Embedded newform 819.2.ew.a.176.4

• Label: 819.2.ew.a.176.4
• 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			176.4
Character	\(\chi\)	\(=\)	819.176
Dual form			819.2.ew.a.470.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86338 + 1.86338i) q^{2} +(0.258684 - 1.71262i) q^{3} -4.94436i q^{4} +(-0.423179 - 1.57933i) q^{5} +(2.70924 + 3.67329i) q^{6} +(-0.258819 - 0.965926i) q^{7} +(5.48647 + 5.48647i) q^{8} +(-2.86617 - 0.886057i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86338	+	1.86338i	−1.31761	+	1.31761i	−0.401944	+	0.915664i	\(0.631665\pi\)
							−0.915664	+	0.401944i	\(0.868335\pi\)
\(3\)	0.258684	−	1.71262i	0.149351	−	0.988784i
							\(5\)	−0.423179	−	1.57933i	−0.189252	−	0.706297i	−0.993680	−	0.112247i	\(-0.964195\pi\)
0.804429	−	0.594049i	\(-0.202472\pi\)
\(7\)	−0.258819	−	0.965926i	−0.0978244	−	0.365086i
							\(11\)	−3.66523	−	3.66523i	−1.10511	−	1.10511i	−0.993784	−	0.111325i	\(-0.964491\pi\)
−0.111325	−	0.993784i	\(-0.535509\pi\)
\(13\)	0.885220	−	3.49519i	0.245516	−	0.969393i
							\(17\)	−1.14096	−	1.97620i	−0.276724	−	0.479300i	0.693845	−	0.720125i	\(-0.255915\pi\)
−0.970569	+	0.240825i	\(0.922582\pi\)
\(19\)	0.707708	−	2.64120i	0.162359	−	0.605934i	−0.836003	−	0.548725i	\(-0.815113\pi\)
							0.998362	−	0.0572085i	\(-0.0182200\pi\)
\(23\)	3.42513	+	5.93250i	0.714189	+	1.23701i	0.963272	+	0.268529i	\(0.0865375\pi\)
							−0.249083	+	0.968482i	\(0.580129\pi\)
\(29\)			7.64621i			1.41986i	0.704270	+	0.709932i	\(0.251275\pi\)
							−0.704270	+	0.709932i	\(0.748725\pi\)
\(31\)	−2.33310	+	0.625153i	−0.419038	+	0.112281i	−0.462176	−	0.886788i	\(-0.652931\pi\)
							0.0431382	+	0.999069i	\(0.486264\pi\)
\(37\)	−1.15114	−	4.29610i	−0.189246	−	0.706275i	−0.993682	−	0.112235i	\(-0.964199\pi\)
							0.804436	−	0.594040i	\(-0.202468\pi\)
\(41\)	−3.22667	−	0.864584i	−0.503921	−	0.135025i	−0.00210208	−	0.999998i	\(-0.500669\pi\)
							−0.501819	+	0.864972i	\(0.667336\pi\)
\(43\)	1.17159	+	0.676415i	0.178665	+	0.103152i	0.586665	−	0.809829i	\(-0.300440\pi\)
							−0.408000	+	0.912982i	\(0.633774\pi\)
\(47\)	1.48694	−	5.54932i	0.216892	−	0.809451i	−0.768600	−	0.639729i	\(-0.779046\pi\)
							0.985492	−	0.169722i	\(-0.0542869\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.176.4	✓	336
9.2	odd	6		819.2.fy.a.722.4	yes	336
13.2	odd	12		819.2.fy.a.743.4	yes	336
117.2	even	12	inner	819.2.ew.a.470.4	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.4	✓	336	1.1	even	1	trivial
819.2.ew.a.470.4	yes	336	117.2	even	12	inner
819.2.fy.a.722.4	yes	336	9.2	odd	6
819.2.fy.a.743.4	yes	336	13.2	odd	12