Embedded newform 819.2.ew.a.176.3

• Label: 819.2.ew.a.176.3
• Level: $819$
• Weight: $2$
• Character: 819.176
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $336$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.ew (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(336\)
Relative dimension:	\(84\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			176.3
Character	\(\chi\)	\(=\)	819.176
Dual form			819.2.ew.a.470.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87295 + 1.87295i) q^{2} +(0.832889 - 1.51865i) q^{3} -5.01588i q^{4} +(0.0815777 + 0.304452i) q^{5} +(1.28440 + 4.40431i) q^{6} +(0.258819 + 0.965926i) q^{7} +(5.64860 + 5.64860i) q^{8} +(-1.61259 - 2.52973i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\)	\(92\)	\(379\)	\(703\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{11}{12}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87295	+	1.87295i	−1.32438	+	1.32438i	−0.414181	+	0.910194i	\(0.635932\pi\)
							−0.910194	+	0.414181i	\(0.864068\pi\)
\(3\)	0.832889	−	1.51865i	0.480869	−	0.876793i
							\(5\)	0.0815777	+	0.304452i	0.0364826	+	0.136155i	0.981765	−	0.190098i	\(-0.0608805\pi\)
−0.945283	+	0.326253i	\(0.894214\pi\)
\(7\)	0.258819	+	0.965926i	0.0978244	+	0.365086i
							\(11\)	3.03644	+	3.03644i	0.915520	+	0.915520i	0.996699	−	0.0811796i	\(-0.0258687\pi\)
−0.0811796	+	0.996699i	\(0.525869\pi\)
\(13\)	−3.60527	+	0.0453460i	−0.999921	+	0.0125767i
							\(17\)	3.01052	+	5.21438i	0.730159	+	1.26467i	0.956815	+	0.290697i	\(0.0938873\pi\)
−0.226656	+	0.973975i	\(0.572779\pi\)
\(19\)	−1.15445	+	4.30847i	−0.264849	+	0.988431i	0.697493	+	0.716591i	\(0.254299\pi\)
							−0.962343	+	0.271840i	\(0.912368\pi\)
\(23\)	−2.47480	−	4.28647i	−0.516031	−	0.893792i	−0.999827	−	0.0186109i	\(-0.994076\pi\)
							0.483796	−	0.875181i	\(-0.339258\pi\)
\(29\)		−	0.320779i		−	0.0595672i	−0.999556	−	0.0297836i	\(-0.990518\pi\)
							0.999556	−	0.0297836i	\(-0.00948181\pi\)
\(31\)	−2.33422	+	0.625451i	−0.419238	+	0.112334i	−0.462270	−	0.886739i	\(-0.652965\pi\)
							0.0430322	+	0.999074i	\(0.486298\pi\)
\(37\)	−0.308563	−	1.15157i	−0.0507274	−	0.189317i	0.935913	−	0.352232i	\(-0.114577\pi\)
							−0.986640	+	0.162915i	\(0.947910\pi\)
\(41\)	11.3389	+	3.03826i	1.77085	+	0.474497i	0.988866	−	0.148807i	\(-0.0475432\pi\)
							0.781980	+	0.623303i	\(0.214210\pi\)
\(43\)	10.1841	+	5.87979i	1.55306	+	0.896659i	0.997890	+	0.0649208i	\(0.0206795\pi\)
							0.555168	+	0.831738i	\(0.312654\pi\)
\(47\)	−0.788045	+	2.94102i	−0.114948	+	0.428992i	−0.999283	−	0.0378609i	\(-0.987946\pi\)
							0.884335	+	0.466853i	\(0.154612\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.176.3	✓	336
9.2	odd	6		819.2.fy.a.722.3	yes	336
13.2	odd	12		819.2.fy.a.743.3	yes	336
117.2	even	12	inner	819.2.ew.a.470.3	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.3	✓	336	1.1	even	1	trivial
819.2.ew.a.470.3	yes	336	117.2	even	12	inner
819.2.fy.a.722.3	yes	336	9.2	odd	6
819.2.fy.a.743.3	yes	336	13.2	odd	12