Newform orbit 390.2.bj.a

• Label: 390.2.bj.a
• Level: $390$
• Weight: $2$
• Character orbit: 390.bj
• Analytic conductor: $3.114$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.bj (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.11416567883\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(28\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−0.965926	+	0.258819i	−1.63384	−	0.574960i	0.866025	−	0.500000i	−1.64463	−	1.51499i	1.72698	+	0.132501i	−0.693378	+	2.58772i	−0.707107	+	0.707107i	2.33884	+	1.87878i	1.98070	+	1.03770i
59.2	−0.965926	+	0.258819i	−1.47500	−	0.907950i	0.866025	−	0.500000i	−1.09695	+	1.94851i	1.65974	+	0.495254i	0.391563	−	1.46133i	−0.707107	+	0.707107i	1.35125	+	2.67845i	0.555259	−	2.16603i
59.3	−0.965926	+	0.258819i	−1.43885	+	0.964217i	0.866025	−	0.500000i	2.20603	+	0.365292i	1.14026	−	1.30376i	0.366472	−	1.36769i	−0.707107	+	0.707107i	1.14057	−	2.77472i	−2.22540	+	0.218117i
59.4	−0.965926	+	0.258819i	−1.35043	+	1.08459i	0.866025	−	0.500000i	−2.23318	−	0.113651i	1.02370	−	1.39715i	0.484679	−	1.80885i	−0.707107	+	0.707107i	0.647313	−	2.92933i	2.18650	−	0.468211i
59.5	−0.965926	+	0.258819i	−1.04185	−	1.38367i	0.866025	−	0.500000i	2.22803	+	0.189470i	1.36447	+	1.06687i	−0.885238	+	3.30375i	−0.707107	+	0.707107i	−0.829082	+	2.88316i	−2.20115	+	0.393641i
59.6	−0.965926	+	0.258819i	−0.681138	−	1.59250i	0.866025	−	0.500000i	1.33218	−	1.79591i	1.07010	+	1.36194i	1.22554	−	4.57378i	−0.707107	+	0.707107i	−2.07210	+	2.16942i	−0.821970	+	2.07951i
59.7	−0.965926	+	0.258819i	−0.264072	+	1.71180i	0.866025	−	0.500000i	−0.113651	−	2.23318i	−0.187973	−	1.72182i	−0.484679	+	1.80885i	−0.707107	+	0.707107i	−2.86053	−	0.904077i	0.687767	+	2.12767i
59.8	−0.965926	+	0.258819i	−0.115611	+	1.72819i	0.866025	−	0.500000i	0.365292	+	2.20603i	−0.335616	−	1.69922i	−0.366472	+	1.36769i	−0.707107	+	0.707107i	−2.97327	−	0.399597i	−0.923807	−	2.03632i
59.9	−0.965926	+	0.258819i	0.710153	−	1.57977i	0.866025	−	0.500000i	−2.23444	−	0.0852486i	−0.277080	+	1.70974i	0.748636	−	2.79395i	−0.707107	+	0.707107i	−1.99137	−	2.24376i	2.18037	−	0.495972i
59.10	−0.965926	+	0.258819i	1.01305	−	1.40490i	0.866025	−	0.500000i	−0.0852486	−	2.23444i	−0.614914	+	1.61922i	−0.748636	+	2.79395i	−0.707107	+	0.707107i	−0.947472	−	2.84645i	0.660660	+	2.13624i
59.11	−0.965926	+	0.258819i	1.31485	+	1.12746i	0.866025	−	0.500000i	−1.51499	−	1.64463i	−1.56185	−	0.748739i	0.693378	−	2.58772i	−0.707107	+	0.707107i	0.457650	+	2.96489i	1.88903	+	1.19649i
