Embedded newform 847.2.u.a.23.12

• Label: 847.2.u.a.23.12
• Level: $847$
• Weight: $2$
• Character: 847.23
• Analytic conductor: $6.763$
• Analytic rank: $0$
• Dimension: $1720$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.u (of order \(33\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.76332905120\)
Analytic rank:	\(0\)
Dimension:	\(1720\)
Relative dimension:	\(86\) over \(\Q(\zeta_{33})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{33}]$

Embedding invariants

Embedding label			23.12
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.26662 - 1.77872i) q^{2} +(-0.0320353 - 0.0554867i) q^{3} +(-0.905377 + 2.61591i) q^{4} +(0.962361 - 0.917609i) q^{5} +(-0.0581187 + 0.127262i) q^{6} +(1.16758 + 2.37418i) q^{7} +(1.60942 - 0.472568i) q^{8} +(1.49795 - 2.59452i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/847\mathbb{Z}\right)^\times\).

\(n\)	\(122\)	\(365\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(e\left(\frac{7}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.26662	−	1.77872i	−0.895635	−	1.25774i	−0.965557	−	0.260191i	\(-0.916214\pi\)
							0.0699222	−	0.997552i	\(-0.477725\pi\)
\(3\)	−0.0320353	−	0.0554867i	−0.0184956	−	0.0320353i	0.856629	−	0.515932i	\(-0.172554\pi\)
							−0.875125	+	0.483897i	\(0.839221\pi\)
\(5\)	0.962361	−	0.917609i	0.430381	−	0.410367i	−0.443679	−	0.896186i	\(-0.646327\pi\)
							0.874060	+	0.485818i	\(0.161478\pi\)
\(7\)	1.16758	+	2.37418i	0.441305	+	0.897357i
							\(11\)	0.756503	−	3.22920i	0.228094	−	0.973639i
\(13\)	1.94947	+	2.24981i	0.540685	+	0.623984i								0.958687	−	0.284462i	\(-0.0918149\pi\)
							−0.418002	+	0.908446i	\(0.637269\pi\)
\(17\)	−0.193516	+	4.06240i	−0.0469345	+	0.985277i	0.845465	+	0.534031i	\(0.179323\pi\)
							−0.892400	+	0.451246i	\(0.850980\pi\)
\(19\)	0.301728	+	6.33405i	0.0692211	+	1.45313i	0.720059	+	0.693913i	\(0.244115\pi\)
							−0.650838	+	0.759217i	\(0.725582\pi\)
\(23\)	−0.977897	−	0.769026i	−0.203906	−	0.160353i	0.510985	−	0.859590i	\(-0.329281\pi\)
							−0.714890	+	0.699237i	\(0.753523\pi\)
\(29\)	6.25974	−	4.02289i	1.16240	−	0.747031i	0.190333	−	0.981720i	\(-0.439043\pi\)
							0.972071	+	0.234688i	\(0.0754069\pi\)
\(31\)	8.16152	+	1.57300i	1.46585	+	0.282520i	0.858869	−	0.512195i	\(-0.171167\pi\)
							0.606983	+	0.794715i	\(0.292380\pi\)
\(37\)	−2.64908	−	7.65401i	−0.435506	−	1.25831i	−0.922869	−	0.385113i	\(-0.874162\pi\)
							0.487363	−	0.873199i	\(-0.337959\pi\)
\(41\)	−1.41206	+	3.09199i	−0.220527	+	0.482888i	−0.987267	−	0.159070i	\(-0.949150\pi\)
							0.766740	+	0.641958i	\(0.221878\pi\)
\(43\)	9.97507	−	2.92894i	1.52118	−	0.446660i	0.588845	−	0.808246i	\(-0.299583\pi\)
							0.932339	+	0.361586i	\(0.117765\pi\)
\(47\)	−4.37800	−	0.418048i	−0.638597	−	0.0609786i	−0.229265	−	0.973364i	\(-0.573632\pi\)
							−0.409332	+	0.912385i	\(0.634238\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.12	✓	1720
7.4	even	3	inner	847.2.u.a.144.75	yes	1720
121.100	even	11	inner	847.2.u.a.100.75	yes	1720
847.221	even	33	inner	847.2.u.a.221.12	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.12	✓	1720	1.1	even	1	trivial
847.2.u.a.100.75	yes	1720	121.100	even	11	inner
847.2.u.a.144.75	yes	1720	7.4	even	3	inner
847.2.u.a.221.12	yes	1720	847.221	even	33	inner