Embedded newform 819.2.du.a.248.81

• Label: 819.2.du.a.248.81
• Level: $819$
• Weight: $2$
• Character: 819.248
• Analytic conductor: $6.540$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.du (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.53974792554\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(96\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			248.81
Character	\(\chi\)	\(=\)	819.248
Dual form			819.2.du.a.677.81

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.78650 + 1.03143i) q^{2} +(0.518751 - 1.65254i) q^{3} +(1.12771 + 1.95325i) q^{4} -3.13197 q^{5} +(2.63123 - 2.41720i) q^{6} +(1.93878 - 1.80031i) q^{7} +0.526900i q^{8} +(-2.46180 - 1.71452i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 96 q^{4} - 4 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.78650	+	1.03143i	1.26324	+	0.729334i	0.973701	−	0.227831i	\(-0.0731635\pi\)
							0.289542	+	0.957165i	\(0.406497\pi\)
\(3\)	0.518751	−	1.65254i	0.299501	−	0.954096i
							\(5\)	−3.13197			−1.40066			−0.700330	−	0.713819i	\(-0.746964\pi\)
−0.700330	+	0.713819i	\(0.746964\pi\)
\(7\)	1.93878	−	1.80031i	0.732790	−	0.680455i
							\(11\)		−	3.36020i		−	1.01314i	−0.862199	−	0.506570i	\(-0.830913\pi\)
0.862199	−	0.506570i	\(-0.169087\pi\)
\(13\)	0.866025	+	0.500000i	0.240192	+	0.138675i
							\(17\)	−1.26417	+	2.18961i	−0.306606	+	0.531057i	−0.977618	−	0.210390i	\(-0.932527\pi\)
0.671012	+	0.741447i	\(0.265860\pi\)
\(19\)	−2.58532	+	1.49264i	−0.593114	+	0.342434i	−0.766328	−	0.642450i	\(-0.777918\pi\)
							0.173214	+	0.984884i	\(0.444585\pi\)
\(23\)		−	5.03360i		−	1.04958i	−0.851232	−	0.524789i	\(-0.824144\pi\)
							0.851232	−	0.524789i	\(-0.175856\pi\)
\(29\)	2.78044	−	1.60529i	0.516315	−	0.298095i	−0.219111	−	0.975700i	\(-0.570316\pi\)
							0.735426	+	0.677605i	\(0.236982\pi\)
\(31\)	8.72596	−	5.03794i	1.56723	−	0.904840i	0.570739	−	0.821132i	\(-0.306657\pi\)
							0.996490	−	0.0837083i	\(-0.0266764\pi\)
\(37\)	−1.32010	−	2.28649i	−0.217024	−	0.375896i	0.736873	−	0.676031i	\(-0.236302\pi\)
							−0.953897	+	0.300135i	\(0.902968\pi\)
\(41\)	−1.80339	+	3.12356i	−0.281641	+	0.487817i	−0.971789	−	0.235851i	\(-0.924212\pi\)
							0.690148	+	0.723669i	\(0.257546\pi\)
\(43\)	1.01465	+	1.75743i	0.154733	+	0.268005i	0.932962	−	0.359976i	\(-0.117215\pi\)
							−0.778229	+	0.627981i	\(0.783882\pi\)
\(47\)	−5.32064	+	9.21562i	−0.776095	+	1.34424i	0.158082	+	0.987426i	\(0.449469\pi\)
							−0.934177	+	0.356810i	\(0.883864\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.du.a.248.81	yes	192
7.5	odd	6		819.2.be.a.131.16	✓	192
9.2	odd	6		819.2.be.a.794.81	yes	192
63.47	even	6	inner	819.2.du.a.677.81	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.be.a.131.16	✓	192	7.5	odd	6
819.2.be.a.794.81	yes	192	9.2	odd	6
819.2.du.a.248.81	yes	192	1.1	even	1	trivial
819.2.du.a.677.81	yes	192	63.47	even	6	inner