Embedded newform 211.3.e.a.197.3

• Label: 211.3.e.a.197.3
• Level: $211$
• Weight: $3$
• Character: 211.197
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 211 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	211.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74933357800\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			197.3
Character	\(\chi\)	\(=\)	211.197
Dual form			211.3.e.a.15.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.96716 - 1.71309i) q^{2} +(-2.59992 - 1.50106i) q^{3} +(3.86935 + 6.70190i) q^{4} +5.80352 q^{5} +(5.14291 + 8.90778i) q^{6} +(-3.19116 - 1.84241i) q^{7} -12.8094i q^{8} +(0.00637874 + 0.0110483i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 3 q^{2} + 55 q^{4} - 4 q^{5} - q^{6} - 24 q^{7} + 82 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/211\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.96716	−	1.71309i	−1.48358	−	0.856544i	−0.483752	−	0.875205i	\(-0.660727\pi\)
							−0.999826	+	0.0186605i	\(0.994060\pi\)
\(3\)	−2.59992	−	1.50106i	−0.866639	−	0.500354i	−0.000409100	−	1.00000i	\(-0.500130\pi\)
							−0.866230	+	0.499646i	\(0.833464\pi\)
\(5\)	5.80352			1.16070			0.580352	−	0.814365i	\(-0.302915\pi\)
							0.580352	+	0.814365i	\(0.302915\pi\)
\(7\)	−3.19116	−	1.84241i	−0.455879	−	0.263202i	0.254431	−	0.967091i	\(-0.418112\pi\)
							−0.710310	+	0.703889i	\(0.751445\pi\)
\(11\)	3.36905			0.306277			0.153139	−	0.988205i	\(-0.451062\pi\)
							0.153139	+	0.988205i	\(0.451062\pi\)
\(13\)	−16.3775			−1.25981			−0.629906	−	0.776672i	\(-0.716906\pi\)
							−0.629906	+	0.776672i	\(0.716906\pi\)
\(17\)	−18.3899	−	10.6174i	−1.08176	−	0.624553i	−0.150388	−	0.988627i	\(-0.548052\pi\)
							−0.931370	+	0.364074i	\(0.881386\pi\)
\(19\)	7.78756	−	13.4885i	0.409872	−	0.709918i	−0.585003	−	0.811031i	\(-0.698907\pi\)
							0.994875	+	0.101112i	\(0.0322402\pi\)
\(23\)			27.9752i			1.21631i	0.793817	+	0.608157i	\(0.208091\pi\)
							−0.793817	+	0.608157i	\(0.791909\pi\)
\(29\)	−15.0581	+	8.69377i	−0.519243	+	0.299785i	−0.736625	−	0.676301i	\(-0.763582\pi\)
							0.217382	+	0.976087i	\(0.430248\pi\)
\(31\)	32.6640	+	18.8586i	1.05368	+	0.608341i	0.923676	−	0.383173i	\(-0.125169\pi\)
							0.130000	+	0.991514i	\(0.458502\pi\)
\(37\)	−1.44396	+	2.50101i	−0.0390259	+	0.0675949i	−0.884879	−	0.465822i	\(-0.845759\pi\)
							0.845853	+	0.533417i	\(0.179092\pi\)
\(41\)	−47.6543	+	27.5132i	−1.16230	+	0.671055i	−0.951854	−	0.306551i	\(-0.900825\pi\)
							−0.210446	+	0.977605i	\(0.567492\pi\)
\(43\)	−25.6148	−	44.3662i	−0.595694	−	1.03177i	−0.993449	−	0.114281i	\(-0.963544\pi\)
							0.397754	−	0.917492i	\(-0.369790\pi\)
\(47\)	−3.93527	+	6.81609i	−0.0837292	+	0.145023i	−0.904849	−	0.425733i	\(-0.860016\pi\)
							0.821120	+	0.570756i	\(0.193350\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	211.3.e.a.197.3	yes	68
211.15	odd	6	inner	211.3.e.a.15.3	✓	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
211.3.e.a.15.3	✓	68	211.15	odd	6	inner
211.3.e.a.197.3	yes	68	1.1	even	1	trivial