Embedded newform 378.2.bf.a.5.16

• Label: 378.2.bf.a.5.16
• 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			5.16
Character	\(\chi\)	\(=\)	378.5
Dual form			378.2.bf.a.227.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.984808 - 0.173648i) q^{2} +(-0.926008 - 1.46373i) q^{3} +(0.939693 - 0.342020i) q^{4} +(0.475930 - 2.69913i) q^{5} +(-1.16611 - 1.28069i) q^{6} +(2.49768 + 0.872702i) q^{7} +(0.866025 - 0.500000i) q^{8} +(-1.28502 + 2.71085i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

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
							\(3\)	−0.926008	−	1.46373i	−0.534631	−	0.845086i
\(5\)	0.475930	−	2.69913i	0.212842	−	1.20709i								−0.671769	−	0.740760i	\(-0.734465\pi\)
							0.884612	−	0.466328i	\(-0.154423\pi\)
\(7\)	2.49768	+	0.872702i	0.944033	+	0.329850i
							\(11\)	−1.07532	+	0.189608i	−0.324221	+	0.0571689i	−0.333390	−	0.942789i	\(-0.608193\pi\)
0.00916889	+	0.999958i	\(0.497081\pi\)
\(13\)	1.90382	−	2.26888i	0.528024	−	0.629275i	−0.434434	−	0.900703i	\(-0.643052\pi\)
							0.962458	+	0.271429i	\(0.0874962\pi\)
\(17\)	−5.57989			−1.35332			−0.676661	−	0.736295i	\(-0.736574\pi\)
							−0.676661	+	0.736295i	\(0.736574\pi\)
\(19\)		−	4.64047i		−	1.06460i	−0.846557	−	0.532299i	\(-0.821328\pi\)
							0.846557	−	0.532299i	\(-0.178672\pi\)
\(23\)	−1.90634	+	2.27189i	−0.397499	+	0.473721i	−0.927256	−	0.374429i	\(-0.877839\pi\)
							0.529757	+	0.848150i	\(0.322283\pi\)
\(29\)	−0.231031	−	0.275331i	−0.0429013	−	0.0511278i	0.744167	−	0.667993i	\(-0.232847\pi\)
							−0.787068	+	0.616866i	\(0.788402\pi\)
\(31\)	3.21758	+	8.84022i	0.577894	+	1.58775i	0.791723	+	0.610880i	\(0.209184\pi\)
							−0.213829	+	0.976871i	\(0.568594\pi\)
\(37\)	−1.69903	−	2.94280i	−0.279318	−	0.483793i	0.691897	−	0.721996i	\(-0.256775\pi\)
							−0.971215	+	0.238203i	\(0.923442\pi\)
\(41\)	2.52193	+	2.11615i	0.393860	+	0.330488i	0.818115	−	0.575055i	\(-0.195019\pi\)
							−0.424255	+	0.905543i	\(0.639464\pi\)
\(43\)	4.64520	+	1.69072i	0.708387	+	0.257832i	0.670988	−	0.741469i	\(-0.265870\pi\)
							0.0373995	+	0.999300i	\(0.488093\pi\)
\(47\)	11.6411	+	4.23701i	1.69803	+	0.618032i	0.995598	−	0.0937219i	\(-0.0298765\pi\)
							0.702429	+	0.711754i	\(0.252099\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.5.16	yes	144
7.3	odd	6		378.2.ba.a.59.8	✓	144
27.11	odd	18		378.2.ba.a.173.8	yes	144
189.38	even	18	inner	378.2.bf.a.227.16	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.59.8	✓	144	7.3	odd	6
378.2.ba.a.173.8	yes	144	27.11	odd	18
378.2.bf.a.5.16	yes	144	1.1	even	1	trivial
378.2.bf.a.227.16	yes	144	189.38	even	18	inner