Embedded newform 855.2.t.a.464.116

• Label: 855.2.t.a.464.116
• Level: $855$
• Weight: $2$
• Character: 855.464
• Analytic conductor: $6.827$
• Analytic rank: $0$
• Dimension: $232$
• 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			464.116
Character	\(\chi\)	\(=\)	855.464
Dual form			855.2.t.a.164.1

$q$-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 \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.73759i			1.93577i	0.251402	+	0.967883i	\(0.419108\pi\)
							−0.251402	+	0.967883i	\(0.580892\pi\)
\(3\)	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								0.394251	−	0.919003i	\(-0.371004\pi\)
							0.993005	+	0.118070i	\(0.0376709\pi\)
\(11\)	−0.874847	+	0.505093i	−0.263776	+	0.152291i	−0.626056	−	0.779778i	\(-0.715332\pi\)
							0.362280	+	0.932069i	\(0.381999\pi\)
\(13\)	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	0.128605	+	0.991696i	\(0.458950\pi\)
							−0.923136	+	0.384473i	\(0.874383\pi\)
\(19\)	−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	0.497858	+	0.867259i	\(0.334120\pi\)
							−0.999997	+	0.00247205i	\(0.999213\pi\)
\(31\)	0.803831	+	0.464092i	0.144372	+	0.0833534i	0.570446	−	0.821335i	\(-0.306770\pi\)
							−0.426074	+	0.904688i	\(0.640104\pi\)
\(37\)	−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	0.999612	−	0.0278610i	\(-0.00886957\pi\)
							−0.523934	+	0.851759i	\(0.675536\pi\)
\(43\)		−	11.2884i		−	1.72147i	−0.509052	−	0.860736i	\(-0.670004\pi\)
							0.509052	−	0.860736i	\(-0.329996\pi\)
\(47\)	3.23468	−	5.60263i	0.471827	−	0.817228i	−0.527653	−	0.849460i	\(-0.676928\pi\)
							0.999480	+	0.0322314i	\(0.0102614\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.t.a.464.116	yes	232
5.4	even	2	inner	855.2.t.a.464.1	yes	232
9.2	odd	6		855.2.bm.a.749.1	yes	232
19.12	odd	6		855.2.bm.a.734.116	yes	232
45.29	odd	6		855.2.bm.a.749.116	yes	232
95.69	odd	6		855.2.bm.a.734.1	yes	232
171.164	even	6	inner	855.2.t.a.164.116	yes	232
855.164	even	6	inner	855.2.t.a.164.1	✓	232

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.t.a.164.1	✓	232	855.164	even	6	inner
855.2.t.a.164.116	yes	232	171.164	even	6	inner
855.2.t.a.464.1	yes	232	5.4	even	2	inner
855.2.t.a.464.116	yes	232	1.1	even	1	trivial
855.2.bm.a.734.1	yes	232	95.69	odd	6
855.2.bm.a.734.116	yes	232	19.12	odd	6
855.2.bm.a.749.1	yes	232	9.2	odd	6
855.2.bm.a.749.116	yes	232	45.29	odd	6