Embedded newform 378.2.bf.a.101.3

• Label: 378.2.bf.a.101.3
• Level: $378$
• Weight: $2$
• Character: 378.101
• 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			101.3
Character	\(\chi\)	\(=\)	378.101
Dual form			378.2.bf.a.131.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.642788 + 0.766044i) q^{2} +(-1.58119 - 0.707001i) q^{3} +(-0.173648 - 0.984808i) q^{4} +(2.18848 - 1.83636i) q^{5} +(1.55796 - 0.756808i) q^{6} +(-1.61790 + 2.09342i) q^{7} +(0.866025 + 0.500000i) q^{8} +(2.00030 + 2.23580i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{7}{18}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.642788	+	0.766044i	−0.454519	+	0.541675i
							\(3\)	−1.58119	−	0.707001i	−0.912898	−	0.408187i
\(5\)	2.18848	−	1.83636i	0.978719	−	0.821243i								−0.00517647	−	0.999987i	\(-0.501648\pi\)
							0.983896	+	0.178744i	\(0.0572033\pi\)
\(7\)	−1.61790	+	2.09342i	−0.611509	+	0.791237i
							\(11\)	−0.926583	+	1.10426i	−0.279375	+	0.332946i	−0.887425	−	0.460953i	\(-0.847508\pi\)
0.608049	+	0.793899i	\(0.291952\pi\)
\(13\)	2.10178	−	5.77458i	0.582928	−	1.60158i	−0.200224	−	0.979750i	\(-0.564167\pi\)
							0.783151	−	0.621831i	\(-0.213611\pi\)
\(17\)	−0.818911			−0.198615			−0.0993075	−	0.995057i	\(-0.531663\pi\)
							−0.0993075	+	0.995057i	\(0.531663\pi\)
\(19\)		−	7.47349i		−	1.71454i	−0.514870	−	0.857268i	\(-0.672160\pi\)
							0.514870	−	0.857268i	\(-0.327840\pi\)
\(23\)	1.97998	−	5.43995i	0.412854	−	1.13431i	−0.542812	−	0.839854i	\(-0.682640\pi\)
							0.955666	−	0.294453i	\(-0.0951375\pi\)
\(29\)	1.98129	+	5.44355i	0.367917	+	1.01084i	0.976152	+	0.217086i	\(0.0696552\pi\)
							−0.608236	+	0.793756i	\(0.708123\pi\)
\(31\)	−1.41888	+	0.250187i	−0.254838	+	0.0449348i	−0.299608	−	0.954063i	\(-0.596856\pi\)
							0.0447695	+	0.998997i	\(0.485745\pi\)
\(37\)	4.32791	−	7.49615i	0.711504	−	1.23236i	−0.252789	−	0.967521i	\(-0.581348\pi\)
							0.964293	−	0.264839i	\(-0.0853188\pi\)
\(41\)	2.74350	+	0.998551i	0.428462	+	0.155947i	0.547244	−	0.836973i	\(-0.315677\pi\)
							−0.118782	+	0.992920i	\(0.537899\pi\)
\(43\)	1.88071	−	10.6660i	0.286806	−	1.62656i	−0.411958	−	0.911203i	\(-0.635155\pi\)
							0.698764	−	0.715352i	\(-0.253734\pi\)
\(47\)	−0.657421	+	3.72842i	−0.0958947	+	0.543846i	0.898575	+	0.438820i	\(0.144604\pi\)
							−0.994470	+	0.105025i	\(0.966508\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.101.3	yes	144
7.5	odd	6		378.2.ba.a.47.16	✓	144
27.23	odd	18		378.2.ba.a.185.16	yes	144
189.131	even	18	inner	378.2.bf.a.131.3	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.47.16	✓	144	7.5	odd	6
378.2.ba.a.185.16	yes	144	27.23	odd	18
378.2.bf.a.101.3	yes	144	1.1	even	1	trivial
378.2.bf.a.131.3	yes	144	189.131	even	18	inner