Embedded newform 847.2.u.a.23.17

• Label: 847.2.u.a.23.17
• 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			23.17
Character	\(\chi\)	\(=\)	847.23
Dual form			847.2.u.a.221.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08714 - 1.52668i) q^{2} +(1.52438 + 2.64031i) q^{3} +(-0.494736 + 1.42945i) q^{4} +(0.554663 - 0.528870i) q^{5} +(2.37369 - 5.19765i) q^{6} +(0.743974 - 2.53900i) q^{7} +(-0.876412 + 0.257338i) q^{8} +(-3.14749 + 5.45162i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1720 q - 11 q^{2} - 20 q^{3} + 73 q^{4} - 7 q^{5} + 8 q^{6} - 24 q^{7} - 32 q^{8} - 844 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08714	−	1.52668i	−0.768728	−	1.07953i	−0.994437	−	0.105332i	\(-0.966409\pi\)
							0.225710	−	0.974195i	\(-0.427530\pi\)
\(3\)	1.52438	+	2.64031i	0.880104	+	1.52438i	0.851225	+	0.524801i	\(0.175860\pi\)
							0.0288789	+	0.999583i	\(0.490806\pi\)
\(5\)	0.554663	−	0.528870i	0.248053	−	0.236518i	−0.555844	−	0.831287i	\(-0.687605\pi\)
							0.803897	+	0.594769i	\(0.202756\pi\)
\(7\)	0.743974	−	2.53900i	0.281196	−	0.959650i
							\(11\)	−1.35180	+	3.02864i	−0.407583	+	0.913168i
\(13\)	1.05686	+	1.21968i	0.293119	+	0.338278i								0.883139	−	0.469111i	\(-0.155426\pi\)
							−0.590020	+	0.807389i	\(0.700880\pi\)
\(17\)	−0.156477	+	3.28487i	−0.0379514	+	0.796697i	0.897023	+	0.441983i	\(0.145725\pi\)
							−0.934975	+	0.354714i	\(0.884578\pi\)
\(19\)	−0.108880	−	2.28568i	−0.0249788	−	0.524370i	−0.976333	−	0.216271i	\(-0.930610\pi\)
							0.951354	−	0.308099i	\(-0.0996927\pi\)
\(23\)	6.27886	+	4.93775i	1.30923	+	1.02959i	0.996823	+	0.0796442i	\(0.0253784\pi\)
							0.312409	+	0.949948i	\(0.398864\pi\)
\(29\)	1.05724	−	0.679447i	0.196325	−	0.126170i	−0.438785	−	0.898592i	\(-0.644591\pi\)
							0.635110	+	0.772422i	\(0.280955\pi\)
\(31\)	0.172596	+	0.0332651i	0.0309991	+	0.00597459i	0.204727	−	0.978819i	\(-0.434369\pi\)
							−0.173728	+	0.984794i	\(0.555581\pi\)
\(37\)	−0.600993	−	1.73646i	−0.0988027	−	0.285472i	0.884758	−	0.466051i	\(-0.154324\pi\)
							−0.983560	+	0.180580i	\(0.942203\pi\)
\(41\)	−0.327525	+	0.717179i	−0.0511508	+	0.112005i	−0.933482	−	0.358625i	\(-0.883246\pi\)
							0.882331	+	0.470629i	\(0.155973\pi\)
\(43\)	5.16621	−	1.51694i	0.787839	−	0.231330i	0.137025	−	0.990568i	\(-0.456246\pi\)
							0.650814	+	0.759237i	\(0.274428\pi\)
\(47\)	8.71240	+	0.831933i	1.27083	+	0.121350i	0.708589	−	0.705621i	\(-0.249332\pi\)
							0.562245	+	0.826971i	\(0.309938\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.u.a.23.17	✓	1720
7.4	even	3	inner	847.2.u.a.144.70	yes	1720
121.100	even	11	inner	847.2.u.a.100.70	yes	1720
847.221	even	33	inner	847.2.u.a.221.17	yes	1720

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.u.a.23.17	✓	1720	1.1	even	1	trivial
847.2.u.a.100.70	yes	1720	121.100	even	11	inner
847.2.u.a.144.70	yes	1720	7.4	even	3	inner
847.2.u.a.221.17	yes	1720	847.221	even	33	inner