Embedded newform 819.2.ew.a.176.5

• Label: 819.2.ew.a.176.5
• 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			176.5
Character	\(\chi\)	\(=\)	819.176
Dual form			819.2.ew.a.470.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.83201 + 1.83201i) q^{2} +(-1.42869 - 0.979200i) q^{3} -4.71250i q^{4} +(-0.819999 - 3.06028i) q^{5} +(4.41128 - 0.823478i) q^{6} +(0.258819 + 0.965926i) q^{7} +(4.96932 + 4.96932i) q^{8} +(1.08234 + 2.79795i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 336 q + 24 q^{6} - 36 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.83201	+	1.83201i	−1.29542	+	1.29542i	−0.364042	+	0.931382i	\(0.618604\pi\)
							−0.931382	+	0.364042i	\(0.881396\pi\)
\(3\)	−1.42869	−	0.979200i	−0.824857	−	0.565341i
							\(5\)	−0.819999	−	3.06028i	−0.366715	−	1.36860i	−0.865081	−	0.501631i	\(-0.832733\pi\)
0.498367	−	0.866966i	\(-0.333933\pi\)
\(7\)	0.258819	+	0.965926i	0.0978244	+	0.365086i
							\(11\)	−0.386859	−	0.386859i	−0.116642	−	0.116642i	0.646376	−	0.763019i	\(-0.276284\pi\)
−0.763019	+	0.646376i	\(0.776284\pi\)
\(13\)	1.19138	+	3.40303i	0.330430	+	0.943831i
							\(17\)	−0.250650	−	0.434139i	−0.0607917	−	0.105294i	0.834028	−	0.551722i	\(-0.186029\pi\)
−0.894820	+	0.446428i	\(0.852696\pi\)
\(19\)	−0.243699	+	0.909496i	−0.0559083	+	0.208653i	−0.988229	−	0.152979i	\(-0.951113\pi\)
							0.932321	+	0.361631i	\(0.117780\pi\)
\(23\)	0.270676	+	0.468825i	0.0564399	+	0.0977568i	0.892865	−	0.450325i	\(-0.148692\pi\)
							−0.836425	+	0.548081i	\(0.815358\pi\)
\(29\)			4.01206i			0.745022i	0.928028	+	0.372511i	\(0.121503\pi\)
							−0.928028	+	0.372511i	\(0.878497\pi\)
\(31\)	8.77689	−	2.35176i	1.57638	−	0.422389i	0.638575	−	0.769560i	\(-0.279524\pi\)
							0.937802	+	0.347171i	\(0.112858\pi\)
\(37\)	1.02996	+	3.84385i	0.169324	+	0.631926i	0.997449	+	0.0713824i	\(0.0227411\pi\)
							−0.828125	+	0.560543i	\(0.810592\pi\)
\(41\)	−4.24029	−	1.13618i	−0.662223	−	0.177442i	−0.0879738	−	0.996123i	\(-0.528039\pi\)
							−0.574249	+	0.818681i	\(0.694706\pi\)
\(43\)	9.33686	+	5.39064i	1.42386	+	0.822065i	0.996626	−	0.0820777i	\(-0.0261556\pi\)
							0.427232	+	0.904142i	\(0.359489\pi\)
\(47\)	0.564530	−	2.10686i	0.0823452	−	0.307317i	−0.912453	−	0.409182i	\(-0.865814\pi\)
							0.994798	+	0.101865i	\(0.0324810\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.ew.a.176.5	✓	336
9.2	odd	6		819.2.fy.a.722.5	yes	336
13.2	odd	12		819.2.fy.a.743.5	yes	336
117.2	even	12	inner	819.2.ew.a.470.5	yes	336

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.ew.a.176.5	✓	336	1.1	even	1	trivial
819.2.ew.a.470.5	yes	336	117.2	even	12	inner
819.2.fy.a.722.5	yes	336	9.2	odd	6
819.2.fy.a.743.5	yes	336	13.2	odd	12