Embedded newform 847.2.u.a.23.1

• Label: 847.2.u.a.23.1
• 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			23.1
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.58299 - 2.22300i) q^{2} +(-0.529306 - 0.916785i) q^{3} +(-1.78173 + 5.14797i) q^{4} +(-2.73161 + 2.60458i) q^{5} +(-1.20013 + 2.62791i) q^{6} +(0.944173 - 2.47155i) q^{7} +(9.02743 - 2.65069i) q^{8} +(0.939671 - 1.62756i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.58299	−	2.22300i	−1.11934	−	1.57190i	−0.773724	−	0.633523i	\(-0.781608\pi\)
							−0.345619	−	0.938375i	\(-0.612331\pi\)
\(3\)	−0.529306	−	0.916785i	−0.305595	−	0.529306i	0.671799	−	0.740734i	\(-0.265522\pi\)
							−0.977394	+	0.211428i	\(0.932189\pi\)
\(5\)	−2.73161	+	2.60458i	−1.22161	+	1.16480i	−0.240648	+	0.970612i	\(0.577360\pi\)
							−0.980964	+	0.194192i	\(0.937792\pi\)
\(7\)	0.944173	−	2.47155i	0.356864	−	0.934156i
							\(11\)	0.0787296	−	3.31569i	0.0237379	−	0.999718i
\(13\)	−0.833698	−	0.962139i	−0.231226	−	0.266849i								0.628265	−	0.777999i	\(-0.283765\pi\)
							−0.859492	+	0.511150i	\(0.829220\pi\)
\(17\)	−0.259824	+	5.45439i	−0.0630167	+	1.32288i	0.716977	+	0.697096i	\(0.245525\pi\)
							−0.779994	+	0.625787i	\(0.784778\pi\)
\(19\)	−0.223947	−	4.70123i	−0.0513770	−	1.07854i	−0.866477	−	0.499216i	\(-0.833621\pi\)
							0.815100	−	0.579320i	\(-0.196682\pi\)
\(23\)	0.00258013	+	0.00202903i	0.000537994	+	0.000423083i	0.618428	−	0.785842i	\(-0.287770\pi\)
							−0.617890	+	0.786265i	\(0.712012\pi\)
\(29\)	1.58933	−	1.02140i	0.295131	−	0.189669i	−0.384693	−	0.923045i	\(-0.625693\pi\)
							0.679824	+	0.733375i	\(0.262056\pi\)
\(31\)	4.11893	+	0.793858i	0.739781	+	0.142581i	0.545201	−	0.838305i	\(-0.316453\pi\)
							0.194580	+	0.980887i	\(0.437666\pi\)
\(37\)	−2.85252	−	8.24183i	−0.468952	−	1.35495i	−0.893305	−	0.449451i	\(-0.851620\pi\)
							0.424353	−	0.905497i	\(-0.360502\pi\)
\(41\)	−4.18386	+	9.16139i	−0.653410	+	1.43077i	0.235128	+	0.971964i	\(0.424449\pi\)
							−0.888538	+	0.458804i	\(0.848278\pi\)
\(43\)	−7.98251	+	2.34388i	−1.21732	+	0.357438i	−0.826451	−	0.563008i	\(-0.809644\pi\)
							−0.390870	+	0.920446i	\(0.627826\pi\)
\(47\)	9.93068	+	0.948265i	1.44854	+	0.138319i	0.789481	−	0.613775i	\(-0.210350\pi\)
							0.659057	+	0.752093i	\(0.270956\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.1	✓	1720
7.4	even	3	inner	847.2.u.a.144.86	yes	1720
121.100	even	11	inner	847.2.u.a.100.86	yes	1720
847.221	even	33	inner	847.2.u.a.221.1	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.1	✓	1720	1.1	even	1	trivial
847.2.u.a.100.86	yes	1720	121.100	even	11	inner
847.2.u.a.144.86	yes	1720	7.4	even	3	inner
847.2.u.a.221.1	yes	1720	847.221	even	33	inner