Embedded newform 819.2.ew.a.470.34

• Label: 819.2.ew.a.470.34
• 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			470.34
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.404390 - 0.404390i) q^{2} +(0.200524 + 1.72040i) q^{3} -1.67294i q^{4} +(0.289920 - 1.08200i) q^{5} +(0.614625 - 0.776805i) q^{6} +(-0.258819 + 0.965926i) q^{7} +(-1.48530 + 1.48530i) q^{8} +(-2.91958 + 0.689964i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.404390	−	0.404390i	−0.285947	−	0.285947i	0.549528	−	0.835475i	\(-0.314808\pi\)
							−0.835475	+	0.549528i	\(0.814808\pi\)
\(3\)	0.200524	+	1.72040i	0.115772	+	0.993276i
							\(5\)	0.289920	−	1.08200i	0.129656	−	0.483883i	−0.870307	−	0.492510i	\(-0.836079\pi\)
0.999963	+	0.00862686i	\(0.00274605\pi\)
\(7\)	−0.258819	+	0.965926i	−0.0978244	+	0.365086i
							\(11\)	−0.192575	+	0.192575i	−0.0580636	+	0.0580636i	−0.735542	−	0.677479i	\(-0.763073\pi\)
0.677479	+	0.735542i	\(0.263073\pi\)
\(13\)	−2.47640	+	2.62058i	−0.686830	+	0.726818i
							\(17\)	−0.531916	+	0.921305i	−0.129009	+	0.223449i	−0.923293	−	0.384097i	\(-0.874513\pi\)
0.794284	+	0.607546i	\(0.207846\pi\)
\(19\)	1.72012	+	6.41959i	0.394624	+	1.47276i	0.822420	+	0.568880i	\(0.192623\pi\)
							−0.427797	+	0.903875i	\(0.640710\pi\)
\(23\)	−0.0829410	+	0.143658i	−0.0172944	+	0.0299548i	−0.874543	−	0.484948i	\(-0.838839\pi\)
							0.857249	+	0.514903i	\(0.172172\pi\)
\(29\)			6.09035i			1.13095i	0.824766	+	0.565475i	\(0.191307\pi\)
							−0.824766	+	0.565475i	\(0.808693\pi\)
\(31\)	1.78215	+	0.477525i	0.320083	+	0.0857661i	0.415283	−	0.909692i	\(-0.363682\pi\)
							−0.0952000	+	0.995458i	\(0.530349\pi\)
\(37\)	−0.977524	+	3.64817i	−0.160704	+	0.599756i	0.837845	+	0.545908i	\(0.183815\pi\)
							−0.998549	+	0.0538476i	\(0.982851\pi\)
\(41\)	7.82341	−	2.09628i	1.22181	−	0.327383i	0.410425	−	0.911894i	\(-0.365380\pi\)
							0.811386	+	0.584511i	\(0.198714\pi\)
\(43\)	−6.27667	+	3.62384i	−0.957183	+	0.552630i	−0.895305	−	0.445454i	\(-0.853042\pi\)
							−0.0618782	+	0.998084i	\(0.519709\pi\)
\(47\)	−2.25651	−	8.42140i	−0.329146	−	1.22839i	−0.910078	−	0.414436i	\(-0.863979\pi\)
							0.580933	−	0.813952i	\(-0.302688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.34	yes	336
9.5	odd	6		819.2.fy.a.743.34	yes	336
13.7	odd	12		819.2.fy.a.722.34	yes	336
117.59	even	12	inner	819.2.ew.a.176.34	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.34	✓	336	117.59	even	12	inner
819.2.ew.a.470.34	yes	336	1.1	even	1	trivial
819.2.fy.a.722.34	yes	336	13.7	odd	12
819.2.fy.a.743.34	yes	336	9.5	odd	6