Embedded newform 819.2.fy.a.344.1

• Label: 819.2.fy.a.344.1
• Level: $819$
• Weight: $2$
• Character: 819.344
• 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.fy (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			344.1
Character	\(\chi\)	\(=\)	819.344
Dual form			819.2.fy.a.50.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.727236 + 2.71408i) q^{2} +(1.68275 - 0.410318i) q^{3} +(-5.10531 - 2.94755i) q^{4} +(-0.0876862 + 0.327249i) q^{5} +(-0.110119 + 4.86551i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(7.73897 - 7.73897i) q^{8} +(2.66328 - 1.38092i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q - 12 q^{6} + 36 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.727236	+	2.71408i	−0.514233	+	1.91914i	−0.146511	+	0.989209i	\(0.546804\pi\)
							−0.367723	+	0.929935i	\(0.619862\pi\)
\(3\)	1.68275	−	0.410318i	0.971535	−	0.236897i
							\(5\)	−0.0876862	+	0.327249i	−0.0392145	+	0.146350i	−0.982757	−	0.184900i	\(-0.940804\pi\)
0.943543	+	0.331251i	\(0.107471\pi\)
\(7\)	−0.707107	−	0.707107i	−0.267261	−	0.267261i
							\(11\)	−0.128056	−	0.0343126i	−0.0386104	−	0.0103456i	0.239462	−	0.970906i	\(-0.423029\pi\)
−0.278073	+	0.960560i	\(0.589696\pi\)
\(13\)	−3.19064	−	1.67924i	−0.884924	−	0.465736i
							\(17\)	−2.30345	−	3.98969i	−0.558668	−	0.967641i	−0.997608	−	0.0691247i	\(-0.977979\pi\)
0.438940	−	0.898516i	\(-0.355354\pi\)
\(19\)	1.40069	+	0.375314i	0.321341	+	0.0861029i	0.415884	−	0.909418i	\(-0.363472\pi\)
							−0.0945431	+	0.995521i	\(0.530139\pi\)
\(23\)	6.68614			1.39416			0.697078	−	0.716995i	\(-0.254483\pi\)
							0.697078	+	0.716995i	\(0.254483\pi\)
\(29\)	8.80141	−	5.08150i	1.63438	−	0.943610i	0.651661	−	0.758510i	\(-0.274072\pi\)
							0.982720	−	0.185100i	\(-0.0592610\pi\)
\(31\)	3.93871	+	1.05538i	0.707414	+	0.189551i	0.594549	−	0.804059i	\(-0.297331\pi\)
							0.112865	+	0.993610i	\(0.463997\pi\)
\(37\)	1.96258	−	0.525872i	0.322646	−	0.0864528i	−0.0938608	−	0.995585i	\(-0.529921\pi\)
							0.416507	+	0.909133i	\(0.363254\pi\)
\(41\)	−3.40817	−	3.40817i	−0.532266	−	0.532266i	0.388980	−	0.921246i	\(-0.372827\pi\)
							−0.921246	+	0.388980i	\(0.872827\pi\)
\(43\)		−	0.860926i		−	0.131290i	−0.997843	−	0.0656450i	\(-0.979090\pi\)
							0.997843	−	0.0656450i	\(-0.0209105\pi\)
\(47\)	2.32971	+	8.69461i	0.339824	+	1.26824i	0.898544	+	0.438883i	\(0.144626\pi\)
							−0.558720	+	0.829356i	\(0.688708\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.fy.a.344.1	yes	336
9.5	odd	6		819.2.ew.a.617.84	yes	336
13.11	odd	12		819.2.ew.a.596.84	✓	336
117.50	even	12	inner	819.2.fy.a.50.1	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.596.84	✓	336	13.11	odd	12
819.2.ew.a.617.84	yes	336	9.5	odd	6
819.2.fy.a.50.1	yes	336	117.50	even	12	inner
819.2.fy.a.344.1	yes	336	1.1	even	1	trivial