Embedded newform 378.2.bf.a.5.4

• Label: 378.2.bf.a.5.4
• Level: $378$
• Weight: $2$
• Character: 378.5
• Analytic conductor: $3.018$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.bf (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.01834519640\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(24\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	378.5
Dual form			378.2.bf.a.227.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.984808 + 0.173648i) q^{2} +(-0.906167 + 1.47610i) q^{3} +(0.939693 - 0.342020i) q^{4} +(0.392637 - 2.22675i) q^{5} +(0.636079 - 1.61103i) q^{6} +(1.14836 + 2.38354i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(-1.35772 - 2.67518i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.984808	+	0.173648i	−0.696364	+	0.122788i
							\(3\)	−0.906167	+	1.47610i	−0.523176	+	0.852225i
\(5\)	0.392637	−	2.22675i	0.175593	−	0.995835i								−0.761865	−	0.647736i	\(-0.775716\pi\)
							0.937458	−	0.348099i	\(-0.113173\pi\)
\(7\)	1.14836	+	2.38354i	0.434039	+	0.900894i
							\(11\)	0.696469	−	0.122806i	0.209993	−	0.0370275i	−0.0676618	−	0.997708i	\(-0.521554\pi\)
0.277655	+	0.960681i	\(0.410443\pi\)
\(13\)	3.55775	−	4.23996i	0.986742	−	1.17595i	0.00234444	−	0.999997i	\(-0.499254\pi\)
							0.984398	−	0.175957i	\(-0.0563018\pi\)
\(17\)	0.529657			0.128461			0.0642304	−	0.997935i	\(-0.479541\pi\)
							0.0642304	+	0.997935i	\(0.479541\pi\)
\(19\)			4.46348i			1.02399i	0.858988	+	0.511996i	\(0.171094\pi\)
							−0.858988	+	0.511996i	\(0.828906\pi\)
\(23\)	3.37278	−	4.01953i	0.703274	−	0.838129i	−0.289619	−	0.957142i	\(-0.593528\pi\)
							0.992893	+	0.119013i	\(0.0379729\pi\)
\(29\)	1.49380	+	1.78024i	0.277392	+	0.330583i	0.886695	−	0.462355i	\(-0.152995\pi\)
							−0.609303	+	0.792937i	\(0.708551\pi\)
\(31\)	3.15712	+	8.67411i	0.567035	+	1.55792i	0.809109	+	0.587658i	\(0.199950\pi\)
							−0.242074	+	0.970258i	\(0.577828\pi\)
\(37\)	1.53359	+	2.65625i	0.252120	+	0.436685i	0.964109	−	0.265505i	\(-0.0855388\pi\)
							−0.711989	+	0.702191i	\(0.752205\pi\)
\(41\)	3.35950	+	2.81895i	0.524666	+	0.440247i	0.866255	−	0.499603i	\(-0.166521\pi\)
							−0.341589	+	0.939849i	\(0.610965\pi\)
\(43\)	2.45567	+	0.893792i	0.374487	+	0.136302i	0.522405	−	0.852698i	\(-0.325035\pi\)
							−0.147918	+	0.989000i	\(0.547257\pi\)
\(47\)	−7.22834	−	2.63090i	−1.05436	−	0.383756i	−0.244055	−	0.969761i	\(-0.578478\pi\)
							−0.810307	+	0.586005i	\(0.800700\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.5.4	yes	144
7.3	odd	6		378.2.ba.a.59.13	✓	144
27.11	odd	18		378.2.ba.a.173.13	yes	144
189.38	even	18	inner	378.2.bf.a.227.4	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.59.13	✓	144	7.3	odd	6
378.2.ba.a.173.13	yes	144	27.11	odd	18
378.2.bf.a.5.4	yes	144	1.1	even	1	trivial
378.2.bf.a.227.4	yes	144	189.38	even	18	inner