Embedded newform 310.2.w.a.117.6

• Label: 310.2.w.a.117.6
• Level: $310$
• Weight: $2$
• Character: 310.117
• Analytic conductor: $2.475$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• 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			117.6
Character	\(\chi\)	\(=\)	310.117
Dual form			310.2.w.a.53.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.453990 + 0.891007i) q^{2} +(0.865591 + 0.562122i) q^{3} +(-0.587785 - 0.809017i) q^{4} +(-0.633251 - 2.14453i) q^{5} +(-0.893824 + 0.516050i) q^{6} +(-0.913031 - 2.37853i) q^{7} +(0.987688 - 0.156434i) q^{8} +(-0.786942 - 1.76750i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 24 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(187\)	\(251\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{30}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.453990	+	0.891007i	−0.321020	+	0.630037i
							\(3\)	0.865591	+	0.562122i	0.499749	+	0.324541i	0.769778	−	0.638311i	\(-0.220367\pi\)
−0.270029	+	0.962852i	\(0.587033\pi\)
\(5\)	−0.633251	−	2.14453i	−0.283198	−	0.959061i
							\(7\)	−0.913031	−	2.37853i	−0.345093	−	0.898999i	−0.990790	−	0.135405i	\(-0.956766\pi\)
0.645697	−	0.763594i	\(-0.276567\pi\)
\(11\)	1.21275	−	0.127465i	0.365658	−	0.0384322i	0.0800821	−	0.996788i	\(-0.474482\pi\)
							0.285576	+	0.958356i	\(0.407815\pi\)
\(13\)	0.164024	−	3.12976i	0.0454920	−	0.868039i	−0.876636	−	0.481154i	\(-0.840218\pi\)
							0.922128	−	0.386885i	\(-0.126449\pi\)
\(17\)	−1.30160	−	1.60734i	−0.315684	−	0.389837i	0.594463	−	0.804123i	\(-0.297365\pi\)
							−0.910147	+	0.414286i	\(0.864031\pi\)
\(19\)	1.02943	+	0.926904i	0.236168	+	0.212646i	0.778713	−	0.627380i	\(-0.215873\pi\)
							−0.542545	+	0.840026i	\(0.682539\pi\)
\(23\)	0.488822	+	3.08630i	0.101926	+	0.643538i	0.984769	+	0.173869i	\(0.0556268\pi\)
							−0.882842	+	0.469669i	\(0.844373\pi\)
\(29\)	1.86983	+	5.75474i	0.347218	+	1.06863i	0.960385	+	0.278675i	\(0.0898952\pi\)
							−0.613167	+	0.789953i	\(0.710105\pi\)
\(31\)	5.43852	−	1.19269i	0.976787	−	0.214213i
							\(37\)	−1.30163	−	0.348770i	−0.213986	−	0.0573375i	0.150233	−	0.988651i	\(-0.451998\pi\)
−0.364220	+	0.931313i	\(0.618664\pi\)
\(41\)	6.62433	+	1.40804i	1.03455	+	0.219900i	0.693734	−	0.720232i	\(-0.255964\pi\)
							0.340813	+	0.940131i	\(0.389298\pi\)
\(43\)	−5.33448	+	0.279568i	−0.813500	+	0.0426337i	−0.454548	−	0.890722i	\(-0.650199\pi\)
							−0.358952	+	0.933356i	\(0.616866\pi\)
\(47\)	5.82891	−	2.96998i	0.850234	−	0.433216i	0.0261326	−	0.999658i	\(-0.491681\pi\)
							0.824101	+	0.566443i	\(0.191681\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
310.2.w.a.53.6	✓	256	155.53	even	60	inner
310.2.w.a.117.6	yes	256	1.1	even	1	trivial
310.2.w.a.177.11	yes	256	31.22	odd	30	inner
310.2.w.a.303.11	yes	256	5.3	odd	4	inner