Embedded newform 378.2.bf.a.101.24

• Label: 378.2.bf.a.101.24
• 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			101.24
Character	\(\chi\)	\(=\)	378.101
Dual form			378.2.bf.a.131.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.642788 - 0.766044i) q^{2} +(1.71953 - 0.207904i) q^{3} +(-0.173648 - 0.984808i) q^{4} +(1.27218 - 1.06748i) q^{5} +(0.946028 - 1.45087i) q^{6} +(-1.28354 - 2.31355i) q^{7} +(-0.866025 - 0.500000i) q^{8} +(2.91355 - 0.714993i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q + 18 q^{6} - 12 q^{9}+O(q^{10}) \)

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
							\(3\)	1.71953	−	0.207904i	0.992770	−	0.120033i
\(5\)	1.27218	−	1.06748i	0.568935	−	0.477393i								−0.312357	−	0.949965i	\(-0.601119\pi\)
							0.881292	+	0.472572i	\(0.156674\pi\)
\(7\)	−1.28354	−	2.31355i	−0.485132	−	0.874441i
							\(11\)	−0.966322	+	1.15162i	−0.291357	+	0.347226i	−0.891790	−	0.452449i	\(-0.850550\pi\)
0.600433	+	0.799675i	\(0.294995\pi\)
\(13\)	−1.85461	+	5.09550i	−0.514376	+	1.41324i	0.362257	+	0.932078i	\(0.382006\pi\)
							−0.876633	+	0.481159i	\(0.840216\pi\)
\(17\)	−0.973015			−0.235991			−0.117995	−	0.993014i	\(-0.537647\pi\)
							−0.117995	+	0.993014i	\(0.537647\pi\)
\(19\)		−	2.22362i		−	0.510134i	−0.966923	−	0.255067i	\(-0.917903\pi\)
							0.966923	−	0.255067i	\(-0.0820975\pi\)
\(23\)	−0.445838	+	1.22493i	−0.0929637	+	0.255416i	−0.977456	−	0.211139i	\(-0.932283\pi\)
							0.884492	+	0.466555i	\(0.154505\pi\)
\(29\)	2.35916	+	6.48174i	0.438085	+	1.20363i	0.940736	+	0.339139i	\(0.110136\pi\)
							−0.502651	+	0.864489i	\(0.667642\pi\)
\(31\)	10.9427	−	1.92949i	1.96537	−	0.346547i	0.971286	−	0.237914i	\(-0.0764637\pi\)
							0.994082	−	0.108633i	\(-0.0346474\pi\)
\(37\)	2.90275	−	5.02770i	0.477208	−	0.826549i	−0.522450	−	0.852670i	\(-0.674982\pi\)
							0.999659	+	0.0261204i	\(0.00831533\pi\)
\(41\)	−8.50383	−	3.09514i	−1.32807	−	0.483380i	−0.422038	−	0.906578i	\(-0.638685\pi\)
							−0.906037	+	0.423198i	\(0.860907\pi\)
\(43\)	0.660765	−	3.74738i	0.100766	−	0.571471i	−0.892062	−	0.451914i	\(-0.850741\pi\)
							0.992827	−	0.119557i	\(-0.0381474\pi\)
\(47\)	−1.70236	+	9.65455i	−0.248314	+	1.40826i	0.564354	+	0.825533i	\(0.309125\pi\)
							−0.812668	+	0.582727i	\(0.801986\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.bf.a.101.24	yes	144
7.5	odd	6		378.2.ba.a.47.8	✓	144
27.23	odd	18		378.2.ba.a.185.8	yes	144
189.131	even	18	inner	378.2.bf.a.131.24	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.ba.a.47.8	✓	144	7.5	odd	6
378.2.ba.a.185.8	yes	144	27.23	odd	18
378.2.bf.a.101.24	yes	144	1.1	even	1	trivial
378.2.bf.a.131.24	yes	144	189.131	even	18	inner