59.12	−0.965926	+	0.258819i	1.52381	+	0.823413i	0.866025	−	0.500000i	1.94851	−	1.09695i	−1.68500	−	0.400965i	−0.391563	+	1.46133i	−0.707107	+	0.707107i	1.64398	+	2.50945i	−1.59821	+	1.56388i
59.13	−0.965926	+	0.258819i	1.71922	+	0.210437i	0.866025	−	0.500000i	0.189470	+	2.22803i	−1.71510	+	0.241700i	0.885238	−	3.30375i	−0.707107	+	0.707107i	2.91143	+	0.723575i	−0.759670	−	2.10307i
59.14	−0.965926	+	0.258819i	1.71971	−	0.206367i	0.866025	−	0.500000i	−1.79591	+	1.33218i	−1.60770	+	0.644429i	−1.22554	+	4.57378i	−0.707107	+	0.707107i	2.91483	−	0.709783i	1.38992	−	1.75160i
59.15	0.965926	−	0.258819i	−1.71971	+	0.206367i	0.866025	−	0.500000i	−1.33218	+	1.79591i	−1.60770	+	0.644429i	1.22554	−	4.57378i	0.707107	−	0.707107i	2.91483	−	0.709783i	−0.821970	+	2.07951i
59.16	0.965926	−	0.258819i	−1.71922	−	0.210437i	0.866025	−	0.500000i	−2.22803	−	0.189470i	−1.71510	+	0.241700i	−0.885238	+	3.30375i	0.707107	−	0.707107i	2.91143	+	0.723575i	−2.20115	+	0.393641i
59.17	0.965926	−	0.258819i	−1.52381	−	0.823413i	0.866025	−	0.500000i	1.09695	−	1.94851i	−1.68500	−	0.400965i	0.391563	−	1.46133i	0.707107	−	0.707107i	1.64398	+	2.50945i	0.555259	−	2.16603i
59.18	0.965926	−	0.258819i	−1.31485	−	1.12746i	0.866025	−	0.500000i	1.64463	+	1.51499i	−1.56185	−	0.748739i	−0.693378	+	2.58772i	0.707107	−	0.707107i	0.457650	+	2.96489i	1.98070	+	1.03770i
59.19	0.965926	−	0.258819i	−1.01305	+	1.40490i	0.866025	−	0.500000i	2.23444	+	0.0852486i	−0.614914	+	1.61922i	0.748636	−	2.79395i	0.707107	−	0.707107i	−0.947472	−	2.84645i	2.18037	−	0.495972i
59.20	0.965926	−	0.258819i	−0.710153	+	1.57977i	0.866025	−	0.500000i	0.0852486	+	2.23444i	−0.277080	+	1.70974i	−0.748636	+	2.79395i	0.707107	−	0.707107i	−1.99137	−	2.24376i	0.660660	+	2.13624i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
13.f	odd	12	1	inner
15.d	odd	2	1	inner
39.k	even	12	1	inner
65.s	odd	12	1	inner
195.bh	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	390.2.bj.a	✓	112
3.b	odd	2	1	inner	390.2.bj.a	✓	112
5.b	even	2	1	inner	390.2.bj.a	✓	112
13.f	odd	12	1	inner	390.2.bj.a	✓	112
15.d	odd	2	1	inner	390.2.bj.a	✓	112
39.k	even	12	1	inner	390.2.bj.a	✓	112
65.s	odd	12	1	inner	390.2.bj.a	✓	112
195.bh	even	12	1	inner	390.2.bj.a	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
390.2.bj.a	✓	112	1.a	even	1	1	trivial
390.2.bj.a	✓	112	3.b	odd	2	1	inner
390.2.bj.a	✓	112	5.b	even	2	1	inner
390.2.bj.a	✓	112	13.f	odd	12	1	inner
390.2.bj.a	✓	112	15.d	odd	2	1	inner
390.2.bj.a	✓	112	39.k	even	12	1	inner
390.2.bj.a	✓	112	65.s	odd	12	1	inner
390.2.bj.a	✓	112	195.bh	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(390, [\chi])\).