Embedded newform 819.2.ew.a.176.9

• Label: 819.2.ew.a.176.9
• 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			176.9
Character	\(\chi\)	\(=\)	819.176
Dual form			819.2.ew.a.470.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.63956 + 1.63956i) q^{2} +(1.73159 - 0.0399945i) q^{3} -3.37635i q^{4} +(-1.02893 - 3.84001i) q^{5} +(-2.77348 + 2.90463i) q^{6} +(0.258819 + 0.965926i) q^{7} +(2.25661 + 2.25661i) q^{8} +(2.99680 - 0.138508i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.63956	+	1.63956i	−1.15935	+	1.15935i	−0.174731	+	0.984616i	\(0.555906\pi\)
							−0.984616	+	0.174731i	\(0.944094\pi\)
\(3\)	1.73159	−	0.0399945i	0.999733	−	0.0230908i
							\(5\)	−1.02893	−	3.84001i	−0.460150	−	1.71730i	−0.672488	−	0.740108i	\(-0.734774\pi\)
0.212338	−	0.977196i	\(-0.431892\pi\)
\(7\)	0.258819	+	0.965926i	0.0978244	+	0.365086i
							\(11\)	−0.994890	−	0.994890i	−0.299971	−	0.299971i	0.541032	−	0.841002i	\(-0.318034\pi\)
−0.841002	+	0.541032i	\(0.818034\pi\)
\(13\)	−2.45328	−	2.64224i	−0.680416	−	0.732826i
							\(17\)	−1.81546	−	3.14447i	−0.440314	−	0.762647i	0.557398	−	0.830245i	\(-0.311800\pi\)
−0.997713	+	0.0675984i	\(0.978466\pi\)
\(19\)	−1.52825	+	5.70351i	−0.350605	+	1.30847i	0.535322	+	0.844648i	\(0.320190\pi\)
							−0.885926	+	0.463826i	\(0.846476\pi\)
\(23\)	1.33402	+	2.31060i	0.278163	+	0.481792i	0.970928	−	0.239371i	\(-0.0769411\pi\)
							−0.692765	+	0.721163i	\(0.743608\pi\)
\(29\)		−	7.46624i		−	1.38645i	−0.720723	−	0.693223i	\(-0.756190\pi\)
							0.720723	−	0.693223i	\(-0.243810\pi\)
\(31\)	−4.03933	+	1.08234i	−0.725486	+	0.194393i	−0.602618	−	0.798030i	\(-0.705876\pi\)
							−0.122868	+	0.992423i	\(0.539209\pi\)
\(37\)	−1.05832	−	3.94969i	−0.173986	−	0.649325i	−0.996722	−	0.0809014i	\(-0.974220\pi\)
							0.822736	−	0.568424i	\(-0.192447\pi\)
\(41\)	−9.93822	−	2.66294i	−1.55209	−	0.415881i	−0.621941	−	0.783064i	\(-0.713656\pi\)
							−0.930149	+	0.367183i	\(0.880322\pi\)
\(43\)	−3.32543	−	1.91994i	−0.507123	−	0.292788i	0.224527	−	0.974468i	\(-0.427916\pi\)
							−0.731650	+	0.681680i	\(0.761250\pi\)
\(47\)	1.64013	−	6.12104i	0.239237	−	0.892845i	−0.736956	−	0.675941i	\(-0.763737\pi\)
							0.976193	−	0.216904i	\(-0.0695960\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.176.9	✓	336
9.2	odd	6		819.2.fy.a.722.9	yes	336
13.2	odd	12		819.2.fy.a.743.9	yes	336
117.2	even	12	inner	819.2.ew.a.470.9	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.9	✓	336	1.1	even	1	trivial
819.2.ew.a.470.9	yes	336	117.2	even	12	inner
819.2.fy.a.722.9	yes	336	9.2	odd	6
819.2.fy.a.743.9	yes	336	13.2	odd	12