Embedded newform 819.2.ew.a.176.10

• Label: 819.2.ew.a.176.10
• 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			176.10
Character	\(\chi\)	\(=\)	819.176
Dual form			819.2.ew.a.470.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.59695 + 1.59695i) q^{2} +(0.0991579 + 1.72921i) q^{3} -3.10050i q^{4} +(0.947622 + 3.53657i) q^{5} +(-2.91981 - 2.60311i) q^{6} +(-0.258819 - 0.965926i) q^{7} +(1.75745 + 1.75745i) q^{8} +(-2.98034 + 0.342930i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.59695	+	1.59695i	−1.12921	+	1.12921i	−0.138910	+	0.990305i	\(0.544360\pi\)
							−0.990305	+	0.138910i	\(0.955640\pi\)
\(3\)	0.0991579	+	1.72921i	0.0572489	+	0.998360i
							\(5\)	0.947622	+	3.53657i	0.423789	+	1.58160i	0.766553	+	0.642182i	\(0.221970\pi\)
−0.342763	+	0.939422i	\(0.611363\pi\)
\(7\)	−0.258819	−	0.965926i	−0.0978244	−	0.365086i
							\(11\)	−1.02299	−	1.02299i	−0.308443	−	0.308443i	0.535862	−	0.844305i	\(-0.319987\pi\)
−0.844305	+	0.535862i	\(0.819987\pi\)
\(13\)	−3.43259	−	1.10331i	−0.952030	−	0.306004i
							\(17\)	2.16028	+	3.74172i	0.523945	+	0.907500i	0.999611	+	0.0278741i	\(0.00887375\pi\)
−0.475666	+	0.879626i	\(0.657793\pi\)
\(19\)	1.71102	−	6.38562i	0.392536	−	1.46496i	−0.433402	−	0.901201i	\(-0.642687\pi\)
							0.825937	−	0.563762i	\(-0.190646\pi\)
\(23\)	−4.12584	−	7.14616i	−0.860296	−	1.49008i	−0.871643	−	0.490141i	\(-0.836945\pi\)
							0.0113467	−	0.999936i	\(-0.496388\pi\)
\(29\)			1.07273i			0.199202i	0.995027	+	0.0996008i	\(0.0317566\pi\)
							−0.995027	+	0.0996008i	\(0.968243\pi\)
\(31\)	−9.92777	+	2.66014i	−1.78308	+	0.477775i	−0.991140	−	0.132822i	\(-0.957596\pi\)
							−0.791941	+	0.610597i	\(0.790929\pi\)
\(37\)	0.196673	+	0.733993i	0.0323328	+	0.120668i	0.980206	−	0.197980i	\(-0.0634381\pi\)
							−0.947873	+	0.318647i	\(0.896771\pi\)
\(41\)	−3.83786	−	1.02835i	−0.599373	−	0.160602i	−0.0536398	−	0.998560i	\(-0.517082\pi\)
							−0.545734	+	0.837959i	\(0.683749\pi\)
\(43\)	−2.96988	−	1.71466i	−0.452902	−	0.261483i	0.256153	−	0.966636i	\(-0.417545\pi\)
							−0.709055	+	0.705153i	\(0.750878\pi\)
\(47\)	0.402190	−	1.50099i	0.0586655	−	0.218943i	−0.930370	−	0.366623i	\(-0.880514\pi\)
							0.989035	+	0.147680i	\(0.0471806\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.176.10	✓	336
9.2	odd	6		819.2.fy.a.722.10	yes	336
13.2	odd	12		819.2.fy.a.743.10	yes	336
117.2	even	12	inner	819.2.ew.a.470.10	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.10	✓	336	1.1	even	1	trivial
819.2.ew.a.470.10	yes	336	117.2	even	12	inner
819.2.fy.a.722.10	yes	336	9.2	odd	6
819.2.fy.a.743.10	yes	336	13.2	odd	12