Newform orbit 380.2.bd.a

• Label: 380.2.bd.a
• Level: $380$
• Weight: $2$
• Character orbit: 380.bd
• Analytic conductor: $3.034$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.bd (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.03431527681\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1		0		−2.21259	−	2.63686i		0		1.24816	+	1.85529i		0		2.51511	+	1.45210i		0		−1.53655	+	8.71419i		0
9.2		0		−1.23733	−	1.47460i		0		1.46729	−	1.68732i		0		−1.29052	−	0.745084i		0		−0.122496	+	0.694710i		0
9.3		0		−0.907311	−	1.08129i		0		−1.92890	+	1.13108i		0		−1.09754	−	0.633667i		0		0.174968	−	0.992290i		0
9.4		0		−0.817452	−	0.974201i		0		−2.23594	−	0.0236276i		0		4.08102	+	2.35618i		0		0.240104	−	1.36170i		0
9.5		0		−0.587809	−	0.700524i		0		0.750619	+	2.10632i		0		−3.44494	−	1.98894i		0		0.375731	−	2.13087i		0
9.6		0		0.587809	+	0.700524i		0		1.92892	−	1.13104i		0		3.44494	+	1.98894i		0		0.375731	−	2.13087i		0
9.7		0		0.817452	+	0.974201i		0		−1.72802	−	1.41914i		0		−4.08102	−	2.35618i		0		0.240104	−	1.36170i		0
9.8		0		0.907311	+	1.08129i		0		−0.750575	−	2.10633i		0		1.09754	+	0.633667i		0		0.174968	−	0.992290i		0
9.9		0		1.23733	+	1.47460i		0		0.0394253	+	2.23572i		0		1.29052	+	0.745084i		0		−0.122496	+	0.694710i		0
9.10		0		2.21259	+	2.63686i		0		2.14870	−	0.618927i		0		−2.51511	−	1.45210i		0		−1.53655	+	8.71419i		0
149.1		0		−1.00692	−	2.76648i		0		−1.18175	−	1.89828i		0		4.39465	−	2.53725i		0		−4.34139	+	3.64286i		0
149.2		0		−0.920334	−	2.52860i		0		−0.766078	+	2.10074i		0		−1.72263	+	0.994561i		0		−3.24865	+	2.72594i		0
149.3		0		−0.490777	−	1.34840i		0		2.12823	+	0.686037i		0		2.27251	−	1.31204i		0		0.720817	−	0.604837i		0
149.4		0		−0.423995	−	1.16492i		0		1.62907	−	1.53171i		0		−2.04336	+	1.17974i		0		1.12087	−	0.940523i		0
149.5		0		−0.240889	−	0.661837i		0		−1.92970	+	1.12972i		0		−0.446593	+	0.257841i		0		1.91813	−	1.60950i		0
149.6		0		0.240889	+	0.661837i		0		1.42693	+	1.72159i		0		0.446593	−	0.257841i		0		1.91813	−	1.60950i		0
149.7		0		0.423995	+	1.16492i		0		−1.00695	−	1.99651i		0		2.04336	−	1.17974i		0		1.12087	−	0.940523i		0
149.8		0		0.490777	+	1.34840i		0		−2.23452	−	0.0832329i		0		−2.27251	+	1.31204i		0		0.720817	−	0.604837i		0
149.9		0		0.920334	+	2.52860i		0		0.00138065	+	2.23607i		0		1.72263	−	0.994561i		0		−3.24865	+	2.72594i		0
149.10		0		1.00692	+	2.76648i		0		1.75973	−	1.37962i		0		−4.39465	+	2.53725i		0		−4.34139	+	3.64286i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
19.e	even	9	1	inner
95.p	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	380.2.bd.a	✓	60
5.b	even	2	1	inner	380.2.bd.a	✓	60
19.e	even	9	1	inner	380.2.bd.a	✓	60
95.p	even	18	1	inner	380.2.bd.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
380.2.bd.a	✓	60	1.a	even	1	1	trivial
380.2.bd.a	✓	60	5.b	even	2	1	inner
380.2.bd.a	✓	60	19.e	even	9	1	inner
380.2.bd.a	✓	60	95.p	even	18	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(380, [\chi])\).