Embedded newform 819.2.ew.a.470.15

• Label: 819.2.ew.a.470.15
• Level: $819$
• Weight: $2$
• Character: 819.470
• 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			470.15
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39334 - 1.39334i) q^{2} +(-1.02425 + 1.39675i) q^{3} +1.88279i q^{4} +(-0.512387 + 1.91225i) q^{5} +(3.37328 - 0.519029i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(-0.163307 + 0.163307i) q^{8} +(-0.901840 - 2.86124i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39334	−	1.39334i	−0.985240	−	0.985240i	0.0146523	−	0.999893i	\(-0.495336\pi\)
							−0.999893	+	0.0146523i	\(0.995336\pi\)
\(3\)	−1.02425	+	1.39675i	−0.591349	+	0.806416i
							\(5\)	−0.512387	+	1.91225i	−0.229146	+	0.855186i	0.751554	+	0.659671i	\(0.229304\pi\)
−0.980701	+	0.195515i	\(0.937362\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	2.85019	−	2.85019i	0.859364	−	0.859364i	−0.131899	−	0.991263i	\(-0.542108\pi\)
0.991263	+	0.131899i	\(0.0421076\pi\)
\(13\)	1.33881	+	3.34777i	0.371320	+	0.928505i
							\(17\)	3.82325	−	6.62206i	0.927273	−	1.60608i	0.139410	−	0.990235i	\(-0.455479\pi\)
0.787863	−	0.615850i	\(-0.211187\pi\)
\(19\)	−0.147832	−	0.551717i	−0.0339150	−	0.126573i	0.946893	−	0.321549i	\(-0.104204\pi\)
							−0.980808	+	0.194977i	\(0.937537\pi\)
\(23\)	−3.82518	+	6.62540i	−0.797605	+	1.38149i	0.123567	+	0.992336i	\(0.460566\pi\)
							−0.921172	+	0.389156i	\(0.872767\pi\)
\(29\)		−	0.0304039i		−	0.00564586i	−0.999996	−	0.00282293i	\(-0.999101\pi\)
							0.999996	−	0.00282293i	\(-0.000898568\pi\)
\(31\)	6.35858	+	1.70378i	1.14204	+	0.306007i	0.779769	−	0.626067i	\(-0.215336\pi\)
							0.362266	+	0.932075i	\(0.382003\pi\)
\(37\)	−2.15914	+	8.05800i	−0.354960	+	1.32473i	0.525576	+	0.850747i	\(0.323850\pi\)
							−0.880535	+	0.473981i	\(0.842817\pi\)
\(41\)	−9.95742	+	2.66808i	−1.55509	+	0.416684i	−0.931104	−	0.364753i	\(-0.881153\pi\)
							−0.623983	+	0.781438i	\(0.714487\pi\)
\(43\)	−8.11766	+	4.68673i	−1.23793	+	0.714720i	−0.968671	−	0.248347i	\(-0.920113\pi\)
							−0.269260	+	0.963067i	\(0.586779\pi\)
\(47\)	−0.357481	−	1.33414i	−0.0521439	−	0.194604i	0.934941	−	0.354804i	\(-0.115453\pi\)
							−0.987084	+	0.160201i	\(0.948786\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.15	yes	336
9.5	odd	6		819.2.fy.a.743.15	yes	336
13.7	odd	12		819.2.fy.a.722.15	yes	336
117.59	even	12	inner	819.2.ew.a.176.15	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.15	✓	336	117.59	even	12	inner
819.2.ew.a.470.15	yes	336	1.1	even	1	trivial
819.2.fy.a.722.15	yes	336	13.7	odd	12
819.2.fy.a.743.15	yes	336	9.5	odd	6