Embedded newform 798.2.bt.a.5.5

• Label: 798.2.bt.a.5.5
• Level: $798$
• Weight: $2$
• Character: 798.5
• Analytic conductor: $6.372$
• Analytic rank: $0$
• Dimension: $324$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	798.bt (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.37206208130\)
Analytic rank:	\(0\)
Dimension:	\(324\)
Relative dimension:	\(54\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.5
Character	\(\chi\)	\(=\)	798.5
Dual form			798.2.bt.a.479.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.984808 - 0.173648i) q^{2} +(-1.48409 + 0.893007i) q^{3} +(0.939693 + 0.342020i) q^{4} +(-1.32965 + 0.483955i) q^{5} +(1.61662 - 0.621730i) q^{6} +(-1.30520 + 2.30140i) q^{7} +(-0.866025 - 0.500000i) q^{8} +(1.40508 - 2.65062i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 324 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).

\(n\)	\(115\)	\(211\)	\(533\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{8}{9}\right)\)	\(-1\)

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
							\(3\)	−1.48409	+	0.893007i	−0.856843	+	0.515578i
\(5\)	−1.32965	+	0.483955i	−0.594640	+	0.216431i								−0.621769	−	0.783201i	\(-0.713586\pi\)
							0.0271292	+	0.999632i	\(0.491363\pi\)
\(7\)	−1.30520	+	2.30140i	−0.493319	+	0.869849i
							\(11\)		−	5.87024i		−	1.76995i	−0.465643	−	0.884973i	\(-0.654177\pi\)
0.465643	−	0.884973i	\(-0.345823\pi\)
\(13\)	1.63643	−	0.288546i	0.453863	−	0.0800283i	0.0579564	−	0.998319i	\(-0.481542\pi\)
							0.395906	+	0.918291i	\(0.370430\pi\)
\(17\)	2.96954	+	2.49174i	0.720219	+	0.604335i	0.927446	−	0.373957i	\(-0.121999\pi\)
							−0.207227	+	0.978293i	\(0.566444\pi\)
\(19\)	−0.795404	−	4.28571i	−0.182478	−	0.983210i
							\(23\)	4.28732	−	0.755970i	0.893968	−	0.157631i	0.292252	−	0.956341i	\(-0.405595\pi\)
0.601715	+	0.798711i	\(0.294484\pi\)
\(29\)	−2.32222	+	6.38025i	−0.431226	+	1.18478i	0.513835	+	0.857889i	\(0.328224\pi\)
							−0.945061	+	0.326894i	\(0.893998\pi\)
\(31\)	1.02321	+	0.590748i	0.183773	+	0.106101i	0.589064	−	0.808086i	\(-0.299497\pi\)
							−0.405291	+	0.914188i	\(0.632830\pi\)
\(37\)	2.99063	−	5.17992i	0.491656	−	0.851573i	−0.508298	−	0.861181i	\(-0.669725\pi\)
							0.999954	+	0.00960833i	\(0.00305847\pi\)
\(41\)	−1.88682	+	10.7007i	−0.294672	+	1.67117i	0.373862	+	0.927484i	\(0.378033\pi\)
							−0.668534	+	0.743682i	\(0.733078\pi\)
\(43\)	2.92316	+	2.45282i	0.445778	+	0.374052i	0.837866	−	0.545875i	\(-0.183803\pi\)
							−0.392088	+	0.919928i	\(0.628247\pi\)
\(47\)	2.06438	−	1.73222i	0.301120	−	0.252670i	−0.479690	−	0.877438i	\(-0.659251\pi\)
							0.780810	+	0.624768i	\(0.214807\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	798.2.bt.a.5.5	✓	324
3.2	odd	2	inner	798.2.bt.a.5.53	yes	324
7.3	odd	6		798.2.cb.a.689.31	yes	324
19.4	even	9		798.2.cb.a.593.20	yes	324
21.17	even	6		798.2.cb.a.689.20	yes	324
57.23	odd	18		798.2.cb.a.593.31	yes	324
133.80	odd	18	inner	798.2.bt.a.479.53	yes	324
399.80	even	18	inner	798.2.bt.a.479.5	yes	324

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
798.2.bt.a.5.5	✓	324	1.1	even	1	trivial
798.2.bt.a.5.53	yes	324	3.2	odd	2	inner
798.2.bt.a.479.5	yes	324	399.80	even	18	inner
798.2.bt.a.479.53	yes	324	133.80	odd	18	inner
798.2.cb.a.593.20	yes	324	19.4	even	9
798.2.cb.a.593.31	yes	324	57.23	odd	18
798.2.cb.a.689.20	yes	324	21.17	even	6
798.2.cb.a.689.31	yes	324	7.3	odd	6