Newform orbit 387.2.g.a

• Label: 387.2.g.a
• Level: $387$
• Weight: $2$
• Character orbit: 387.g
• Analytic conductor: $3.090$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	387.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.09021055822\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(42\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + q^{3} - 42 q^{4} - 10 q^{5} - 3 q^{6} - 6 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
178.1	−1.35683	+	2.35010i	−1.62705	+	0.593895i	−2.68199	−	4.64534i	−1.79547			0.811919	−	4.62955i	1.12179			9.12870			2.29458	−	1.93259i	2.43615	−	4.21953i
178.2	−1.34205	+	2.32450i	1.50349	+	0.859959i	−2.60220	−	4.50714i	2.54776			−4.01673	+	2.34075i	0.503300			8.60092			1.52094	+	2.58587i	−3.41922	+	5.92227i
178.3	−1.23721	+	2.14291i	1.28321	−	1.16335i	−2.06138	−	3.57042i	0.394496			0.905361	+	4.18911i	−4.52183			5.25261			0.293234	−	2.98563i	−0.488074	+	0.845369i
178.4	−1.12402	+	1.94685i	1.48161	−	0.897132i	−1.52683	−	2.64454i	−3.26640			0.0812361	+	3.89286i	2.37170			2.36866			1.39031	−	2.65839i	3.67149	−	6.35920i
178.5	−1.12334	+	1.94569i	1.04100	+	1.38431i	−1.52380	−	2.63930i	−2.41424			−3.86284	+	0.470397i	−0.234135			2.35364			−0.832648	+	2.88213i	2.71202	−	4.69736i
178.6	−1.11348	+	1.92861i	−0.551029	+	1.64206i	−1.47969	−	2.56290i	2.62048			−2.55333	−	2.89113i	4.18251			2.13651			−2.39273	−	1.80965i	−2.91786	+	5.05389i
178.7	−1.09085	+	1.88941i	−1.32687	−	1.11329i	−1.37992	−	2.39009i	−1.15954			3.55089	−	1.29257i	−3.44675			1.65775			0.521168	+	2.95438i	1.26489	−	2.19085i
178.8	−1.06090	+	1.83754i	−1.26225	+	1.18605i	−1.25103	−	2.16685i	0.844171			−0.840286	−	3.57772i	−2.65924			1.06528			0.186566	−	2.99419i	−0.895584	+	1.55120i
178.9	−1.02788	+	1.78034i	0.219741	−	1.71806i	−1.11308	−	1.92791i	2.47709			2.83286	+	2.15717i	2.23018			0.464922			−2.90343	−	0.755055i	−2.54615	+	4.41007i
178.10	−0.876392	+	1.51796i	−1.51412	−	0.841099i	−0.536127	−	0.928599i	−1.51554			2.60371	−	1.56123i	4.17682			−1.62614			1.58511	+	2.54705i	1.32821	−	2.30052i
178.11	−0.754398	+	1.30666i	−0.269833	+	1.71090i	−0.138233	−	0.239426i	−2.70250			−2.03200	−	1.64328i	−0.922861			−2.60046			−2.85438	−	0.923318i	2.03876	−	3.53123i
178.12	−0.713242	+	1.23537i	1.70792	−	0.288099i	−0.0174285	−	0.0301871i	3.46026			−0.862253	+	2.31540i	−0.126709			−2.80325			2.83400	−	0.984100i	−2.46800	+	4.27471i
178.13	−0.564408	+	0.977583i	0.634303	+	1.61173i	0.362887	+	0.628539i	2.56488			−1.93360	−	0.289587i	−2.28504			−3.07690			−2.19532	+	2.04464i	−1.44764	+	2.50739i
178.14	−0.563899	+	0.976702i	−0.404912	−	1.68406i	0.364035	+	0.630527i	−0.296732			1.87315	+	0.554160i	−2.79375			−3.07671			−2.67209	+	1.36379i	0.167327	−	0.289819i
178.15	−0.517340	+	0.896059i	−1.71049	+	0.272435i	0.464719	+	0.804916i	1.82857			0.640788	−	1.67364i	−2.60700			−3.03103			2.85156	−	0.931995i	−0.945992	+	1.63851i
178.16	−0.459591	+	0.796035i	1.72551	−	0.150431i	0.577552	+	1.00035i	−0.751580			−0.673279	+	1.44270i	3.47682			−2.90012			2.95474	−	0.519139i	0.345419	−	0.598284i
178.17	−0.412823	+	0.715030i	0.378967	−	1.69008i	0.659154	+	1.14169i	−2.84177			1.05202	+	0.968678i	0.541001			−2.73975			−2.71277	−	1.28097i	1.17315	−	2.03195i
178.18	−0.238922	+	0.413825i	1.68676	+	0.393513i	0.885833	+	1.53431i	−3.71924			−0.565849	+	0.604003i	−4.19510			−1.80227			2.69029	+	1.32752i	0.888608	−	1.53911i
178.19	−0.229061	+	0.396746i	1.07270	+	1.35990i	0.895062	+	1.55029i	−0.426499			−0.785246	+	0.114089i	4.16397			−1.73634			−0.698638	+	2.91752i	0.0976944	−	0.169212i
178.20	−0.101278	+	0.175419i	−1.70773	−	0.289232i	0.979485	+	1.69652i	−3.97981			0.223693	−	0.270276i	−0.501783			−0.801917			2.83269	+	0.987862i	0.403069	−	0.698136i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
387.g	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	387.2.g.a	yes	84
3.b	odd	2	1		1161.2.g.a		84
9.c	even	3	1		387.2.e.a	✓	84
9.d	odd	6	1		1161.2.e.a		84
43.c	even	3	1		387.2.e.a	✓	84
129.f	odd	6	1		1161.2.e.a		84
387.g	even	3	1	inner	387.2.g.a	yes	84
387.o	odd	6	1		1161.2.g.a		84

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
387.2.e.a	✓	84	9.c	even	3	1
387.2.e.a	✓	84	43.c	even	3	1
387.2.g.a	yes	84	1.a	even	1	1	trivial
387.2.g.a	yes	84	387.g	even	3	1	inner
1161.2.e.a		84	9.d	odd	6	1
1161.2.e.a		84	129.f	odd	6	1
1161.2.g.a		84	3.b	odd	2	1
1161.2.g.a		84	387.o	odd	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(387, [\chi])\).