Embedded newform 930.2.bt.a.13.10

• Label: 930.2.bt.a.13.10
• Level: $930$
• Weight: $2$
• Character: 930.13
• Analytic conductor: $7.426$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.bt (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.42608738798\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(16\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			13.10
Character	\(\chi\)	\(=\)	930.13
Dual form			930.2.bt.a.787.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.156434 - 0.987688i) q^{2} +(-0.358368 + 0.933580i) q^{3} +(-0.951057 - 0.309017i) q^{4} +(-1.90371 + 1.17298i) q^{5} +(0.866025 + 0.500000i) q^{6} +(1.91384 + 0.100300i) q^{7} +(-0.453990 + 0.891007i) q^{8} +(-0.743145 - 0.669131i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q + 8 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/930\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(311\)	\(871\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(1\)	\(e\left(\frac{11}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.156434	−	0.987688i	0.110616	−	0.698401i
							\(3\)	−0.358368	+	0.933580i	−0.206904	+	0.539003i
\(5\)	−1.90371	+	1.17298i	−0.851367	+	0.524571i
							\(7\)	1.91384	+	0.100300i	0.723364	+	0.0379099i	0.410465	−	0.911876i	\(-0.365366\pi\)
0.312899	+	0.949786i	\(0.398700\pi\)
\(11\)	0.252290	+	1.18693i	0.0760683	+	0.357873i	0.999676	−	0.0254666i	\(-0.00810715\pi\)
							−0.923607	+	0.383340i	\(0.874774\pi\)
\(13\)	−2.38063	−	2.93983i	−0.660268	−	0.815363i	0.331494	−	0.943457i	\(-0.392447\pi\)
							−0.991762	+	0.128094i	\(0.959114\pi\)
\(17\)	2.13622	+	3.28949i	0.518109	+	0.797818i	0.996781	−	0.0801763i	\(-0.0255483\pi\)
							−0.478672	+	0.877994i	\(0.658882\pi\)
\(19\)	−7.53524	+	0.791985i	−1.72870	+	0.181694i	−0.916052	−	0.401059i	\(-0.868642\pi\)
							−0.812649	+	0.582753i	\(0.801976\pi\)
\(23\)	−1.12583	−	0.573640i	−0.234752	−	0.119612i	0.332658	−	0.943048i	\(-0.392055\pi\)
							−0.567410	+	0.823435i	\(0.692055\pi\)
\(29\)	−1.18480	+	0.860806i	−0.220011	+	0.159848i	−0.692332	−	0.721579i	\(-0.743417\pi\)
							0.472320	+	0.881427i	\(0.343417\pi\)
\(31\)	0.168472	−	5.56521i	0.0302585	−	0.999542i
							\(37\)	−11.4876	+	3.07810i	−1.88856	+	0.506037i	−0.889790	+	0.456370i	\(0.849149\pi\)
−0.998766	+	0.0496667i	\(0.984184\pi\)
\(41\)	−2.09998	−	0.934971i	−0.327962	−	0.146018i	0.236150	−	0.971717i	\(-0.424114\pi\)
							−0.564111	+	0.825699i	\(0.690781\pi\)
\(43\)	−5.30497	−	4.29588i	−0.809000	−	0.655116i	0.132854	−	0.991136i	\(-0.457586\pi\)
							−0.941855	+	0.336020i	\(0.890919\pi\)
\(47\)	−7.68802	+	1.21766i	−1.12141	+	0.177614i	−0.689489	−	0.724296i	\(-0.742165\pi\)
							−0.431923	+	0.901910i	\(0.642165\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.bt.a.13.10	✓	256
5.2	odd	4		930.2.bt.b.757.15	yes	256
31.12	odd	30		930.2.bt.b.43.15	yes	256
155.12	even	60	inner	930.2.bt.a.787.10	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.bt.a.13.10	✓	256	1.1	even	1	trivial
930.2.bt.a.787.10	yes	256	155.12	even	60	inner
930.2.bt.b.43.15	yes	256	31.12	odd	30
930.2.bt.b.757.15	yes	256	5.2	odd	4