Embedded newform 819.2.ew.a.470.46

• Label: 819.2.ew.a.470.46
• 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			470.46
Character	\(\chi\)	\(=\)	819.470
Dual form			819.2.ew.a.176.46

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.216928 + 0.216928i) q^{2} +(-0.956404 - 1.44405i) q^{3} -1.90588i q^{4} +(-0.804360 + 3.00191i) q^{5} +(0.105785 - 0.520726i) q^{6} +(0.258819 - 0.965926i) q^{7} +(0.847295 - 0.847295i) q^{8} +(-1.17058 + 2.76220i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.216928	+	0.216928i	0.153391	+	0.153391i	0.779631	−	0.626240i	\(-0.215407\pi\)
							−0.626240	+	0.779631i	\(0.715407\pi\)
\(3\)	−0.956404	−	1.44405i	−0.552180	−	0.833725i
							\(5\)	−0.804360	+	3.00191i	−0.359721	+	1.34250i	0.514718	+	0.857359i	\(0.327897\pi\)
−0.874439	+	0.485136i	\(0.838770\pi\)
\(7\)	0.258819	−	0.965926i	0.0978244	−	0.365086i
							\(11\)	−1.96343	+	1.96343i	−0.591996	+	0.591996i	−0.938170	−	0.346174i	\(-0.887481\pi\)
0.346174	+	0.938170i	\(0.387481\pi\)
\(13\)	2.65621	+	2.43815i	0.736699	+	0.676221i
							\(17\)	−1.32271	+	2.29101i	−0.320805	+	0.555651i	−0.980654	−	0.195747i	\(-0.937287\pi\)
0.659849	+	0.751398i	\(0.270620\pi\)
\(19\)	−0.321554	−	1.20005i	−0.0737695	−	0.275311i	0.919182	−	0.393833i	\(-0.128851\pi\)
							−0.992951	+	0.118522i	\(0.962185\pi\)
\(23\)	−1.55707	+	2.69692i	−0.324672	+	0.562348i	−0.981446	−	0.191740i	\(-0.938587\pi\)
							0.656774	+	0.754087i	\(0.271920\pi\)
\(29\)			8.35679i			1.55182i	0.630846	+	0.775908i	\(0.282708\pi\)
							−0.630846	+	0.775908i	\(0.717292\pi\)
\(31\)	8.01949	+	2.14882i	1.44034	+	0.385939i	0.892654	−	0.450743i	\(-0.148841\pi\)
							0.547689	+	0.836682i	\(0.315508\pi\)
\(37\)	0.903152	−	3.37061i	0.148477	−	0.554124i	−0.851099	−	0.525006i	\(-0.824063\pi\)
							0.999576	−	0.0291188i	\(-0.00927012\pi\)
\(41\)	1.07144	−	0.287092i	0.167331	−	0.0448362i	−0.174181	−	0.984714i	\(-0.555728\pi\)
							0.341512	+	0.939877i	\(0.389061\pi\)
\(43\)	−0.599070	+	0.345873i	−0.0913573	+	0.0527451i	−0.544983	−	0.838447i	\(-0.683464\pi\)
							0.453625	+	0.891192i	\(0.350130\pi\)
\(47\)	1.27357	+	4.75302i	0.185769	+	0.693300i	0.994465	+	0.105072i	\(0.0335074\pi\)
							−0.808695	+	0.588228i	\(0.799826\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.470.46	yes	336
9.5	odd	6		819.2.fy.a.743.46	yes	336
13.7	odd	12		819.2.fy.a.722.46	yes	336
117.59	even	12	inner	819.2.ew.a.176.46	✓	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.46	✓	336	117.59	even	12	inner
819.2.ew.a.470.46	yes	336	1.1	even	1	trivial
819.2.fy.a.722.46	yes	336	13.7	odd	12
819.2.fy.a.743.46	yes	336	9.5	odd	6