815i	−2.89511	−	1.58474i	2.81479	−	1.22264i	2.84006	+	0.937108i	−1.42314	−	1.15781i
5.9	0.630379	−	1.54671i	−0.339182	+	2.11040i	−0.564712	−	0.552000i	−1.13281	+	3.71534i	3.05037	+	1.85497i	−0.288561	−	0.157954i	1.85415	−	0.805371i	−1.48984	−	0.491590i	5.03245	+	4.09420i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
139.g	even	69	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	139.2.g.a	✓	484
139.g	even	69	1	inner	139.2.g.a	✓	484

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
139.2.g.a	✓	484	1.a	even	1	1	trivial
139.2.g.a	✓	484	139.g	even	69	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(139, [\chi])\).