Embedded newform 310.2.w.a.177.11

• Label: 310.2.w.a.177.11
• Level: $310$
• Weight: $2$
• Character: 310.177
• Analytic conductor: $2.475$
• Analytic rank: $0$
• Dimension: $256$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 310 = 2 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	310.w (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.47536246266\)
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			177.11
Character	\(\chi\)	\(=\)	310.177
Dual form			310.2.w.a.303.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.891007 - 0.453990i) q^{2} +(-0.562122 - 0.865591i) q^{3} +(0.587785 - 0.809017i) q^{4} +(2.17384 + 0.523852i) q^{5} +(-0.893824 - 0.516050i) q^{6} +(-2.37853 - 0.913031i) q^{7} +(0.156434 - 0.987688i) q^{8} +(0.786942 - 1.76750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 24 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/310\mathbb{Z}\right)^\times\).

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{17}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.891007	−	0.453990i	0.630037	−	0.321020i
							\(3\)	−0.562122	−	0.865591i	−0.324541	−	0.499749i	0.638311	−	0.769778i	\(-0.279633\pi\)
−0.962852	+	0.270029i	\(0.912967\pi\)
\(5\)	2.17384	+	0.523852i	0.972171	+	0.234274i
							\(7\)	−2.37853	−	0.913031i	−0.898999	−	0.345093i	−0.135405	−	0.990790i	\(-0.543234\pi\)
−0.763594	+	0.645697i	\(0.776567\pi\)
\(11\)	1.21275	+	0.127465i	0.365658	+	0.0384322i	0.285576	−	0.958356i	\(-0.407815\pi\)
							0.0800821	+	0.996788i	\(0.474482\pi\)
\(13\)	3.12976	−	0.164024i	0.868039	−	0.0454920i	0.386885	−	0.922128i	\(-0.373551\pi\)
							0.481154	+	0.876636i	\(0.340218\pi\)
\(17\)	−1.60734	−	1.30160i	−0.389837	−	0.315684i	0.414286	−	0.910147i	\(-0.364031\pi\)
							−0.804123	+	0.594463i	\(0.797365\pi\)
\(19\)	−1.02943	+	0.926904i	−0.236168	+	0.212646i	−0.778713	−	0.627380i	\(-0.784127\pi\)
							0.542545	+	0.840026i	\(0.317461\pi\)
\(23\)	−3.08630	−	0.488822i	−0.643538	−	0.101926i	−0.173869	−	0.984769i	\(-0.555627\pi\)
							−0.469669	+	0.882842i	\(0.655627\pi\)
\(29\)	−1.86983	+	5.75474i	−0.347218	+	1.06863i	0.613167	+	0.789953i	\(0.289895\pi\)
							−0.960385	+	0.278675i	\(0.910105\pi\)
\(31\)	5.43852	+	1.19269i	0.976787	+	0.214213i
							\(37\)	−0.348770	−	1.30163i	−0.0573375	−	0.213986i	0.931313	−	0.364220i	\(-0.118664\pi\)
−0.988651	+	0.150233i	\(0.951998\pi\)
\(41\)	6.62433	−	1.40804i	1.03455	−	0.219900i	0.340813	−	0.940131i	\(-0.389298\pi\)
							0.693734	+	0.720232i	\(0.255964\pi\)
\(43\)	−0.279568	+	5.33448i	−0.0426337	+	0.813500i	0.890722	+	0.454548i	\(0.150199\pi\)
							−0.933356	+	0.358952i	\(0.883134\pi\)
\(47\)	−2.96998	+	5.82891i	−0.433216	+	0.850234i	0.566443	+	0.824101i	\(0.308319\pi\)
							−0.999658	+	0.0261326i	\(0.991681\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	310.2.w.a.177.11	yes	256
5.3	odd	4	inner	310.2.w.a.53.6	✓	256
31.24	odd	30	inner	310.2.w.a.117.6	yes	256
155.148	even	60	inner	310.2.w.a.303.11	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.w.a.53.6	✓	256	5.3	odd	4	inner
310.2.w.a.117.6	yes	256	31.24	odd	30	inner
310.2.w.a.177.11	yes	256	1.1	even	1	trivial
310.2.w.a.303.11	yes	256	155.148	even	60	inner