Embedded newform 847.2.u.a.23.20

• Label: 847.2.u.a.23.20
• 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			23.20
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.02678 - 1.44191i) q^{2} +(0.607575 + 1.05235i) q^{3} +(-0.370686 + 1.07103i) q^{4} +(2.00114 - 1.90808i) q^{5} +(0.893547 - 1.95660i) q^{6} +(0.986575 + 2.45493i) q^{7} +(-1.47193 + 0.432196i) q^{8} +(0.761704 - 1.31931i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.02678	−	1.44191i	−0.726040	−	1.01958i	−0.998353	−	0.0573721i	\(-0.981728\pi\)
							0.272313	−	0.962209i	\(-0.412212\pi\)
\(3\)	0.607575	+	1.05235i	0.350784	+	0.607575i	0.986387	−	0.164441i	\(-0.0525819\pi\)
							−0.635603	+	0.772016i	\(0.719249\pi\)
\(5\)	2.00114	−	1.90808i	0.894936	−	0.853320i	−0.0949067	−	0.995486i	\(-0.530255\pi\)
							0.989843	+	0.142166i	\(0.0454068\pi\)
\(7\)	0.986575	+	2.45493i	0.372890	+	0.927875i
							\(11\)	−1.30719	+	3.04816i	−0.394133	+	0.919053i
\(13\)	−2.60065	−	3.00130i	−0.721289	−	0.832412i								0.270172	−	0.962812i	\(-0.412919\pi\)
							−0.991461	+	0.130400i	\(0.958374\pi\)
\(17\)	0.275441	−	5.78221i	0.0668042	−	1.40239i	−0.677977	−	0.735084i	\(-0.737143\pi\)
							0.744781	−	0.667309i	\(-0.232554\pi\)
\(19\)	−0.334006	−	7.01166i	−0.0766263	−	1.60858i	−0.630639	−	0.776076i	\(-0.717207\pi\)
							0.554013	−	0.832508i	\(-0.313096\pi\)
\(23\)	−1.65865	−	1.30437i	−0.345852	−	0.271981i	0.430089	−	0.902786i	\(-0.358482\pi\)
							−0.775941	+	0.630806i	\(0.782725\pi\)
\(29\)	4.52146	−	2.90577i	0.839614	−	0.539587i	−0.0487054	−	0.998813i	\(-0.515510\pi\)
							0.888319	+	0.459226i	\(0.151873\pi\)
\(31\)	4.49884	+	0.867080i	0.808016	+	0.155732i	0.576498	−	0.817098i	\(-0.304419\pi\)
							0.231518	+	0.972831i	\(0.425631\pi\)
\(37\)	2.61397	+	7.55257i	0.429734	+	1.24163i	0.927336	+	0.374229i	\(0.122093\pi\)
							−0.497602	+	0.867405i	\(0.665786\pi\)
\(41\)	2.14293	−	4.69236i	0.334669	−	0.732823i	−0.665235	−	0.746634i	\(-0.731669\pi\)
							0.999905	+	0.0138103i	\(0.00439609\pi\)
\(43\)	7.86525	−	2.30945i	1.19944	−	0.352187i	0.379801	−	0.925068i	\(-0.375992\pi\)
							0.819639	+	0.572881i	\(0.194174\pi\)
\(47\)	−0.682923	−	0.0652112i	−0.0996145	−	0.00951203i	0.0451297	−	0.998981i	\(-0.485630\pi\)
							−0.144744	+	0.989469i	\(0.546236\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.20	✓	1720
7.4	even	3	inner	847.2.u.a.144.67	yes	1720
121.100	even	11	inner	847.2.u.a.100.67	yes	1720
847.221	even	33	inner	847.2.u.a.221.20	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.20	✓	1720	1.1	even	1	trivial
847.2.u.a.100.67	yes	1720	121.100	even	11	inner
847.2.u.a.144.67	yes	1720	7.4	even	3	inner
847.2.u.a.221.20	yes	1720	847.221	even	33	inner