Embedded newform 378.2.bf.a.5.3

• Label: 378.2.bf.a.5.3
• 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			5.3
Character	\(\chi\)	\(=\)	378.5
Dual form			378.2.bf.a.227.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.984808 + 0.173648i) q^{2} +(-1.31364 - 1.12887i) q^{3} +(0.939693 - 0.342020i) q^{4} +(-0.00506474 + 0.0287236i) q^{5} +(1.48971 + 0.883607i) q^{6} +(-2.31258 + 1.28530i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(0.451313 + 2.96586i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

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
							\(3\)	−1.31364	−	1.12887i	−0.758432	−	0.651752i
\(5\)	−0.00506474	+	0.0287236i	−0.00226502	+	0.0128456i								−0.985919	−	0.167222i	\(-0.946520\pi\)
							0.983654	+	0.180067i	\(0.0576315\pi\)
\(7\)	−2.31258	+	1.28530i	−0.874072	+	0.485797i
							\(11\)	5.86925	−	1.03491i	1.76965	−	0.312036i	0.808586	−	0.588378i	\(-0.200233\pi\)
0.961059	+	0.276342i	\(0.0891221\pi\)
\(13\)	−2.43284	+	2.89935i	−0.674748	+	0.804134i	−0.989422	−	0.145066i	\(-0.953660\pi\)
							0.314674	+	0.949200i	\(0.398105\pi\)
\(17\)	1.17665			0.285379			0.142689	−	0.989768i	\(-0.454425\pi\)
							0.142689	+	0.989768i	\(0.454425\pi\)
\(19\)		−	1.06443i		−	0.244198i	−0.992518	−	0.122099i	\(-0.961037\pi\)
							0.992518	−	0.122099i	\(-0.0389625\pi\)
\(23\)	5.54404	−	6.60712i	1.15601	−	1.37768i	0.242859	−	0.970061i	\(-0.421915\pi\)
							0.913152	−	0.407619i	\(-0.133641\pi\)
\(29\)	0.310282	+	0.369779i	0.0576178	+	0.0686663i	0.794084	−	0.607809i	\(-0.207951\pi\)
							−0.736466	+	0.676475i	\(0.763507\pi\)
\(31\)	2.36014	+	6.48443i	0.423894	+	1.16464i	0.949461	+	0.313886i	\(0.101631\pi\)
							−0.525567	+	0.850752i	\(0.676147\pi\)
\(37\)	−4.41580	−	7.64839i	−0.725953	−	1.25739i	−0.958580	−	0.284822i	\(-0.908065\pi\)
							0.232627	−	0.972566i	\(-0.425268\pi\)
\(41\)	9.35723	+	7.85165i	1.46135	+	1.22622i	0.923734	+	0.383035i	\(0.125121\pi\)
							0.537621	+	0.843187i	\(0.319323\pi\)
\(43\)	10.6165	+	3.86408i	1.61900	+	0.589267i	0.983191	−	0.182582i	\(-0.0584454\pi\)
							0.635807	+	0.771848i	\(0.280668\pi\)
\(47\)	5.88846	+	2.14322i	0.858920	+	0.312621i	0.733672	−	0.679504i	\(-0.237805\pi\)
							0.125248	+	0.992125i	\(0.460027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.5.3	yes	144
7.3	odd	6		378.2.ba.a.59.19	✓	144
27.11	odd	18		378.2.ba.a.173.19	yes	144
189.38	even	18	inner	378.2.bf.a.227.3	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.59.19	✓	144	7.3	odd	6
378.2.ba.a.173.19	yes	144	27.11	odd	18
378.2.bf.a.5.3	yes	144	1.1	even	1	trivial
378.2.bf.a.227.3	yes	144	189.38	even	18	inner