Embedded newform 31.8.d.a.8.1

• Label: 31.8.d.a.8.1
• Level: $31$
• Weight: $8$
• Character: 31.8
• Analytic conductor: $9.684$
• Analytic rank: $0$
• Dimension: $68$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 31 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	31.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.68393579001\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(17\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			8.1
Character	\(\chi\)	\(=\)	31.8
Dual form			31.8.d.a.4.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-6.21729 + 19.1348i) q^{2} +(-6.66015 - 20.4978i) q^{3} +(-223.933 - 162.697i) q^{4} -211.276 q^{5} +433.631 q^{6} +(148.175 + 107.656i) q^{7} +(2421.98 - 1759.67i) q^{8} +(1393.52 - 1012.45i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 3 q^{2} - 31 q^{3} - 847 q^{4} - 150 q^{5} + 1910 q^{6} - 3083 q^{7} - 4145 q^{8} - 11638 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−6.21729	+	19.1348i	−0.549536	+	1.69130i	0.160418	+	0.987049i	\(0.448716\pi\)
							−0.709954	+	0.704248i	\(0.751284\pi\)
\(3\)	−6.66015	−	20.4978i	−0.142416	−	0.438312i	0.854253	−	0.519857i	\(-0.174015\pi\)
							−0.996670	+	0.0815445i	\(0.974015\pi\)
\(5\)	−211.276			−0.755884			−0.377942	−	0.925829i	\(-0.623368\pi\)
							−0.377942	+	0.925829i	\(0.623368\pi\)
\(7\)	148.175	+	107.656i	0.163280	+	0.118630i	0.666425	−	0.745572i	\(-0.267824\pi\)
							−0.503145	+	0.864202i	\(0.667824\pi\)
\(11\)	953.532	+	692.782i	0.216004	+	0.156936i	0.690526	−	0.723308i	\(-0.257379\pi\)
							−0.474522	+	0.880243i	\(0.657379\pi\)
\(13\)	643.974	+	1981.95i	0.0812955	+	0.250202i	0.983441	−	0.181231i	\(-0.0580082\pi\)
							−0.902145	+	0.431433i	\(0.858008\pi\)
\(17\)	4764.43	−	3461.56i	0.235201	−	0.170884i	−0.463941	−	0.885866i	\(-0.653565\pi\)
							0.699143	+	0.714982i	\(0.253565\pi\)
\(19\)	434.447	−	1337.09i	0.0145311	−	0.0447222i	−0.943528	−	0.331293i	\(-0.892515\pi\)
							0.958059	+	0.286570i	\(0.0925153\pi\)
\(23\)	80386.2	−	58404.0i	1.37763	−	1.00091i	0.380538	−	0.924765i	\(-0.375739\pi\)
							0.997097	−	0.0761447i	\(-0.0242611\pi\)
\(29\)	58290.5	−	179400.i	0.443818	−	1.36593i	−0.439957	−	0.898019i	\(-0.645006\pi\)
							0.883775	−	0.467912i	\(-0.154994\pi\)
\(31\)	164087.	−	24248.0i	0.989257	−	0.146188i
							\(37\)	223946.			0.726836			0.363418	−	0.931626i	\(-0.381610\pi\)
0.363418	+	0.931626i	\(0.381610\pi\)
\(41\)	68470.1	−	210729.i	0.155152	−	0.477509i	−0.843024	−	0.537876i	\(-0.819227\pi\)
							0.998176	+	0.0603667i	\(0.0192270\pi\)
\(43\)	−185193.	+	569967.i	−0.355211	+	1.09323i	0.600677	+	0.799492i	\(0.294898\pi\)
							−0.955887	+	0.293734i	\(0.905102\pi\)
\(47\)	−224974.	−	692399.i	−0.316075	−	0.972778i	−0.975310	−	0.220841i	\(-0.929120\pi\)
							0.659235	−	0.751937i	\(-0.270880\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	31.8.d.a.8.1	yes	68
31.4	even	5	inner	31.8.d.a.4.1	✓	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.8.d.a.4.1	✓	68	31.4	even	5	inner
31.8.d.a.8.1	yes	68	1.1	even	1	trivial