Embedded newform 378.2.bf.a.5.5

• Label: 378.2.bf.a.5.5
• 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			5.5
Character	\(\chi\)	\(=\)	378.5
Dual form			378.2.bf.a.227.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.984808 + 0.173648i) q^{2} +(-0.755599 + 1.55855i) q^{3} +(0.939693 - 0.342020i) q^{4} +(-0.0946220 + 0.536628i) q^{5} +(0.473481 - 1.66608i) q^{6} +(-2.19872 - 1.47161i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(-1.85814 - 2.35527i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.755599	+	1.55855i	−0.436245	+	0.899828i
\(5\)	−0.0946220	+	0.536628i	−0.0423162	+	0.239987i								−0.998628	−	0.0523607i	\(-0.983325\pi\)
							0.956312	+	0.292348i	\(0.0944366\pi\)
\(7\)	−2.19872	−	1.47161i	−0.831038	−	0.556216i
							\(11\)	−0.0131106	+	0.00231175i	−0.00395299	+	0.000697018i	−0.175624	−	0.984457i	\(-0.556194\pi\)
0.171671	+	0.985154i	\(0.445083\pi\)
\(13\)	−0.598741	+	0.713551i	−0.166061	+	0.197904i	−0.842657	−	0.538451i	\(-0.819010\pi\)
							0.676596	+	0.736354i	\(0.263454\pi\)
\(17\)	−7.52510			−1.82511			−0.912553	−	0.408960i	\(-0.865892\pi\)
							−0.912553	+	0.408960i	\(0.865892\pi\)
\(19\)		−	7.77163i		−	1.78293i	−0.453086	−	0.891467i	\(-0.649677\pi\)
							0.453086	−	0.891467i	\(-0.350323\pi\)
\(23\)	−0.167527	+	0.199651i	−0.0349319	+	0.0416302i	−0.783228	−	0.621735i	\(-0.786428\pi\)
							0.748296	+	0.663365i	\(0.230872\pi\)
\(29\)	4.97143	+	5.92472i	0.923171	+	1.10019i	0.994707	+	0.102755i	\(0.0327657\pi\)
							−0.0715354	+	0.997438i	\(0.522790\pi\)
\(31\)	−3.17360	−	8.71941i	−0.569996	−	1.56605i	−0.804510	−	0.593939i	\(-0.797572\pi\)
							0.234514	−	0.972113i	\(-0.424650\pi\)
\(37\)	−4.61508	−	7.99355i	−0.758714	−	1.31413i	−0.943506	−	0.331354i	\(-0.892495\pi\)
							0.184792	−	0.982778i	\(-0.440839\pi\)
\(41\)	−4.50657	−	3.78146i	−0.703808	−	0.590565i	0.219046	−	0.975714i	\(-0.429705\pi\)
							−0.922854	+	0.385150i	\(0.874150\pi\)
\(43\)	−5.61845	−	2.04495i	−0.856806	−	0.311852i	−0.123994	−	0.992283i	\(-0.539570\pi\)
							−0.732812	+	0.680431i	\(0.761793\pi\)
\(47\)	−8.08793	−	2.94377i	−1.17975	−	0.429393i	−0.323634	−	0.946182i	\(-0.604905\pi\)
							−0.856112	+	0.516790i	\(0.827127\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.5.5	yes	144
7.3	odd	6		378.2.ba.a.59.14	✓	144
27.11	odd	18		378.2.ba.a.173.14	yes	144
189.38	even	18	inner	378.2.bf.a.227.5	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.59.14	✓	144	7.3	odd	6
378.2.ba.a.173.14	yes	144	27.11	odd	18
378.2.bf.a.5.5	yes	144	1.1	even	1	trivial
378.2.bf.a.227.5	yes	144	189.38	even	18	inner