Embedded newform 855.2.t.a.164.1

• Label: 855.2.t.a.164.1
• Level: $855$
• Weight: $2$
• Character: 855.164
• Analytic conductor: $6.827$
• Analytic rank: $0$
• Dimension: $232$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.82720937282\)
Analytic rank:	\(0\)
Dimension:	\(232\)
Relative dimension:	\(116\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			164.1
Character	\(\chi\)	\(=\)	855.164
Dual form			855.2.t.a.464.116

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.73759i q^{2} +(1.63815 - 0.562551i) q^{3} -5.49438 q^{4} +(1.89416 - 1.18833i) q^{5} +(-1.54003 - 4.48458i) q^{6} +(3.67033 + 2.11907i) q^{7} +9.56616i q^{8} +(2.36707 - 1.84309i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 232 q - 228 q^{4} - 3 q^{5} + 14 q^{6} - 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\)	\(172\)	\(191\)	\(496\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.73759i		−	1.93577i	−0.251402	−	0.967883i	\(-0.580892\pi\)
							0.251402	−	0.967883i	\(-0.419108\pi\)
\(3\)	1.63815	−	0.562551i	0.945786	−	0.324789i
							\(5\)	1.89416	−	1.18833i	0.847096	−	0.531439i
\(7\)	3.67033	+	2.11907i	1.38726	+	0.800933i								0.993005	−	0.118070i	\(-0.0376709\pi\)
							0.394251	+	0.919003i	\(0.371004\pi\)
\(11\)	−0.874847	−	0.505093i	−0.263776	−	0.152291i	0.362280	−	0.932069i	\(-0.381999\pi\)
							−0.626056	+	0.779778i	\(0.715332\pi\)
\(13\)	1.34809			0.373893			0.186946	−	0.982370i	\(-0.440141\pi\)
							0.186946	+	0.982370i	\(0.440141\pi\)
\(17\)	−3.27594	−	5.67409i	−0.794531	−	1.37617i	−0.923136	−	0.384473i	\(-0.874383\pi\)
							0.128605	−	0.991696i	\(-0.458950\pi\)
\(19\)	−3.21150	+	2.94725i	−0.736768	+	0.676146i
							\(23\)	−1.22920			−0.256306			−0.128153	−	0.991754i	\(-0.540905\pi\)
−0.128153	+	0.991754i	\(0.540905\pi\)
\(29\)	−2.70410	−	4.68364i	−0.502139	−	0.869731i	−0.999997	−	0.00247205i	\(-0.999213\pi\)
							0.497858	−	0.867259i	\(-0.334120\pi\)
\(31\)	0.803831	−	0.464092i	0.144372	−	0.0833534i	−0.426074	−	0.904688i	\(-0.640104\pi\)
							0.570446	+	0.821335i	\(0.306770\pi\)
\(37\)	−0.549265			−0.0902985			−0.0451493	−	0.998980i	\(-0.514376\pi\)
							−0.0451493	+	0.998980i	\(0.514376\pi\)
\(41\)	3.04582	−	5.27552i	0.475678	−	0.823898i	−0.523934	−	0.851759i	\(-0.675536\pi\)
							0.999612	+	0.0278610i	\(0.00886957\pi\)
\(43\)			11.2884i			1.72147i	0.509052	+	0.860736i	\(0.329996\pi\)
							−0.509052	+	0.860736i	\(0.670004\pi\)
\(47\)	3.23468	+	5.60263i	0.471827	+	0.817228i	0.999480	−	0.0322314i	\(-0.0102614\pi\)
							−0.527653	+	0.849460i	\(0.676928\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.t.a.164.1	✓	232
5.4	even	2	inner	855.2.t.a.164.116	yes	232
9.5	odd	6		855.2.bm.a.734.1	yes	232
19.8	odd	6		855.2.bm.a.749.116	yes	232
45.14	odd	6		855.2.bm.a.734.116	yes	232
95.84	odd	6		855.2.bm.a.749.1	yes	232
171.122	even	6	inner	855.2.t.a.464.1	yes	232
855.464	even	6	inner	855.2.t.a.464.116	yes	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.t.a.164.1	✓	232	1.1	even	1	trivial
855.2.t.a.164.116	yes	232	5.4	even	2	inner
855.2.t.a.464.1	yes	232	171.122	even	6	inner
855.2.t.a.464.116	yes	232	855.464	even	6	inner
855.2.bm.a.734.1	yes	232	9.5	odd	6
855.2.bm.a.734.116	yes	232	45.14	odd	6
855.2.bm.a.749.1	yes	232	95.84	odd	6
855.2.bm.a.749.116	yes	232	19.8	odd	6